# Chapter 9. Graph generators

Graph generators create graphs.

Almost all functions which create graph objects are documented here. The exceptions are `igraph_induced_subgraph()` and alike, these create graphs based on another graph.

## 1. Deterministic graph generators

### 1.1. `igraph_create` — Creates a graph with the specified edges.

    igraph_error_t igraph_create(igraph_t *graph, const igraph_vector_int_t *edges,
                      igraph_integer_t n, igraph_bool_t directed);

Arguments:

`graph`: | An uninitialized graph object.
`edges`: | The edges to add, the first two elements are the first edge, etc.
`n`: | The number of vertices in the graph, if smaller or equal to the highest vertex ID in the `edges` vector it will be increased automatically. So it is safe to give 0 here.
`directed`: | Boolean, whether to create a directed graph or not. If yes, then the first edge points from the first vertex ID in `edges` to the second, etc.

Returns:

`` | Error code: `IGRAPH_EINVEVECTOR`: invalid edges vector (odd number of vertices). `IGRAPH_EINVVID`: invalid (negative) vertex ID.

Time complexity: O(|V|+|E|), |V| is the number of vertices, |E| the number of edges in the graph.

Example 9.1. File `examples/simple/igraph_create.c`

    #include <igraph.h>

    int main(void) {

        igraph_t g;
        igraph_vector_int_t v1, v2;

        /* simple use */
        igraph_vector_int_init(&v1, 8);
        VECTOR(v1)[0] = 0;
        VECTOR(v1)[1] = 1;
        VECTOR(v1)[2] = 1;
        VECTOR(v1)[3] = 2;
        VECTOR(v1)[4] = 2;
        VECTOR(v1)[5] = 3;
        VECTOR(v1)[6] = 2;
        VECTOR(v1)[7] = 2;
        igraph_create(&g, &v1, 0, 0);
        if (igraph_vcount(&g) != 4) {
            return 1;
        }
        igraph_vector_int_init(&v2, 0);
        igraph_get_edgelist(&g, &v2, 0);
        igraph_vector_int_sort(&v1);
        igraph_vector_int_sort(&v2);
        if (!igraph_vector_int_all_e(&v1, &v2)) {
            return 2;
        }
        igraph_destroy(&g);

        /* higher number of vertices */
        igraph_create(&g, &v1, 10, 0);
        if (igraph_vcount(&g) != 10) {
            return 1;
        }
        igraph_get_edgelist(&g, &v2, 0);
        igraph_vector_int_sort(&v1);
        igraph_vector_int_sort(&v2);
        if (!igraph_vector_int_all_e(&v1, &v2)) {
            return 3;
        }
        igraph_destroy(&g);
        igraph_vector_int_destroy(&v1);
        igraph_vector_int_destroy(&v2);

        return 0;
    }

### 1.2. `igraph_small` — Shorthand to create a small graph, giving the edges as arguments.

    igraph_error_t igraph_small(igraph_t *graph, igraph_integer_t n, igraph_bool_t directed,
                                int first, ...);

This function is handy when a relatively small graph needs to be created. Instead of giving the edges as a vector, they are given simply as arguments and a `-1` needs to be given after the last meaningful edge argument.

This function is intended to be used with vertex IDs that are entered as literal integers. If you use a variable instead of a literal, make sure that it is of type int, as this is the type that this function assumes for all variadic arguments. Using a different integer type is undefined behaviour and likely to cause platform-specific issues.

Arguments:

`graph`: | Pointer to an uninitialized graph object. The result will be stored here.
`n`: | The number of vertices in the graph; a non-negative integer.
`directed`: | Logical constant; gives whether the graph should be directed. Supported values are:
`IGRAPH_DIRECTED` | The graph to be created will be directed.
`IGRAPH_UNDIRECTED` | The graph to be created will be undirected.
`...`: | The additional arguments giving the edges of the graph, and must be of type int. Don't forget to supply an additional `-1` after the last (meaningful) argument. The `first` parameter is present for technical reasons and represents the first variadic argument.

Returns:

`` | Error code.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph to create.

Example 9.2. File `examples/simple/igraph_small.c`

    #include <igraph.h>

    int main(void) {

        igraph_t g;

        igraph_small(&g, 0, IGRAPH_DIRECTED, 0, 1, 1, 2, 2, 3, 3, 4, 6, 1, -1);
        igraph_write_graph_edgelist(&g, stdout);
        igraph_destroy(&g);

        return 0;
    }

### 1.3. `igraph_adjacency` — Creates a graph from an adjacency matrix.

    igraph_error_t igraph_adjacency(
        igraph_t *graph, const igraph_matrix_t *adjmatrix, igraph_adjacency_t mode,
        igraph_loops_t loops
    );

The order of the vertices in the matrix is preserved, i.e. the vertex corresponding to the first row/column will be vertex with id 0, the next row is for vertex 1, etc.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`adjmatrix`: | The adjacency matrix. How it is interpreted depends on the `mode` argument.
`mode`: | Constant to specify how the given matrix is interpreted as an adjacency matrix. Possible values (A(i,j) is the element in row i and column j in the adjacency matrix `adjmatrix`):
`IGRAPH_ADJ_DIRECTED` | The graph will be directed and an element gives the number of edges between two vertices.
`IGRAPH_ADJ_UNDIRECTED` | The graph will be undirected and an element gives the number of edges between two vertices. If the input matrix is not symmetric, an error is thrown.
`IGRAPH_ADJ_MAX` | An undirected graph will be created and the number of edges between vertices i and j is max(A(i,j), A(j,i)).
`IGRAPH_ADJ_MIN` | An undirected graph will be created with min(A(i,j), A(j,i)) edges between vertices i and j.
`IGRAPH_ADJ_PLUS` | An undirected graph will be created with A(i,j)+A(j,i) edges between vertices i and j.
`IGRAPH_ADJ_UPPER` | An undirected graph will be created. Only the upper right triangle (including the diagonal) is used for the number of edges.
`IGRAPH_ADJ_LOWER` | An undirected graph will be created. Only the lower left triangle (including the diagonal) is used for the number of edges.
`loops`: | Constant to specify how the diagonal of the matrix should be treated when creating loop edges.
`IGRAPH_NO_LOOPS` | Ignore the diagonal of the input matrix and do not create loops.
`IGRAPH_LOOPS_ONCE` | Treat the diagonal entries as the number of loop edges incident on the corresponding vertex.
`IGRAPH_LOOPS_TWICE` | Treat the diagonal entries as twice the number of loop edges incident on the corresponding vertex. Odd numbers in the diagonal will return an error code.

Returns:

`` | Error code, `IGRAPH_NONSQUARE`: non-square matrix. `IGRAPH_EINVAL`: Negative entry was found in adjacency matrix, or an odd number was found in the diagonal with `IGRAPH_LOOPS_TWICE`

Time complexity: O(|V||V|), |V| is the number of vertices in the graph.

Example 9.3. File `examples/simple/igraph_adjacency.c`

    #include <igraph.h>

    int main(void) {

        return 0;
    }

### 1.4. `igraph_weighted_adjacency` — Creates a graph from a weighted adjacency matrix.

    igraph_error_t igraph_weighted_adjacency(
        igraph_t *graph, const igraph_matrix_t *adjmatrix, igraph_adjacency_t mode,
        igraph_vector_t *weights, igraph_loops_t loops
    );

The order of the vertices in the matrix is preserved, i.e. the vertex corresponding to the first row/column will be vertex with id 0, the next row is for vertex 1, etc.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`adjmatrix`: | The weighted adjacency matrix. How it is interpreted depends on the `mode` argument. The common feature is that edges with zero weights are considered nonexistent (however, negative weights are permitted).
`mode`: | Constant to specify how the given matrix is interpreted as an adjacency matrix. Possible values (A(i,j) is the element in row i and column j in the adjacency matrix `adjmatrix`):
`IGRAPH_ADJ_DIRECTED` | the graph will be directed and an element gives the weight of the edge between two vertices.
`IGRAPH_ADJ_UNDIRECTED` | this is the same as `IGRAPH_ADJ_MAX`, for convenience.
`IGRAPH_ADJ_MAX` | undirected graph will be created and the weight of the edge between vertices i and j is max(A(i,j), A(j,i)).
`IGRAPH_ADJ_MIN` | undirected graph will be created with edge weight min(A(i,j), A(j,i)) between vertices i and j.
`IGRAPH_ADJ_PLUS` | undirected graph will be created with edge weight A(i,j)+A(j,i) between vertices i and j.
`IGRAPH_ADJ_UPPER` | undirected graph will be created, only the upper right triangle (including the diagonal) is used for the edge weights.
`IGRAPH_ADJ_LOWER` | undirected graph will be created, only the lower left triangle (including the diagonal) is used for the edge weights.
`weights`: | Pointer to an initialized vector, the weights will be stored here.
`loops`: | Constant to specify how the diagonal of the matrix should be treated when creating loop edges.
`IGRAPH_NO_LOOPS` | Ignore the diagonal of the input matrix and do not create loops.
`IGRAPH_LOOPS_ONCE` | Treat the diagonal entries as the weight of the loop edge incident on the corresponding vertex.
`IGRAPH_LOOPS_TWICE` | Treat the diagonal entries as twice the weight of the loop edge incident on the corresponding vertex.

Returns:

`` | Error code, `IGRAPH_NONSQUARE`: non-square matrix.

Time complexity: O(|V||V|), |V| is the number of vertices in the graph.

Example 9.4. File `examples/simple/igraph_weighted_adjacency.c`

    #include <igraph.h>
    #include <stdarg.h>

    int main(void) {
        igraph_matrix_t mat;
        igraph_t g;
        int m[4][4] = { { 0, 1, 2, 0 }, { 2, 0, 0, 1 }, { 0, 0, 1, 0 }, { 0, 1, 0, 0 } };
        igraph_vector_t weights;
        igraph_vector_int_t el;
        igraph_integer_t i, j, n;
        igraph_vector_int_init(&el, 0);
        igraph_vector_init(&weights, 0);

        igraph_matrix_init(&mat, 4, 4);
        for (i = 0; i < 4; i++) for (j = 0; j < 4; j++) {
            MATRIX(mat, i, j) = m[i][j];
        }

        igraph_weighted_adjacency(&g, &mat, IGRAPH_ADJ_DIRECTED, &weights, IGRAPH_LOOPS_ONCE);

        igraph_get_edgelist(&g, &el, 0);
        n = igraph_ecount(&g);

        for (i = 0, j = 0; i < n; i++, j += 2) {
            printf("%" IGRAPH_PRId " --> %" IGRAPH_PRId ": %g\n",
                   VECTOR(el)[j], VECTOR(el)[j + 1], VECTOR(weights)[i]);
        }

        igraph_matrix_destroy(&mat);
        igraph_vector_destroy(&weights);
        igraph_vector_int_destroy(&el);
        igraph_destroy(&g);

    }

### 1.5. `igraph_sparse_adjacency` — Creates a graph from a sparse adjacency matrix.

    igraph_error_t igraph_sparse_adjacency(igraph_t *graph, igraph_sparsemat_t *adjmatrix,
            igraph_adjacency_t mode, igraph_loops_t loops);

This has the same functionality as `igraph_adjacency()`, but uses a column-compressed adjacency matrix. Time complexity: O(|E|), where |E| is the number of edges in the graph.

### 1.6. `igraph_sparse_weighted_adjacency` — Creates a graph from a weighted sparse adjacency matrix.

    igraph_error_t igraph_sparse_weighted_adjacency(
        igraph_t *graph, igraph_sparsemat_t *adjmatrix, igraph_adjacency_t mode,
        igraph_vector_t *weights, igraph_loops_t loops
    );

This has the same functionality as `igraph_weighted_adjacency()`, but uses a column-compressed adjacency matrix. Time complexity: O(|E|), where |E| is the number of edges in the graph.

### 1.7. `igraph_adjlist` — Creates a graph from an adjacency list.

    igraph_error_t igraph_adjlist(igraph_t *graph, const igraph_adjlist_t *adjlist,
                       igraph_neimode_t mode, igraph_bool_t duplicate);

An adjacency list is a list of vectors, containing the neighbors of all vertices. For operations that involve many changes to the graph structure, it is recommended that you convert the graph into an adjacency list via `igraph_adjlist_init()`, perform the modifications (these are cheap for an adjacency list) and then recreate the igraph graph via this function.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`adjlist`: | The adjacency list.
`mode`: | Whether or not to create a directed graph. `IGRAPH_ALL` means an undirected graph, `IGRAPH_OUT` means a directed graph from an out-adjacency list (i.e. each list contains the successors of the corresponding vertices), `IGRAPH_IN` means a directed graph from an in-adjacency list
`duplicate`: | Logical, for undirected graphs this specified whether each edge is included twice, in the vectors of both adjacent vertices. If this is false (0), then it is assumed that every edge is included only once. This argument is ignored for directed graphs.

Returns:

`` | Error code.

See also:

`` | `igraph_adjlist_init()` for the opposite operation.

Time complexity: O(|V|+|E|).

### 1.8. `igraph_star` — Creates a star graph, every vertex connects only to the center.

    igraph_error_t igraph_star(igraph_t *graph, igraph_integer_t n, igraph_star_mode_t mode,
                    igraph_integer_t center);

Arguments:

`graph`: | Pointer to an uninitialized graph object, this will be the result.
`n`: | Integer constant, the number of vertices in the graph.
`mode`: | Constant, gives the type of the star graph to create. Possible values:
`IGRAPH_STAR_OUT` | directed star graph, edges point from the center to the other vertices.
`IGRAPH_STAR_IN` | directed star graph, edges point to the center from the other vertices.
`IGRAPH_STAR_MUTUAL` | directed star graph with mutual edges.
`IGRAPH_STAR_UNDIRECTED` | an undirected star graph is created.
`center`: | Id of the vertex which will be the center of the graph.

Returns:

`` | Error code:
`IGRAPH_EINVVID` | invalid number of vertices.
`IGRAPH_EINVAL` | invalid center vertex.
`IGRAPH_EINVMODE` | invalid mode argument.

Time complexity: O(|V|), the number of vertices in the graph.

See also:

`` | `igraph_square_lattice()`, `igraph_ring()`, `igraph_kary_tree()` for creating other regular structures.

Example 9.5. File `examples/simple/igraph_star.c`

    #include <igraph.h>
    #include <stdio.h>

    int main(void) {
        igraph_t graph;

        /* Create an undirected 6-star, with the 0th node as the centre. */
        igraph_star(&graph, 7, IGRAPH_STAR_UNDIRECTED, 0);

        /* Output the edge list of the graph. */
        igraph_write_graph_edgelist(&graph, stdout);

        /* Destroy the graph when we are done using it. */
        igraph_destroy(&graph);

        return 0;
    }

### 1.9. `igraph_wheel` — Creates a wheel graph, a union of a star and a cycle graph.

    igraph_error_t igraph_wheel(igraph_t *graph, igraph_integer_t n, igraph_wheel_mode_t mode,
                    igraph_integer_t center);

A wheel graph on `n` vertices can be thought of as a wheel with `n - 1` spokes. The cycle graph part makes up the rim, while the star graph part adds the spokes.

Note that the two and three-vertex wheel graphs are non-simple: The two-vertex wheel graph contains a self-loop, while the three-vertex wheel graph contains parallel edges (a 1-cycle and a 2-cycle, respectively).

Arguments:

`graph`: | Pointer to an uninitialized graph object, this will be the result.
`n`: | Integer constant, the number of vertices in the graph.
`mode`: | Constant, gives the type of the star graph to create. Possible values:
`IGRAPH_WHEEL_OUT` | directed wheel graph, edges point from the center to the other vertices.
`IGRAPH_WHEEL_IN` | directed wheel graph, edges point to the center from the other vertices.
`IGRAPH_WHEEL_MUTUAL` | directed wheel graph with mutual edges.
`IGRAPH_WHEEL_UNDIRECTED` | an undirected wheel graph is created.
`center`: | Id of the vertex which will be the center of the graph.

Returns:

`` | Error code:
`IGRAPH_EINVVID` | invalid number of vertices.
`IGRAPH_EINVAL` | invalid center vertex.
`IGRAPH_EINVMODE` | invalid mode argument.

Time complexity: O(|V|), the number of vertices in the graph.

See also:

`` | `igraph_square_lattice()`, `igraph_ring()`, `igraph_star()`, `igraph_kary_tree()` for creating other regular structures.

### 1.10. `igraph_square_lattice` — Arbitrary dimensional square lattices.

    igraph_error_t igraph_square_lattice(
        igraph_t *graph, const igraph_vector_int_t *dimvector, igraph_integer_t nei,
        igraph_bool_t directed, igraph_bool_t mutual, const igraph_vector_bool_t *periodic
    );

Creates d-dimensional square lattices of the given size. Optionally, the lattice can be made periodic, and the neighbors within a given graph distance can be connected.

In the zero-dimensional case, the singleton graph is returned.

The vertices of the resulting graph are ordered such that the index of the vertex at position `(i_1, i_2, i_3, ..., i_d)` in a lattice of size `(n_1, n_2, ..., n_d)` will be `i_1 + n_1 * i_2 + n_1 * n_2 * i_3 + ...`.

Arguments:

`graph`: | An uninitialized graph object.
`dimvector`: | Vector giving the sizes of the lattice in each of its dimensions. The dimension of the lattice will be the same as the length of this vector.
`nei`: | Integer value giving the distance (number of steps) within which two vertices will be connected.
`directed`: | Boolean, whether to create a directed graph. If the `mutual` and `circular` arguments are not set to true, edges will be directed from lower-index vertices towards higher-index ones.
`mutual`: | Boolean, if the graph is directed this gives whether to create all connections as mutual.
`periodic`: | Boolean vector, defines whether the generated lattice is periodic along each dimension. The length of this vector must match the length of `dimvector`. This parameter may also be `NULL`, which implies that the lattice will not be periodic.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid (negative) dimension vector or mismatch between the length of the dimension vector and the periodicity vector.

Time complexity: If `nei` is less than two then it is O(|V|+|E|) (as far as I remember), |V| and |E| are the number of vertices and edges in the generated graph. Otherwise it is O(|V|*d^k+|E|), d is the average degree of the graph, k is the `nei` argument.

### 1.11. `igraph_triangular_lattice` — A triangular lattice with the given shape.

    igraph_error_t igraph_triangular_lattice(
        igraph_t *graph, const igraph_vector_int_t *dims, igraph_bool_t directed,
        igraph_bool_t mutual);

### Warning

This function is experimental and its signature is not considered final yet. We reserve the right to change the function signature without changing the major version of igraph. Use it at your own risk.

Creates a triangular lattice whose vertices have the form (i, j) for non-negative integers i and j and (i, j) is generally connected with (i + 1, j), (i, j + 1), and (i - 1, j + 1). The function constructs a planar dual of the graph constructed by `igraph_hexagonal_lattice()`. In particular, there a one-to-one correspondence between the vertices in the constructed graph and the cycles of length 6 in the graph constructed by `igraph_hexagonal_lattice()` with the same `dims` parameter.

The vertices of the resulting graph are ordered lexicographically with the 2nd coordinate being more significant, e.g., (i, j) < (i + 1, j) and (i + 1, j) < (i, j + 1)

Arguments:

`graph`: | An uninitialized graph object.
`dims`: | Integer vector, defines the shape of the lattice. (Below the "edge length"s are in terms of graph theoretical path lengths.) If `dims` is of length 1, the resulting lattice has a triangular shape where each side of the triangle contains `dims[0]` vertices. If `dims` is of length 2, the resulting lattice has a "quasi rectangular" shape with the sides containing `dims[0]` and `dims[1]` vertices, respectively. If `dims` is of length 3, the resulting lattice has a hexagonal shape where the sides of the hexagon contain `dims[0]`, `dims[1]` and `dims[2]` vertices. All coordinates must be non-negative.
`directed`: | Boolean, whether to create a directed graph. If the `mutual` argument is not set to true, edges will be directed from lower-index vertices towards higher-index ones.
`mutual`: | Boolean, if the graph is directed this gives whether to create all connections as mutual.

Returns:

`` | Error code: `IGRAPH_EINVAL`: The size of `dims` must be either 1, 2, or 3 with all the components at least 1.

See also:

`` | `igraph_hexagonal_lattice()` for creating a triangular lattice.

Time complexity: O(|V|), where |V| is the number of vertices in the generated graph.

### 1.12. `igraph_hexagonal_lattice` — A hexagonal lattice with the given shape.

    igraph_error_t igraph_hexagonal_lattice(
        igraph_t *graph, const igraph_vector_int_t *dims, igraph_bool_t directed,
        igraph_bool_t mutual
    );

### Warning

This function is experimental and its signature is not considered final yet. We reserve the right to change the function signature without changing the major version of igraph. Use it at your own risk.

Creates a hexagonal lattice whose vertices have the form (i, j) for non-negative integers i and j and (i, j) is generally connected with (i + 1, j), and if i is odd also with (i - 1, j + 1). The function constructs a planar dual of the graph constructed by `igraph_triangular_lattice()`. In particular, there a one-to-one correspondence between the cycles of length 6 in the constructed graph and the vertices of the graph constructed by `igraph_triangular_lattice()` function with the same `dims` parameter.

The vertices of the resulting graph are ordered lexicographically with the 2nd coordinate being more significant, e.g., (i, j) < (i + 1, j) and (i + 1, j) < (i, j + 1)

Arguments:

`graph`: | An uninitialized graph object.
`dims`: | Integer vector, defines the shape of the lattice. (Below the "edge length"s are in terms of graph theoretical path lengths.) If `dims` is of length 1, the resulting lattice has a triangular shape where each side of the triangle contains `dims[0]` vertices. If `dims` is of length 2, the resulting lattice has a "quasi rectangular" shape with the sides containing `dims[0]` and `dims[1]` vertices, respectively. If `dims` is of length 3, the resulting lattice has a hexagonal shape where the sides of the hexagon contain `dims[0]`, `dims[1]` and `dims[2]` vertices. All coordinates must be non-negative.
`directed`: | Boolean, whether to create a directed graph. If the `mutual` argument is not set to true, edges will be directed from lower-index vertices towards higher-index ones.
`mutual`: | Boolean, if the graph is directed this gives whether to create all connections as mutual.

Returns:

`` | Error code: `IGRAPH_EINVAL`: The size of `dims` must be either 1, 2, or 3 with all the components at least 1.

See also:

`` | `igraph_triangular_lattice()` for creating a triangular lattice.

Time complexity: O(|V|), where |V| is the number of vertices in the generated graph.

### 1.13. `igraph_ring` — Creates a cycle graph or a path graph.

    igraph_error_t igraph_ring(igraph_t *graph, igraph_integer_t n, igraph_bool_t directed,
                    igraph_bool_t mutual, igraph_bool_t circular);

A circular ring on `n` vertices is commonly known in graph theory as the cycle graph, and often denoted by `C_n`. Removing a single edge from the cycle graph `C_n` results in the path graph `P_n`. This function can generate both.

When `n` is 1 or 2, the result may not be a simple graph: the one-cycle contains a self-loop and the undirected or reciprocally connected directed two-cycle contains parallel edges.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`n`: | The number of vertices in the graph.
`directed`: | Logical, whether to create a directed graph. All edges will be oriented in the same direction along the cycle or path.
`mutual`: | Logical, whether to create mutual edges in directed graphs. It is ignored for undirected graphs.
`circular`: | Logical, whether to create a closed ring (a cycle) or an open path.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid number of vertices.

Time complexity: O(|V|), the number of vertices in the graph.

See also:

`` | `igraph_lattice()` for generating more general lattices.

Example 9.6. File `examples/simple/igraph_ring.c`

    #include <igraph.h>
    #include <stdio.h>

    int main(void) {
        igraph_t graph;

        /* Create a directed path graph on 10 vertices. */
        igraph_ring(&graph, 10, IGRAPH_DIRECTED, /* mutual= */ 0, /* circular= */ 0);

        /* Output the edge list of the graph. */
        printf("10-path graph:\n");
        igraph_write_graph_edgelist(&graph, stdout);

        /* Destroy the graph. */
        igraph_destroy(&graph);

        /* Create a 4-cycle graph. */
        igraph_ring(&graph, 4, IGRAPH_UNDIRECTED, /* mutual= */ 0, /* circular= */ 1);

        /* Output the edge list of the graph. */
        printf("\n4-cycle graph:\n");
        igraph_write_graph_edgelist(&graph, stdout);

        /* Destroy the graph. */
        igraph_destroy(&graph);

        return 0;
    }

### 1.14. `igraph_kary_tree` — Creates a k-ary tree in which almost all vertices have k children.

    igraph_error_t igraph_kary_tree(igraph_t *graph, igraph_integer_t n, igraph_integer_t children,
                    igraph_tree_mode_t type);

To obtain a completely symmetric tree with `l` layers, where each vertex has precisely `children` descendants, use `n = (children^(l+1) - 1) / (children - 1)`. Such trees are often called `k`-ary trees, where `k` refers to the number of children.

Note that for `n=0`, the null graph is returned, which is not considered to be a tree by `igraph_is_tree()`.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`n`: | Integer, the number of vertices in the graph.
`children`: | Integer, the number of children of a vertex in the tree.
`type`: | Constant, gives whether to create a directed tree, and if this is the case, also its orientation. Possible values:
`IGRAPH_TREE_OUT` | directed tree, the edges point from the parents to their children.
`IGRAPH_TREE_IN` | directed tree, the edges point from the children to their parents.
`IGRAPH_TREE_UNDIRECTED` | undirected tree.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid number of vertices. `IGRAPH_INVMODE`: invalid mode argument.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.

See also:

`` | `igraph_lattice()`, `igraph_star()` for creating other regular structures; `igraph_from_prufer()` for creating arbitrary trees; `igraph_tree_game()` for uniform random sampling of trees.

Example 9.7. File `examples/simple/igraph_kary_tree.c`

    #include <igraph.h>

    int main(void) {
        igraph_t graph;
        igraph_bool_t res;

        /* Create a directed binary tree on 15 nodes,
           with edges pointing towards the root. */
        igraph_kary_tree(&graph, 15, 2, IGRAPH_TREE_IN);

        igraph_is_tree(&graph, &res, NULL, IGRAPH_IN);
        printf("Is it an in-tree? %s\n", res ? "Yes" : "No");

        igraph_is_tree(&graph, &res, NULL, IGRAPH_OUT);
        printf("Is it an out-tree? %s\n", res ? "Yes" : "No");

        igraph_destroy(&graph);

        return 0;
    }

### 1.15. `igraph_symmetric_tree` — Creates a symmetric tree with the specified number of branches at each level.

    igraph_error_t igraph_symmetric_tree(igraph_t *graph, const igraph_vector_int_t *branches,
                    igraph_tree_mode_t type);

This function creates a tree in which all vertices at distance `d` from the root have `branching_counts`[d] children.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`branches`: | Vector detailing the number of branches at each level.
`type`: | Constant, gives whether to create a directed tree, and if this is the case, also its orientation. Possible values:
`IGRAPH_TREE_OUT` | directed tree, the edges point from the parents to their children.
`IGRAPH_TREE_IN` | directed tree, the edges point from the children to their parents.
`IGRAPH_TREE_UNDIRECTED` | undirected tree.

Returns:

`` | Error code: `IGRAPH_INVMODE`: invalid mode argument. `IGRAPH_EINVAL`: invalid number of children.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.

See also:

`` | `igraph_kary_tree()`, `igraph_regular_tree()` and `igraph_star()` for creating other regular tree structures; `igraph_from_prufer()` for creating arbitrary trees; `igraph_tree_game()` for uniform random sampling of trees.

Example 9.8. File `examples/simple/igraph_symmetric_tree.c`

    #include <igraph.h>

    int main(void) {

        igraph_t graph;
        igraph_bool_t res;
        igraph_vector_int_t v;
        igraph_vector_int_init_int(&v, 3, 3, 4, 5);

        /* Create a directed symmetric tree with 2 levels -
           3 children in first and 4 children in second level,
           5 children in third level
           with edges pointing towards the root. */
        igraph_symmetric_tree(&graph, &v, IGRAPH_TREE_IN);

        igraph_is_tree(&graph, &res, NULL, IGRAPH_IN);
        printf("Is it an in-tree? %s\n", res ? "Yes" : "No");

        igraph_is_tree(&graph, &res, NULL, IGRAPH_OUT);
        printf("Is it an out-tree? %s\n", res ? "Yes" : "No");

        igraph_destroy(&graph);
        igraph_vector_int_destroy(&v);

        return 0;
    }

### 1.16. `igraph_regular_tree` — Creates a regular tree.

    igraph_error_t igraph_regular_tree(igraph_t *graph, igraph_integer_t h, igraph_integer_t k, igraph_tree_mode_t type);

All vertices of a regular tree, except its leaves, have the same total degree `k`. This is different from a k-ary tree (`igraph_kary_tree()`), where all vertices have the same number of children, thus the degre of the root is one less than the degree of the other internal vertices. Regular trees are also referred to as Bethe lattices.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`h`: | The height of the tree, i.e. the distance between the root and the leaves.
`k`: | The degree of the regular tree.
`type`: | Constant, gives whether to create a directed tree, and if this is the case, also its orientation. Possible values:
`IGRAPH_TREE_OUT` | directed tree, the edges point from the parents to their children.
`IGRAPH_TREE_IN` | directed tree, the edges point from the children to their parents.
`IGRAPH_TREE_UNDIRECTED` | undirected tree.

Returns:

`` | Error code.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.

See also:

`` | `igraph_kary_tree()` to create k-ary tree where each vertex has the same number of children, i.e. out-degree, instead of the same total degree. `igraph_symmetric_tree()` to use a different number of children at each level.

Example 9.9. File `examples/simple/igraph_regular_tree.c`

    #include <igraph.h>

    int main(void) {
        igraph_t tree;
        igraph_vector_t eccentricity;
        igraph_bool_t is_tree;

        /* Create a Bethe lattice with 5 levels, i.e. height 4. */
        igraph_regular_tree(&tree, 4, 3, IGRAPH_TREE_UNDIRECTED);

        /* Bethe lattices are trees. */
        igraph_is_tree(&tree, &is_tree, NULL, IGRAPH_ALL);
        printf("Is it a tree? %s\n", is_tree ? "Yes." : "No.");

        /* Compute and print eccentricities. The root is the most central. */
        igraph_vector_init(&eccentricity, 0);
        igraph_eccentricity(&tree, &eccentricity, igraph_vss_all(), IGRAPH_ALL);
        printf("Vertex eccentricities:\n");
        igraph_vector_print(&eccentricity);
        igraph_vector_destroy(&eccentricity);

        /* Clean up. */
        igraph_destroy(&tree);

        return 0;
    }

### 1.17. `igraph_full` — Creates a full graph (complete graph).

    igraph_error_t igraph_full(igraph_t *graph, igraph_integer_t n, igraph_bool_t directed,
                    igraph_bool_t loops);

In a full graph every possible edge is present: every vertex is connected to every other vertex. igraph generalizes the usual concept of complete graphs in graph theory to graphs with self-loops as well as to directed graphs.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`n`: | Integer, the number of vertices in the graph.
`directed`: | Logical, whether to create a directed graph.
`loops`: | Logical, whether to include self-edges (loops).

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid number of vertices.

Time complexity: O(|V|^2) = O(|E|), where |V| is the number of vertices and |E| is the number of edges.

See also:

`` | `igraph_square_lattice()`, `igraph_star()`, `igraph_kary_tree()` for creating other regular structures.

Example 9.10. File `examples/simple/igraph_full.c`

    #include <igraph.h>
    #include <stdio.h>

    int main(void) {
        igraph_t graph;
        igraph_integer_t n_vertices = 10;

        /* Create an undirected complete graph. */
        /* Use IGRAPH_UNDIRECTED and IGRAPH_NO_LOOPS instead of 1/TRUE and 0/FALSE for better readability. */
        igraph_full(&graph, n_vertices, IGRAPH_UNDIRECTED, IGRAPH_NO_LOOPS);
        printf("The undirected complete graph on %" IGRAPH_PRId " vertices has %" IGRAPH_PRId " edges.\n",
              igraph_vcount(&graph), igraph_ecount(&graph));

        /* Remember to destroy the object at the end. */
        igraph_destroy(&graph);

        /* Create a directed complete graph. */
        igraph_full(&graph, n_vertices, IGRAPH_DIRECTED, IGRAPH_NO_LOOPS);
        printf("The directed complete graph on %" IGRAPH_PRId " vertices has %" IGRAPH_PRId " edges.\n",
              igraph_vcount(&graph), igraph_ecount(&graph));

        igraph_destroy(&graph);

        /* Create an undirected complete graph with self-loops. */
        igraph_full(&graph, n_vertices, IGRAPH_UNDIRECTED, IGRAPH_LOOPS);
        printf("The undirected complete graph on %" IGRAPH_PRId " vertices with self-loops has %" IGRAPH_PRId " edges.\n",
              igraph_vcount(&graph), igraph_ecount(&graph));

        igraph_destroy(&graph);

        /* Create a directed graph with self-loops. */
        igraph_full(&graph, n_vertices, IGRAPH_DIRECTED, IGRAPH_LOOPS);
        printf("The directed complete graph on %" IGRAPH_PRId " vertices with self-loops has %" IGRAPH_PRId " edges.\n",
              igraph_vcount(&graph), igraph_ecount(&graph));

        igraph_destroy(&graph);

        return 0;

    }

### 1.18. `igraph_full_citation` — Creates a full citation graph (a complete directed acyclic graph).

    igraph_error_t igraph_full_citation(igraph_t *graph, igraph_integer_t n,
                             igraph_bool_t directed);

This is a directed graph, where every `i->j` edge is present if and only if `j<i`. If the `directed` argument is false then an undirected graph is created, and it is just a complete graph.

Arguments:

`graph`: | Pointer to an uninitialized graph object, the result is stored here.
`n`: | The number of vertices.
`directed`: | Whether to created a directed graph. If false an undirected graph is created.

Returns:

`` | Error code.

See also:

`` | `igraph_full()`

Time complexity: O(|V|^2) = O(|E|), where |V| is the number of vertices and |E| is the number of edges.

### 1.19. `igraph_full_multipartite` — Creates a full multipartite graph.

    igraph_error_t igraph_full_multipartite(igraph_t *graph,
                              igraph_vector_int_t *types,
                              const igraph_vector_int_t *n,
                              igraph_bool_t directed,
                              igraph_neimode_t mode);

A multipartite graph contains two or more types of vertices and connections are only possible between two vertices of different types. This function creates a complete multipartite graph.

Arguments:

`graph`: | Pointer to an uninitialized graph object, the graph will be created here.
`types`: | Pointer to an integer vector. If not a null pointer, the type of each vertex will be stored here.
`n`: | Pointer to an integer vector, the number of vertices of each type.
`directed`: | Boolean, whether to create a directed graph.
`mode`: | A constant that gives the type of connections for directed graphs. If `IGRAPH_OUT`, then edges point from vertices of low-index vertices to high-index vertices; if `IGRAPH_IN`, then the opposite direction is realized; `IGRAPH_ALL`, then mutual edges will be created.

Returns:

`` | Error code.

Time complexity: O(|V|+|E|), linear in the number of vertices and edges.

See also:

`` | `igraph_full_bipartite()` for complete bipartite graphs, `igraph_turan()` for Turán graphs.

### 1.20. `igraph_turan` — Creates a Turán graph.

    igraph_error_t igraph_turan(igraph_t *graph,
                                igraph_vector_int_t *types,
                                igraph_integer_t n,
                                igraph_integer_t r);

Turán graphs are complete multipartite graphs with the property that the sizes of the partitions are as close to equal as possible.

The Turán graph with `n` vertices and `r` partitions is the densest graph on `n` vertices that does not contain a clique of size `r+1`.

This function generates undirected graphs. The null graph is returned when the number of vertices is zero. A complete graph is returned if the number of partitions is greater than the number of vertices.

Arguments:

`graph`: | Pointer to an igraph_t object, the graph will be created here.
`types`: | Pointer to an integer vector. If not a null pointer, the type (partition index) of each vertex will be stored here.
`n`: | Integer, the number of vertices in the graph.
`r`: | Integer, the number of partitions of the graph, must be positive.

Returns:

`` | Error code.

Time complexity: O(|V|+|E|), linear in the number of vertices and edges.

See also:

`` | `igraph_full_multipartite()` for full multipartite graphs.

### 1.21. `igraph_realize_degree_sequence` — Generates a graph with the given degree sequence.

    igraph_error_t igraph_realize_degree_sequence(
            igraph_t *graph,
            const igraph_vector_int_t *outdeg, const igraph_vector_int_t *indeg,
            igraph_edge_type_sw_t allowed_edge_types,
            igraph_realize_degseq_t method);

This function generates an undirected graph that realizes a given degree sequence, or a directed graph that realized a given pair of out- and in-degree sequences.

Simple undirected graphs are constructed using the Havel-Hakimi algorithm (undirected case), or the analogous Kleitman-Wang algorithm (directed case). These algorithms work by choosing an arbitrary vertex and connecting all its stubs to other vertices of highest degree. In the directed case, the "highest" (in, out) degree pairs are determined based on lexicographic ordering. This step is repeated until all degrees have been connected up.

Loopless multigraphs are generated using an analogous algorithm: an arbitrary vertex is chosen, and it is connected with a single connection to a highest remaining degee vertex. If self-loops are also allowed, the same algorithm is used, but if a non-zero vertex remains at the end of the procedure, the graph is completed by adding self-loops to it. Thus, the result will contain at most one vertex with self-loops.

The `method` parameter controls the order in which the vertices to be connected are chosen.

References:

V. Havel, Poznámka o existenci konečných grafů (A remark on the existence of finite graphs), Časopis pro pěstování matematiky 80, 477-480 (1955). http://eudml.org/doc/19050

S. L. Hakimi, On Realizability of a Set of Integers as Degrees of the Vertices of a Linear Graph, Journal of the SIAM 10, 3 (1962). https://www.jstor.org/stable/2098770

D. J. Kleitman and D. L. Wang, Algorithms for Constructing Graphs and Digraphs with Given Valences and Factors, Discrete Mathematics 6, 1 (1973). https://doi.org/10.1016/0012-365X%2873%2990037-X

Sz. Horvát and C. D. Modes, Connectedness matters: construction and exact random sampling of connected networks (2021). https://doi.org/10.1088/2632-072X/abced5

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`outdeg`: | The degree sequence of an undirected graph (if `indeg` is NULL), or the out-degree sequence of a directed graph (if `indeg` is given).
`indeg`: | The in-degree sequence of a directed graph. Pass `NULL` to generate an undirected graph.
`allowed_edge_types`: | The types of edges to allow in the graph. For directed graphs, only `IGRAPH_SIMPLE_SW` is implemented at this moment. For undirected graphs, the following values are valid:
`IGRAPH_SIMPLE_SW` | simple graphs (i.e. no self-loops or multi-edges allowed).
`IGRAPH_LOOPS_SW` | single self-loops are allowed, but not multi-edges; currently not implemented.
`IGRAPH_MULTI_SW` | multi-edges are allowed, but not self-loops.
`IGRAPH_LOOPS_SW | IGRAPH_MULTI_SW` | both self-loops and multi-edges are allowed.
`method`: | The method to generate the graph. Possible values:
`IGRAPH_REALIZE_DEGSEQ_SMALLEST` | The vertex with smallest remaining degree is selected first. The result is usually a graph with high negative degree assortativity. In the undirected case, this method is guaranteed to generate a connected graph, regardless of whether multi-edges are allowed, provided that a connected realization exists (see Horvát and Modes, 2021, as well as http://szhorvat.net/pelican/hh-connected-graphs.html). In the directed case it tends to generate weakly connected graphs, but this is not guaranteed.
`IGRAPH_REALIZE_DEGSEQ_LARGEST` | The vertex with the largest remaining degree is selected first. The result is usually a graph with high positive degree assortativity, and is often disconnected.
`IGRAPH_REALIZE_DEGSEQ_INDEX` | The vertices are selected in order of their index (i.e. their position in the degree vector). Note that sorting the degree vector and using the `INDEX` method is not equivalent to the `SMALLEST` method above, as `SMALLEST` uses the smallest remaining degree for selecting vertices, not the smallest initial degree.

Returns:

`` | Error code:
`IGRAPH_UNIMPLEMENTED` | The requested method is not implemented.
`IGRAPH_ENOMEM` | There is not enough memory to perform the operation.
`IGRAPH_EINVAL` | Invalid method parameter, or invalid in- and/or out-degree vectors. The degree vectors should be non-negative, the length and sum of `outdeg` and `indeg` should match for directed graphs.

See also:

`` | `igraph_is_graphical()` to test graphicality without generating a graph; `igraph_degree_sequence_game()` to generate random graphs with a given degree sequence; `igraph_k_regular_game()` to generate random regular graphs; `igraph_rewire()` to randomly rewire the edges of a graph while preserving its degree sequence.

Example 9.11. File `examples/simple/igraph_realize_degree_sequence.c`

    #include <igraph.h>
    #include <stdio.h>

    int main(void){
        igraph_t g1, g2, g3;
        igraph_integer_t nodes = 500, A = 0, power = 1, m = 1;
        igraph_real_t assortativity;

        igraph_rng_seed(igraph_rng_default(), 42);
        printf("Demonstration of difference in assortativities of graphs with the same degree sequence but different linkages:\n\nInitial graph based on the Barabasi-Albert model with %" IGRAPH_PRId " nodes.\n", nodes);

        /* Graph 1 generated by a randomized graph generator */
        igraph_barabasi_game(&g1, nodes, power, m, NULL, /* outpref */ 0, A, IGRAPH_UNDIRECTED, IGRAPH_BARABASI_PSUMTREE, /* start from */ NULL);

        igraph_vector_int_t degree;
        igraph_vector_int_init(&degree, nodes);
        igraph_degree(&g1, &degree, igraph_vss_all(), IGRAPH_ALL, IGRAPH_NO_LOOPS);

        /* Measuring assortativity of the first graph */
        igraph_assortativity_degree(&g1, &assortativity, IGRAPH_UNDIRECTED);
        printf("Assortativity of initial graph = %g\n\n", assortativity);
        igraph_destroy(&g1);

        /* Graph 2 (with the same degree sequence) generated by selecting vertices with the smallest degree first */
        igraph_realize_degree_sequence(&g2, &degree, NULL, IGRAPH_SIMPLE_SW, IGRAPH_REALIZE_DEGSEQ_SMALLEST);
        igraph_assortativity_degree(&g2, &assortativity, IGRAPH_UNDIRECTED);
        printf("Assortativity after choosing vertices with the smallest degrees first = %g\n\n", assortativity);
        igraph_destroy(&g2);

        /* Graph 3 (with the same degree sequence) generated by selecting vertices with the largest degree first */
        igraph_realize_degree_sequence(&g3, &degree, NULL, IGRAPH_SIMPLE_SW, IGRAPH_REALIZE_DEGSEQ_LARGEST);
        igraph_assortativity_degree(&g3, &assortativity, IGRAPH_UNDIRECTED);
        printf("Assortativity after choosing vertices with the largest degrees first = %g\n", assortativity);
        igraph_destroy(&g3);
        igraph_vector_int_destroy(&degree);

        return 0;
    }

### 1.22. `igraph_realize_bipartite_degree_sequence` — Generates a bipartite graph with the given bidegree sequence.

    igraph_error_t igraph_realize_bipartite_degree_sequence(
        igraph_t *graph,
        const igraph_vector_int_t *degrees1, const igraph_vector_int_t *degrees2,
        const igraph_edge_type_sw_t allowed_edge_types, const igraph_realize_degseq_t method
    );

### Warning

This function is experimental and its signature is not considered final yet. We reserve the right to change the function signature without changing the major version of igraph. Use it at your own risk.

This function generates a bipartite graph with the given bidegree sequence, using a Havel-Hakimi-like construction algorithm. The order in which vertices are connected up is controlled by the `method` parameter. When using the `IGRAPH_REALIZE_DEGSEQ_SMALLEST` method, it is ensured that the graph will be connected if and only if the given bidegree sequence is potentially connected.

The vertices of the graph will be ordered so that those having `degrees1` come first, followed by `degrees2`.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`degrees1`: | The degree sequence of the first partition.
`degrees2`: | The degree sequence of the second partition.
`allowed_edge_types`: | The types of edges to allow in the graph.
`IGRAPH_SIMPLE_SW` | simple graph (i.e. no multi-edges allowed).
`IGRAPH_MULTI_SW` | multi-edges are allowed
`method`: | Controls the order in which vertices are selected for connection. Possible values:
`IGRAPH_REALIZE_DEGSEQ_SMALLEST` | The vertex with smallest remaining degree is selected first, from either partition. The result is usually a graph with high negative degree assortativity. This method is guaranteed to generate a connected graph, if one exists.
`IGRAPH_REALIZE_DEGSEQ_LARGEST` | The vertex with the largest remaining degree is selected first, from either parition. The result is usually a graph with high positive degree assortativity, and is often disconnected.
`IGRAPH_REALIZE_DEGSEQ_INDEX` | The vertices are selected in order of their index.

Returns:

`` | Error code.

See also:

`` | `igraph_is_bigraphical()` to test bigraphicality without generating a graph.

### 1.23. `igraph_famous` — Create a famous graph by simply providing its name.

    igraph_error_t igraph_famous(igraph_t *graph, const char *name);

The name of the graph can be simply supplied as a string. Note that this function creates graphs which don't take any parameters, there are separate functions for graphs with parameters, e.g. `igraph_full()` for creating a full graph.

The following graphs are supported:

`Bull` | The bull graph, 5 vertices, 5 edges, resembles the head of a bull if drawn properly.
`Chvatal` | This is the smallest triangle-free graph that is both 4-chromatic and 4-regular. According to the Grunbaum conjecture there exists an m-regular, m-chromatic graph with n vertices for every m>1 and n>2. The Chvatal graph is an example for m=4 and n=12. It has 24 edges.
`Coxeter` | A non-Hamiltonian cubic symmetric graph with 28 vertices and 42 edges.
`Cubical` | The Platonic graph of the cube. A convex regular polyhedron with 8 vertices and 12 edges.
`Diamond` | A graph with 4 vertices and 5 edges, resembles a schematic diamond if drawn properly.
`Dodecahedral, Dodecahedron` | Another Platonic solid with 20 vertices and 30 edges.
`Folkman` | The semisymmetric graph with minimum number of vertices, 20 and 40 edges. A semisymmetric graph is regular, edge transitive and not vertex transitive.
`Franklin` | This is a graph whose embedding to the Klein bottle can be colored with six colors, it is a counterexample to the necessity of the Heawood conjecture on a Klein bottle. It has 12 vertices and 18 edges.
`Frucht` | The Frucht Graph is the smallest cubical graph whose automorphism group consists only of the identity element. It has 12 vertices and 18 edges.
`Grotzsch` | The Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, and chromatic number 4. It is named after German mathematician Herbert Grötzsch, and its existence demonstrates that the assumption of planarity is necessary in Grötzsch's theorem that every triangle-free planar graph is 3-colorable.
`Heawood` | The Heawood graph is an undirected graph with 14 vertices and 21 edges. The graph is cubic, and all cycles in the graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6-cage, the smallest cubic graph of girth 6.
`Herschel` | The Herschel graph is the smallest nonhamiltonian polyhedral graph. It is the unique such graph on 11 nodes, and has 18 edges.
`House` | The house graph is a 5-vertex, 6-edge graph, the schematic draw of a house if drawn properly, basically a triangle on top of a square.
`HouseX` | The same as the house graph with an X in the square. 5 vertices and 8 edges.
`Icosahedral, Icosahedron` | A Platonic solid with 12 vertices and 30 edges.
`Krackhardt_Kite` | A social network with 10 vertices and 18 edges. Krackhardt, D. Assessing the Political Landscape: Structure, Cognition, and Power in Organizations. Admin. Sci. Quart. 35, 342-369, 1990.
`Levi` | The graph is a 4-arc transitive cubic graph, it has 30 vertices and 45 edges.
`McGee` | The McGee graph is the unique 3-regular 7-cage graph, it has 24 vertices and 36 edges.
`Meredith` | The Meredith graph is a quartic graph on 70 nodes and 140 edges that is a counterexample to the conjecture that every 4-regular 4-connected graph is Hamiltonian.
`Noperfectmatching` | A connected graph with 16 vertices and 27 edges containing no perfect matching. A matching in a graph is a set of pairwise non-incident edges; that is, no two edges share a common vertex. A perfect matching is a matching which covers all vertices of the graph.
`Nonline` | A graph whose connected components are the 9 graphs whose presence as a vertex-induced subgraph in a graph makes a nonline graph. It has 50 vertices and 72 edges.
`Octahedral, Octahedron` | Platonic solid with 6 vertices and 12 edges.
`Petersen` | A 3-regular graph with 10 vertices and 15 edges. It is the smallest hypohamiltonian graph, i.e. it is non-hamiltonian but removing any single vertex from it makes it Hamiltonian.
`Robertson` | The unique (4,5)-cage graph, i.e. a 4-regular graph of girth 5. It has 19 vertices and 38 edges.
`Smallestcyclicgroup` | A smallest nontrivial graph whose automorphism group is cyclic. It has 9 vertices and 15 edges.
`Tetrahedral, Tetrahedron` | Platonic solid with 4 vertices and 6 edges.
`Thomassen` | The smallest hypotraceable graph, on 34 vertices and 52 edges. A hypotracable graph does not contain a Hamiltonian path but after removing any single vertex from it the remainder always contains a Hamiltonian path. A graph containing a Hamiltonian path is called traceable.
`Tutte` | Tait's Hamiltonian graph conjecture states that every 3-connected 3-regular planar graph is Hamiltonian. This graph is a counterexample. It has 46 vertices and 69 edges.
`Uniquely3colorable` | Returns a 12-vertex, triangle-free graph with chromatic number 3 that is uniquely 3-colorable.
`Walther` | An identity graph with 25 vertices and 31 edges. An identity graph has a single graph automorphism, the trivial one.
`Zachary` | Social network of friendships between 34 members of a karate club at a US university in the 1970s. See W. W. Zachary, An information flow model for conflict and fission in small groups, Journal of Anthropological Research 33, 452-473 (1977).

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`name`: | Character constant, the name of the graph to be created, it is case insensitive.

Returns:

`` | Error code, `IGRAPH_EINVAL` if there is no graph with the given name.

See also:

`` | Other functions for creating graph structures: `igraph_ring()`, `igraph_kary_tree()`, `igraph_square_lattice()`, `igraph_full()`.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.

### 1.24. `igraph_lcf` — Creates a graph from LCF notation.

    igraph_error_t igraph_lcf(igraph_t *graph, igraph_integer_t n, ...);

LCF is short for Lederberg-Coxeter-Frucht, it is a concise notation for 3-regular Hamiltonian graphs. It consists of three parameters: the number of vertices in the graph, a list of shifts giving additional edges to a cycle backbone, and another integer giving how many times the shifts should be performed. See https://mathworld.wolfram.com/LCFNotation.html for details.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`n`: | Integer, the number of vertices in the graph.
`...`: | The shifts and the number of repeats for the shifts, plus an additional 0 to mark the end of the arguments.

Returns:

`` | Error code.

See also:

`` | See `igraph_lcf_vector()` for a similar function using a vector_t instead of the variable length argument list.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.

Example 9.12. File `examples/simple/igraph_lcf.c`

    #include <igraph.h>

    int main(void) {

        igraph_t g, g2;
        igraph_bool_t iso;

        // Franklin graph
        igraph_lcf(&g, 12, 5, -5, 6, 0);
        igraph_famous(&g2, "franklin");

        igraph_isomorphic_vf2(&g, &g2,
                              /*vertex.color1=*/ 0, /*vertex.color2=*/ 0,
                              /*edge.color1=*/ 0, /*edge.color2=*/ 0,
                              &iso, 0, 0, 0, 0, 0);
        if (!iso) {
            printf("Failure: Franklin\n");
            return 1;
        }

        igraph_destroy(&g);
        igraph_destroy(&g2);

        // [3, -2]^4, n=8
        igraph_lcf(&g, 8, 3, -2, 4, 0);

        if (igraph_ecount(&g) != 16) {
            printf("Failure: [3, -2]^4, n=8\n");
            return 1;
        }

        igraph_destroy(&g);

        // [2, -2]^2, n=2
        igraph_lcf(&g, 2, 2, -2, 2, 0);

        if (igraph_ecount(&g) != 1) {
            printf("Failure: [2, -2]^2, n=2\n");
            return 1;
        }

        igraph_destroy(&g);

        // [2]^2, n=2
        igraph_lcf(&g, 2, 2, 2, 0);

        if (igraph_ecount(&g) != 1) {
            printf("Failure: [2]^2, n=2\n");
            return 1;
        }

        igraph_destroy(&g);

        // Regression test for bug #996
        igraph_lcf(&g, 0, 0);
        if (igraph_vcount(&g) != 0 || igraph_ecount(&g) != 0) {
            printf("Failure: regression test for #996\n");
            return 1;
        }

        igraph_destroy(&g);

        return 0;
    }

### 1.25. `igraph_lcf_vector` — Creates a graph from LCF notation.

    igraph_error_t igraph_lcf_vector(igraph_t *graph, igraph_integer_t n,
                          const igraph_vector_int_t *shifts,
                          igraph_integer_t repeats);

This function is essentially the same as `igraph_lcf()`, only the way for giving the arguments is different. See `igraph_lcf()` for details.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`n`: | Integer constant giving the number of vertices.
`shifts`: | A vector giving the shifts.
`repeats`: | An integer constant giving the number of repeats for the shifts.

Returns:

`` | Error code.

See also:

`` | `igraph_lcf()`, `igraph_extended_chordal_ring()`

Time complexity: O(|V|+|E|), linear in the number of vertices plus the number of edges.

### 1.26. `igraph_from_prufer` — Generates a tree from a Prüfer sequence.

    igraph_error_t igraph_from_prufer(igraph_t *graph, const igraph_vector_int_t *prufer);

A Prüfer sequence is a unique sequence of integers associated with a labelled tree. A tree on `n` vertices can be represented by a sequence of `n-2` integers, each between `0` and `n-1` (inclusive). The algorithm used by this function is based on Paulius Micikevičius, Saverio Caminiti, Narsingh Deo: Linear-time Algorithms for Encoding Trees as Sequences of Node Labels

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`prufer`: | The Prüfer sequence

Returns:

`` | Error code:
`IGRAPH_ENOMEM` | there is not enough memory to perform the operation.
`IGRAPH_EINVAL` | invalid Prüfer sequence given

See also:

`` | `igraph_to_prufer()`, `igraph_kary_tree()`, `igraph_tree_game()`

Time complexity: O(|V|), where |V| is the number of vertices in the tree.

### 1.27. `igraph_atlas` — Create a small graph from the “Graph Atlas”.

    igraph_error_t igraph_atlas(igraph_t *graph, igraph_integer_t number);

The number of the graph is given as a parameter. The graphs are listed:

1. in increasing order of number of nodes;

2. for a fixed number of nodes, in increasing order of the number of edges;

3. for fixed numbers of nodes and edges, in increasing order of the degree sequence, for example 111223 < 112222;

4. for fixed degree sequence, in increasing number of automorphisms.

The data was converted from the NetworkX software package, see http://networkx.github.io .

See An Atlas of Graphs by Ronald C. Read and Robin J. Wilson, Oxford University Press, 1998.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`number`: | The number of the graph to generate.

Added in version 0.2.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.

Example 9.13. File `examples/simple/igraph_atlas.c`

    #include <igraph.h>

    int main(void) {

        igraph_t g;

        igraph_atlas(&g, 45);
        igraph_write_graph_edgelist(&g, stdout);
        printf("\n");
        igraph_destroy(&g);

        igraph_atlas(&g, 0);
        igraph_write_graph_edgelist(&g, stdout);
        printf("\n");
        igraph_destroy(&g);

        igraph_atlas(&g, 1252);
        igraph_write_graph_edgelist(&g, stdout);
        printf("\n");
        igraph_destroy(&g);

        return 0;
    }

### 1.28. `igraph_de_bruijn` — Generate a de Bruijn graph.

    igraph_error_t igraph_de_bruijn(igraph_t *graph, igraph_integer_t m, igraph_integer_t n);

A de Bruijn graph represents relationships between strings. An alphabet of `m` letters are used and strings of length `n` are considered. A vertex corresponds to every possible string and there is a directed edge from vertex `v` to vertex `w` if the string of `v` can be transformed into the string of `w` by removing its first letter and appending a letter to it.

Please note that the graph will have `m` to the power `n` vertices and even more edges, so probably you don't want to supply too big numbers for `m` and `n`.

De Bruijn graphs have some interesting properties, please see another source, e.g. Wikipedia for details.

Arguments:

`graph`: | Pointer to an uninitialized graph object, the result will be stored here.
`m`: | Integer, the number of letters in the alphabet.
`n`: | Integer, the length of the strings.

Returns:

`` | Error code.

See also:

`` | `igraph_kautz()`.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.

### 1.29. `igraph_kautz` — Generate a Kautz graph.

    igraph_error_t igraph_kautz(igraph_t *graph, igraph_integer_t m, igraph_integer_t n);

A Kautz graph is a labeled graph, vertices are labeled by strings of length `n`+1 above an alphabet with `m`+1 letters, with the restriction that every two consecutive letters in the string must be different. There is a directed edge from a vertex `v` to another vertex `w` if it is possible to transform the string of `v` into the string of `w` by removing the first letter and appending a letter to it. For string length 1 the new letter cannot equal the old letter, so there are no loops.

Kautz graphs have some interesting properties, see e.g. Wikipedia for details.

Vincent Matossian wrote the first version of this function in R, thanks.

Arguments:

`graph`: | Pointer to an uninitialized graph object, the result will be stored here.
`m`: | Integer, `m`+1 is the number of letters in the alphabet.
`n`: | Integer, `n`+1 is the length of the strings.

Returns:

`` | Error code.

See also:

`` | `igraph_de_bruijn()`.

Time complexity: O(|V|* [(m+1)/m]^n +|E|), in practice it is more like O(|V|+|E|). |V| is the number of vertices, |E| is the number of edges and `m` and `n` are the corresponding arguments.

### 1.30. `igraph_circulant` — Creates a circulant graph.

    igraph_error_t igraph_circulant(igraph_t *graph, igraph_integer_t n, const igraph_vector_int_t *shifts, igraph_bool_t directed);

A circulant graph `G(n, shifts)` consists of `n` vertices `v_0`, ..., `v_(n-1)` such that for each `s_i` in the list of offsets `shifts`, `v_j` is connected to `v_((j + s_i) mod n)` for all j.

The function can generate either directed or undirected graphs. It does not generate multi-edges or self-loops.

Arguments:

`graph`: | Pointer to an uninitialized graph object, the result will be stored here.
`n`: | Integer, the number of vertices in the circulant graph.
`shifts`: | Integer vector, a list of the offsets within the circulant graph.
`directed`: | Boolean, whether to create a directed graph.

Returns:

`` | Error code.

See also:

`` | `igraph_ring()`, `igraph_generalized_petersen()`, `igraph_extended_chordal_ring()`

Time complexity: O(|V||shifts|), the number of vertices in the graph times the number of shifts.

### 1.31. `igraph_generalized_petersen` — Creates a Generalized Petersen graph.

    igraph_error_t igraph_generalized_petersen(igraph_t *graph, igraph_integer_t n, igraph_integer_t k);

The generalized Petersen graph `G(n, k)` consists of `n` vertices `v_0`, ..., `v_n` forming an "outer" cycle graph, and `n` additional vertices `u_0`, ..., `u_n` forming an "inner" circulant graph where `u_i` is connected to `u_(i + k mod n)`. Additionally, all `v_i` are connected to `u_i`.

`G(n, k)` has `2n` vertices and `3n` edges. The Petersen graph itself is `G(5, 2)`.

Reference:

M. E. Watkins, A Theorem on Tait Colorings with an Application to the Generalized Petersen Graphs, Journal of Combinatorial Theory 6, 152-164 (1969). https://doi.org/10.1016%2FS0021-9800%2869%2980116-X

Arguments:

`graph`: | Pointer to an uninitialized graph object, the result will be stored here.
`n`: | Integer, `n` is the number of vertices in the inner and outer cycle/circulant graphs. It must be at least 3.
`k`: | Integer, `k` is the shift of the circulant graph. It must be positive and less than `n/2`.

Returns:

`` | Error code.

See also:

`` | `igraph_famous()` for the original Petersen graph.

Time complexity: O(|V|), the number of vertices in the graph.

### 1.32. `igraph_extended_chordal_ring` — Create an extended chordal ring.

    igraph_error_t igraph_extended_chordal_ring(
        igraph_t *graph, igraph_integer_t nodes, const igraph_matrix_int_t *W,
        igraph_bool_t directed);

An extended chordal ring is a cycle graph with additional chords connecting its vertices. Each row `L` of the matrix `W` specifies a set of chords to be inserted, in the following way: vertex `i` will connect to a vertex `L[(i mod p)]` steps ahead of it along the cycle, where `p` is the length of `L`. In other words, vertex `i` will be connected to vertex `(i + L[(i mod p)]) mod nodes`. If multiple edges are defined in this way, this will output a non-simple graph. The result can be simplified using `igraph_simplify()`.

See also Kotsis, G: Interconnection Topologies for Parallel Processing Systems, PARS Mitteilungen 11, 1-6, 1993. The igraph extended chordal rings are not identical to the ones in the paper. In igraph the matrix specifies which edges to add. In the paper, a condition is specified which should simultaneously hold between two endpoints and the reverse endpoints.

Arguments:

`graph`: | Pointer to an uninitialized graph object, the result will be stored here.
`nodes`: | Integer constant, the number of vertices in the graph. It must be at least 3.
`W`: | The matrix specifying the extra edges. The number of columns should divide the number of total vertices. The elements are allowed to be negative.
`directed`: | Whether the graph should be directed.

Returns:

`` | Error code.

See also:

`` | `igraph_ring()`, `igraph_lcf()`, `igraph_lcf_vector()`.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.

## 2. Games: Randomized graph generators

Games are randomized graph generators. Randomization means that they generate a different graph every time you call them.

### 2.1. `igraph_grg_game` — Generates a geometric random graph.

    igraph_error_t igraph_grg_game(igraph_t *graph, igraph_integer_t nodes,
                        igraph_real_t radius, igraph_bool_t torus,
                        igraph_vector_t *x, igraph_vector_t *y);

A geometric random graph is created by dropping points (i.e. vertices) randomly on the unit square and then connecting all those pairs which are strictly less than `radius` apart in Euclidean distance.

Original code contributed by Keith Briggs, thanks Keith.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`nodes`: | The number of vertices in the graph.
`radius`: | The radius within which the vertices will be connected.
`torus`: | Logical constant. If true, periodic boundary conditions will be used, i.e. the vertices are assumed to be on a torus instead of a square.
`x`: | An initialized vector or `NULL`. If not `NULL`, the points' x coordinates will be returned here.
`y`: | An initialized vector or `NULL`. If not `NULL`, the points' y coordinates will be returned here.

Returns:

`` | Error code.

Time complexity: TODO, less than O(|V|^2+|E|).

Example 9.14. File `examples/simple/igraph_grg_game.c`

    #include <igraph.h>
    #include <math.h>

    int main(void) {
        igraph_t graph;
        igraph_vector_t x, y;
        igraph_vector_t weights;
        igraph_eit_t eit;
        igraph_real_t avg_dist;

        /* Set random seed for reproducible results */

        igraph_rng_seed(igraph_rng_default(), 42);

        /* Create a random geometric graph and retrieve vertex coordinates */

        igraph_vector_init(&x, 0);
        igraph_vector_init(&y, 0);

        igraph_grg_game(&graph, 200, 0.1, /* torus */ 0, &x, &y);

        /* Compute edge weights as geometric distance */

        igraph_vector_init(&weights, igraph_ecount(&graph));
        igraph_eit_create(&graph, igraph_ess_all(IGRAPH_EDGEORDER_ID), &eit);
        for (; ! IGRAPH_EIT_END(eit); IGRAPH_EIT_NEXT(eit)) {
            igraph_integer_t e = IGRAPH_EIT_GET(eit);
            igraph_integer_t u = IGRAPH_FROM(&graph, e);
            igraph_integer_t v = IGRAPH_TO(&graph, e);

            VECTOR(weights)[e] = hypot(VECTOR(x)[u] - VECTOR(x)[v], VECTOR(y)[u] - VECTOR(y)[v]);
        }
        igraph_eit_destroy(&eit);

        /* Compute average path length */

        igraph_average_path_length_dijkstra(&graph, &avg_dist, NULL, &weights, IGRAPH_UNDIRECTED, /* unconn */ 1);

        printf("Average distance in the geometric graph: %g.\n", avg_dist);

        /* Destroy data structures when no longer needed */

        igraph_vector_destroy(&weights);
        igraph_destroy(&graph);
        igraph_vector_destroy(&x);
        igraph_vector_destroy(&y);

        return 0;
    }

### 2.2. `igraph_barabasi_game` — Generates a graph based on the Barabási-Albert model.

    igraph_error_t igraph_barabasi_game(igraph_t *graph, igraph_integer_t n,
                             igraph_real_t power,
                             igraph_integer_t m,
                             const igraph_vector_int_t *outseq,
                             igraph_bool_t outpref,
                             igraph_real_t A,
                             igraph_bool_t directed,
                             igraph_barabasi_algorithm_t algo,
                             const igraph_t *start_from);

This function implements several variants of the preferential attachment process, including linear and non-linear varieties of the Barabási-Albert and Price models. The graph construction starts with a single vertex, or an existing graph given by the `start_from` parameter. Then new vertices are added one at a time. Each new vertex connects to `m` existing vertices, choosing them with probabilities proportional to

`d^power + A`,

where `d` is the in- or total degree of the existing vertex (controlled by the `outpref` argument), while `power` and `A` are given by parameters. The constant attractiveness `A` is used to ensure that vertices with zero in-degree can also be connected to with non-zero probability.

Barabási, A.-L. and Albert R. 1999. Emergence of scaling in random networks, Science, 286 509--512. https://doi.org/10.1126/science.286.5439.509

de Solla Price, D. J. 1965. Networks of Scientific Papers, Science, 149 510--515. https://doi.org/10.1126/science.149.3683.510

Arguments:

`graph`: | An uninitialized graph object.
`n`: | The number of vertices in the graph.
`power`: | Power of the preferential attachment. In the classic preferential attachment model `power=1`. Other values allow for sampling from a non-linear preferential attachment model. Negative values are only allowed when no zero-degree vertices are present during the construction process, i.e. when the starting graph has no isolated vertices and `outpref` is set to `true`.
`m`: | The number of outgoing edges generated for each vertex. Only used when `outseq` is `NULL`.
`outseq`: | Gives the (out-)degrees of the vertices. If this is constant, this can be a `NULL` pointer or an empty vector. In this case `m` contains the constant out-degree. The very first vertex has by definition no outgoing edges, so the first number in this vector is ignored.
`outpref`: | Boolean, if true not only the in- but also the out-degree of a vertex increases its citation probability. I.e., the citation probability is determined by the total degree of the vertices. Ignored and assumed to be true if the graph being generated is undirected.
`A`: | The constant attractiveness of vertices. When `outpref` is set to `false`, it should be positive to ensure that zero in-degree vertices can be connected to as well.
`directed`: | Boolean, whether to generate a directed graph. When set to `false`, outpref is assumed to be `true`.
`algo`: | The algorithm to use to generate the network. Possible values:
`IGRAPH_BARABASI_BAG` | This is the algorithm that was previously (before version 0.6) solely implemented in igraph. It works by putting the IDs of the vertices into a bag (multiset, really), exactly as many times as their (in-)degree, plus once more. Then the required number of cited vertices are drawn from the bag, with replacement. This method might generate multiple edges. It only works if power=1 and A=1.
`IGRAPH_BARABASI_PSUMTREE` | This algorithm uses a partial prefix-sum tree to generate the graph. It does not generate multiple edges and works for any power and A values.
`IGRAPH_BARABASI_PSUMTREE_MULTIPLE` | This algorithm also uses a partial prefix-sum tree to generate the graph. The difference is, that now multiple edges are allowed. This method was implemented under the name `igraph_nonlinear_barabasi_game` before version 0.6.
`start_from`: | Either a `NULL` pointer, or a graph. In the former case, the starting configuration is a clique of size `m`. In the latter case, the graph is a starting configuration. The graph must be non-empty, i.e. it must have at least one vertex. If a graph is supplied here and the `outseq` argument is also given, then `outseq` should only contain information on the vertices that are not in the `start_from` graph.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid `n`, `m`, `A` or `outseq` parameter.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.

Example 9.15. File `examples/simple/igraph_barabasi_game.c`

    #include <igraph.h>

    int main(void) {

        igraph_t g;
        igraph_vector_int_t v;
        igraph_vector_int_t v2, v3;

        igraph_barabasi_game(&g, 10, /*power=*/ 1, 2, 0, 0, /*A=*/ 1, 1,
                             IGRAPH_BARABASI_BAG, /*start_from=*/ 0);
        if (igraph_ecount(&g) != 18) {
            return 1;
        }
        if (igraph_vcount(&g) != 10) {
            return 2;
        }
        if (!igraph_is_directed(&g)) {
            return 3;
        }

        igraph_vector_int_init(&v, 0);
        igraph_get_edgelist(&g, &v, 0);
        for (igraph_integer_t i = 0; i < igraph_ecount(&g); i++) {
            if (VECTOR(v)[2 * i] <= VECTOR(v)[2 * i + 1]) {
                return 4;
            }
        }
        igraph_vector_int_destroy(&v);
        igraph_destroy(&g);

        /* out-degree sequence */
        igraph_vector_int_init_int(&v3, 10, 0, 1, 3, 3, 4, 5, 6, 7, 8, 9);

        igraph_barabasi_game(&g, 10, /*power=*/ 1, 0, &v3, 0, /*A=*/ 1, 1,
                             IGRAPH_BARABASI_BAG, /*start_from=*/ 0);
        if (igraph_ecount(&g) != igraph_vector_int_sum(&v3)) {
            return 5;
        }
        igraph_vector_int_init(&v2, 0);
        igraph_degree(&g, &v2, igraph_vss_all(), IGRAPH_OUT, 1);
        for (igraph_integer_t i = 0; i < igraph_vcount(&g); i++) {
            if (VECTOR(v3)[i] != VECTOR(v2)[i]) {
                igraph_vector_int_print(&v3);
                printf("\n");
                igraph_vector_int_print(&v2);
                return 6;
            }
        }
        igraph_vector_int_destroy(&v3);
        igraph_vector_int_destroy(&v2);
        igraph_destroy(&g);

        /* outpref, we cannot really test this quantitatively,
           would need to set random seed */
        igraph_barabasi_game(&g, 10, /*power=*/ 1, 2, 0, 1, /*A=*/ 1, 1,
                             IGRAPH_BARABASI_BAG, /*start_from=*/ 0);
        igraph_vector_int_init(&v, 0);
        igraph_get_edgelist(&g, &v, 0);
        for (igraph_integer_t i = 0; i < igraph_ecount(&g); i++) {
            if (VECTOR(v)[2 * i] <= VECTOR(v)[2 * i + 1]) {
                return 7;
            }
        }
        if (!igraph_is_directed(&g)) {
            return 8;
        }
        igraph_vector_int_destroy(&v);
        igraph_destroy(&g);

        return 0;
    }

Example 9.16. File `examples/simple/igraph_barabasi_game2.c`

    #include <igraph.h>
    #include <stdio.h>

    int main(void) {

        igraph_t g;
        igraph_bool_t simple;

        igraph_barabasi_game(/* graph=    */ &g,
                                             /* n=        */ 100,
                                             /* power=    */ 1.0,
                                             /* m=        */ 2,
                                             /* outseq=   */ 0,
                                             /* outpref=  */ 0,
                                             /* A=        */ 1.0,
                                             /* directed= */ IGRAPH_DIRECTED,
                                             /* algo=     */ IGRAPH_BARABASI_PSUMTREE,
                                             /* start_from= */ 0);

        if (igraph_ecount(&g) != 197) {
            return 1;
        }
        if (igraph_vcount(&g) != 100) {
            return 2;
        }
        igraph_is_simple(&g, &simple);
        if (!simple) {
            return 3;
        }

        igraph_destroy(&g);

        /* ============================== */

        igraph_barabasi_game(/* graph=    */ &g,
                                             /* n=        */ 100,
                                             /* power=    */ 1.0,
                                             /* m=        */ 2,
                                             /* outseq=   */ 0,
                                             /* outpref=  */ 0,
                                             /* A=        */ 1.0,
                                             /* directed= */ IGRAPH_DIRECTED,
                                             /* algo=     */ IGRAPH_BARABASI_PSUMTREE_MULTIPLE,
                                             /* start_from= */ 0);

        if (igraph_ecount(&g) != 198) {
            return 4;
        }
        if (igraph_vcount(&g) != 100) {
            return 5;
        }
        igraph_is_simple(&g, &simple);
        if (simple) {
            return 6;
        }

        igraph_destroy(&g);

        /* ============================== */

        igraph_barabasi_game(/* graph=    */ &g,
                                             /* n=        */ 100,
                                             /* power=    */ 1.0,
                                             /* m=        */ 2,
                                             /* outseq=   */ 0,
                                             /* outpref=  */ 0,
                                             /* A=        */ 1.0,
                                             /* directed= */ IGRAPH_DIRECTED,
                                             /* algo=     */ IGRAPH_BARABASI_BAG,
                                             /* start_from= */ 0);

        if (igraph_ecount(&g) != 198) {
            return 7;
        }
        if (igraph_vcount(&g) != 100) {
            return 8;
        }
        igraph_is_simple(&g, &simple);
        if (simple) {
            return 9;
        }

        igraph_destroy(&g);

        return 0;
    }

### 2.3. `igraph_erdos_renyi_game_gnm` — Generates a random (Erdős-Rényi) graph with a fixed number of edges.

    igraph_error_t igraph_erdos_renyi_game_gnm(
        igraph_t *graph, igraph_integer_t n, igraph_integer_t m,
        igraph_bool_t directed, igraph_bool_t loops
    );

In this model, a graph with n vertices and m edges is generated such that the edges are selected uniformly at random.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`n`: | The number of vertices in the graph.
`m`: | The number of edges in the graph.
`directed`: | Logical, whether to generate a directed graph.
`loops`: | Logical, whether to generate self-loops.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid `n` or `m` parameter. `IGRAPH_ENOMEM`: there is not enough memory for the operation.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.

See also:

`` | `igraph_barabasi_game()`, `igraph_growing_random_game()`, `igraph_erdos_renyi_game_gnp()`

Example 9.17. File `examples/simple/igraph_erdos_renyi_game_gnm.c`

    #include <igraph.h>

    int main(void) {
        igraph_t graph;
        igraph_vector_int_t component_sizes;

        igraph_rng_seed(igraph_rng_default(), 42); /* make program deterministic */

        /* Sample a graph from the Erdős-Rényi G(n,m) model */

        igraph_erdos_renyi_game_gnm(
            &graph, /* n= */ 100, /* m= */ 100,
            IGRAPH_UNDIRECTED, IGRAPH_NO_LOOPS
        );

        /* Compute the fraction of vertices contained within the largest connected component */

        igraph_vector_int_init(&component_sizes, 0);
        igraph_connected_components(&graph, NULL, &component_sizes, NULL, IGRAPH_STRONG);

        printf(
            "Fraction of vertices in giant component: %g\n",
            ((double) igraph_vector_int_max(&component_sizes)) / igraph_vcount(&graph)
        );

        /* Clean up data structures when no longer needed */

        igraph_vector_int_destroy(&component_sizes);
        igraph_destroy(&graph);

        return 0;
    }

### 2.4. `igraph_erdos_renyi_game_gnp` — Generates a random (Erdős-Rényi) graph with fixed edge probabilities.

    igraph_error_t igraph_erdos_renyi_game_gnp(
        igraph_t *graph, igraph_integer_t n, igraph_real_t p,
        igraph_bool_t directed, igraph_bool_t loops
    );

In this model, a graph with n vertices is generated such that every possible edge is included in the graph with probability p.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`n`: | The number of vertices in the graph.
`p`: | The probability of the existence of an edge in the graph.
`directed`: | Logical, whether to generate a directed graph.
`loops`: | Logical, whether to generate self-loops.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid `n` or `p` parameter. `IGRAPH_ENOMEM`: there is not enough memory for the operation.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.

See also:

`` | `igraph_barabasi_game()`, `igraph_growing_random_game()`, `igraph_erdos_renyi_game_gnm()`

Example 9.18. File `examples/simple/igraph_erdos_renyi_game_gnp.c`

    #include <igraph.h>

    int main(void) {
        igraph_t graph;
        igraph_vector_int_t component_sizes;

        igraph_rng_seed(igraph_rng_default(), 42); /* make program deterministic */

        /* Sample a graph from the Erdős-Rényi G(n,p) model */

        igraph_erdos_renyi_game_gnp(
            &graph, /* n= */ 100, /* p= */ 0.01,
            IGRAPH_UNDIRECTED, IGRAPH_NO_LOOPS
        );

        /* Compute the fraction of vertices contained within the largest connected component */

        igraph_vector_int_init(&component_sizes, 0);
        igraph_connected_components(&graph, NULL, &component_sizes, NULL, IGRAPH_STRONG);

        printf(
            "Fraction of vertices in giant component: %g\n",
            ((double) igraph_vector_int_max(&component_sizes)) / igraph_vcount(&graph)
        );

        /* Clean up data structures when no longer needed */

        igraph_vector_int_destroy(&component_sizes);
        igraph_destroy(&graph);

        return 0;
    }

### 2.5. `igraph_watts_strogatz_game` — The Watts-Strogatz small-world model.

    igraph_error_t igraph_watts_strogatz_game(igraph_t *graph, igraph_integer_t dim,
                                   igraph_integer_t size, igraph_integer_t nei,
                                   igraph_real_t p, igraph_bool_t loops,
                                   igraph_bool_t multiple);

This function generates networks with the small-world property based on a variant of the Watts-Strogatz model. The network is obtained by first creating a periodic undirected lattice, then rewiring both endpoints of each edge with probability `p`, while avoiding the creation of multi-edges.

This process differs from the original model of Watts and Strogatz (see reference) in that it rewires both endpoints of edges. Thus in the limit of `p=1`, we obtain a G(n,m) random graph with the same number of vertices and edges as the original lattice. In comparison, the original Watts-Strogatz model only rewires a single endpoint of each edge, thus the network does not become fully random even for `p=1`. For appropriate choices of `p`, both models exhibit the property of simultaneously having short path lengths and high clustering.

Reference:

Duncan J Watts and Steven H Strogatz: Collective dynamics of “small world” networks, Nature 393, 440-442, 1998.

Arguments:

`graph`: | The graph to initialize.
`dim`: | The dimension of the lattice.
`size`: | The size of the lattice along each dimension.
`nei`: | The size of the neighborhood for each vertex. This is the same as the `nei` argument of `igraph_connect_neighborhood()`.
`p`: | The rewiring probability. A real number between zero and one (inclusive).
`loops`: | Logical, whether to generate loop edges.
`multiple`: | Logical, whether to allow multiple edges in the generated graph.

Returns:

`` | Error code.

See also:

`` | `igraph_square_lattice()`, `igraph_connect_neighborhood()` and `igraph_rewire_edges()` can be used if more flexibility is needed, e.g. a different type of lattice.

Time complexity: O(|V|*d^o+|E|), |V| and |E| are the number of vertices and edges, d is the average degree, o is the `nei` argument.

### 2.6. `igraph_rewire_edges` — Rewires the edges of a graph with constant probability.

    igraph_error_t igraph_rewire_edges(igraph_t *graph, igraph_real_t prob,
                            igraph_bool_t loops, igraph_bool_t multiple);

This function rewires the edges of a graph with a constant probability. More precisely each end point of each edge is rewired to a uniformly randomly chosen vertex with constant probability `prob`.

Note that this function modifies the input `graph`, call `igraph_copy()` if you want to keep it.

Arguments:

`graph`: | The input graph, this will be rewired, it can be directed or undirected.
`prob`: | The rewiring probability a constant between zero and one (inclusive).
`loops`: | Boolean, whether loop edges are allowed in the new graph, or not.
`multiple`: | Boolean, whether multiple edges are allowed in the new graph.

Returns:

`` | Error code.

See also:

`` | `igraph_watts_strogatz_game()` uses this function for the rewiring.

Time complexity: O(|V|+|E|).

### 2.7. `igraph_rewire_directed_edges` — Rewires the chosen endpoint of directed edges.

    igraph_error_t igraph_rewire_directed_edges(igraph_t *graph, igraph_real_t prob,
                                     igraph_bool_t loops, igraph_neimode_t mode);

This function rewires either the start or end of directed edges in a graph with a constant probability. Correspondingly, either the in-degree sequence or the out-degree sequence of the graph will be preserved.

Note that this function modifies the input `graph`, call `igraph_copy()` if you want to keep it.

This function can produce multiple edges between two vertices.

Arguments:

`graph`: | The input graph, this will be rewired, it can be directed or undirected. If it is undirected or `mode` is set to IGRAPH_ALL, `igraph_rewire_edges()` will be called.
`prob`: | The rewiring probability, a constant between zero and one (inclusive).
`loops`: | Boolean, whether loop edges are allowed in the new graph, or not.
`mode`: | The endpoints of directed edges to rewire. It is ignored for undirected graphs. Possible values:
`IGRAPH_OUT` | rewire the end of each directed edge
`IGRAPH_IN` | rewire the start of each directed edge
`IGRAPH_ALL` | rewire both endpoints of each edge

Returns:

`` | Error code.

See also:

`` | `igraph_rewire_edges()`, `igraph_rewire()`

Time complexity: O(|E|).

### 2.8. `igraph_degree_sequence_game` — Generates a random graph with a given degree sequence.

    igraph_error_t igraph_degree_sequence_game(igraph_t *graph, const igraph_vector_int_t *out_deg,
                                    const igraph_vector_int_t *in_deg,
                                    igraph_degseq_t method);

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`out_deg`: | The degree sequence for an undirected graph (if `in_seq` is `NULL` or of length zero), or the out-degree sequence of a directed graph (if `in_deq` is not of length zero).
`in_deg`: | It is either a zero-length vector or `NULL` (if an undirected graph is generated), or the in-degree sequence.
`method`: | The method to generate the graph. Possible values:
`IGRAPH_DEGSEQ_CONFIGURATION` | This method implements the configuration model. For undirected graphs, it puts all vertex IDs in a bag such that the multiplicity of a vertex in the bag is the same as its degree. Then it draws pairs from the bag until the bag becomes empty. This method may generate both loop (self) edges and multiple edges. For directed graphs, the algorithm is basically the same, but two separate bags are used for the in- and out-degrees. Undirected graphs are generated with probability proportional to `(\prod_{i<j} A_{ij} ! \prod_i A_{ii} !!)^{-1}`, where `A` denotes the adjacency matrix and `!!` denotes the double factorial. Here `A` is assumed to have twice the number of self-loops on its diagonal. The corresponding expression for directed graphs is `(\prod_{i,j} A_{ij}!)^{-1}`. Thus the probability of all simple graphs (which only have 0s and 1s in the adjacency matrix) is the same, while that of non-simple ones depends on their edge and self-loop multiplicities.
`IGRAPH_DEGSEQ_CONFIGURATION_SIMPLE` | This method is identical to `IGRAPH_DEGSEQ_CONFIGURATION`, but if the generated graph is not simple, it rejects it and re-starts the generation. It generates all simple graphs with the same probability.
`IGRAPH_DEGSEQ_FAST_HEUR_SIMPLE` | This method generates simple graphs. It is similar to `IGRAPH_DEGSEQ_CONFIGURATION` but tries to avoid multiple and loop edges and restarts the generation from scratch if it gets stuck. It can generate all simple realizations of a degree sequence, but it is not guaranteed to sample them uniformly. This method is relatively fast and it will eventually succeed if the provided degree sequence is graphical, but there is no upper bound on the number of iterations.
`IGRAPH_DEGSEQ_EDGE_SWITCHING_SIMPLE` | This is an MCMC sampler based on degree-preserving edge switches. It generates simple undirected or directed graphs. It uses `igraph_realize_degree_sequence()` to construct an initial graph, then rewires it using `igraph_rewire()`.
`IGRAPH_DEGSEQ_VL` | This method samples undirected connected graphs approximately uniformly. It is a Monte Carlo method based on degree-preserving edge switches. This generator should be favoured if undirected and connected graphs are to be generated and execution time is not a concern. igraph uses the original implementation of Fabien Viger; for the algorithm, see https://www-complexnetworks.lip6.fr/~latapy/FV/generation.html and the paper https://arxiv.org/abs/cs/0502085

Returns:

`` | Error code: `IGRAPH_ENOMEM`: there is not enough memory to perform the operation. `IGRAPH_EINVAL`: invalid method parameter, or invalid in- and/or out-degree vectors. The degree vectors should be non-negative, `out_deg` should sum up to an even integer for undirected graphs; the length and sum of `out_deg` and `in_deg` should match for directed graphs.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges for `IGRAPH_DEGSEQ_SIMPLE`. The time complexity of the other modes is not known.

See also:

`` | `igraph_barabasi_game()`, `igraph_erdos_renyi_game_gnm()`, `igraph_erdos_renyi_game_gnp()`, `igraph_is_graphical()`

Example 9.19. File `examples/simple/igraph_degree_sequence_game.c`

    #include <igraph.h>

    int main(void) {
        igraph_t g;
        igraph_vector_int_t outdeg, indeg;
        igraph_vector_int_t vec;
        igraph_bool_t is_simple;

        /* Set random seed for reproducibility */
        igraph_rng_seed(igraph_rng_default(), 42);

        igraph_vector_int_init_int(&outdeg, 10, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3);
        igraph_vector_int_init_int(&indeg, 10, 4, 4, 2, 2, 4, 4, 2, 2, 3, 3);
        igraph_vector_int_init(&vec, 0);

        /* checking the configuration model, undirected graphs */
        igraph_degree_sequence_game(&g, &outdeg, 0, IGRAPH_DEGSEQ_CONFIGURATION);
        if (igraph_is_directed(&g) || igraph_vcount(&g) != 10) {
            return 1;
        }
        if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_OUT, 1)) {
            return 2;
        }
        igraph_vector_int_print(&vec);
        igraph_destroy(&g);

        /* checking the Viger-Latapy method, undirected graphs */
        igraph_degree_sequence_game(&g, &outdeg, 0, IGRAPH_DEGSEQ_VL);
        if (igraph_is_directed(&g) || igraph_vcount(&g) != 10) {
            return 3;
        }
        if (igraph_is_simple(&g, &is_simple) || !is_simple) {
            return 4;
        }
        if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_OUT, 0)) {
            return 5;
        }
        igraph_vector_int_print(&vec);
        igraph_destroy(&g);

        /* checking the configuration model, directed graphs */
        igraph_degree_sequence_game(&g, &outdeg, &indeg, IGRAPH_DEGSEQ_CONFIGURATION);
        if (!igraph_is_directed(&g) || igraph_vcount(&g) != 10) {
            return 6;
        }
        if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_OUT, 1)) {
            return 7;
        }
        igraph_vector_int_print(&vec);
        if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_IN, 1)) {
            return 8;
        }
        igraph_vector_int_print(&vec);
        igraph_destroy(&g);

        /* checking the fast heuristic method, undirected graphs */
        igraph_degree_sequence_game(&g, &outdeg, 0, IGRAPH_DEGSEQ_FAST_HEUR_SIMPLE);
        if (igraph_is_directed(&g) || igraph_vcount(&g) != 10) {
            return 9;
        }
        if (igraph_is_simple(&g, &is_simple) || !is_simple) {
            return 10;
        }
        if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_OUT, 1)) {
            return 11;
        }
        igraph_vector_int_print(&vec);
        igraph_destroy(&g);

        /* checking the fast heuristic method, directed graphs */
        igraph_degree_sequence_game(&g, &outdeg, &indeg, IGRAPH_DEGSEQ_FAST_HEUR_SIMPLE);
        if (!igraph_is_directed(&g) || igraph_vcount(&g) != 10) {
            return 12;
        }
        if (igraph_is_simple(&g, &is_simple) || !is_simple) {
            return 13;
        }
        if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_OUT, 1)) {
            return 14;
        }
        igraph_vector_int_print(&vec);
        if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_IN, 1)) {
            return 15;
        }
        igraph_vector_int_print(&vec);
        igraph_destroy(&g);

        igraph_vector_int_destroy(&vec);
        igraph_vector_int_destroy(&outdeg);
        igraph_vector_int_destroy(&indeg);

        return 0;
    }

### 2.9. `igraph_k_regular_game` — Generates a random graph where each vertex has the same degree.

    igraph_error_t igraph_k_regular_game(igraph_t *graph,
                              igraph_integer_t no_of_nodes, igraph_integer_t k,
                              igraph_bool_t directed, igraph_bool_t multiple);

This game generates a directed or undirected random graph where the degrees of vertices are equal to a predefined constant k. For undirected graphs, at least one of k and the number of vertices must be even.

Currently, this game simply uses `igraph_degree_sequence_game` with the `IGRAPH_DEGSEQ_CONFIGURATION` or the `IGRAPH_DEGSEQ_FAST_SIMPLE` method and appropriately constructed degree sequences. Thefore, it does not sample uniformly: while it can generate all k-regular graphs with the given number of vertices, it does not generate each one with the same probability.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`no_of_nodes`: | The number of nodes in the generated graph.
`k`: | The degree of each vertex in an undirected graph, or the out-degree and in-degree of each vertex in a directed graph.
`directed`: | Whether the generated graph will be directed.
`multiple`: | Whether to allow multiple edges in the generated graph.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid parameter; e.g., negative number of nodes, or odd number of nodes and odd k for undirected graphs. `IGRAPH_ENOMEM`: there is not enough memory for the operation.

Time complexity: O(|V|+|E|) if `multiple` is true, otherwise not known.

### 2.10. `igraph_static_fitness_game` — Non-growing random graph with edge probabilities proportional to node fitness scores.

    igraph_error_t igraph_static_fitness_game(igraph_t *graph, igraph_integer_t no_of_edges,
                                   const igraph_vector_t *fitness_out, const igraph_vector_t *fitness_in,
                                   igraph_bool_t loops, igraph_bool_t multiple);

This game generates a directed or undirected random graph where the probability of an edge between vertices i and j depends on the fitness scores of the two vertices involved. For undirected graphs, each vertex has a single fitness score. For directed graphs, each vertex has an out- and an in-fitness, and the probability of an edge from i to j depends on the out-fitness of vertex i and the in-fitness of vertex j.

The generation process goes as follows. We start from N disconnected nodes (where N is given by the length of the fitness vector). Then we randomly select two vertices i and j, with probabilities proportional to their fitnesses. (When the generated graph is directed, i is selected according to the out-fitnesses and j is selected according to the in-fitnesses). If the vertices are not connected yet (or if multiple edges are allowed), we connect them; otherwise we select a new pair. This is repeated until the desired number of links are created.

It can be shown that the expected degree of each vertex will be proportional to its fitness, although the actual, observed degree will not be. If you need to generate a graph with an exact degree sequence, consider `igraph_degree_sequence_game` instead.

This model is commonly used to generate static scale-free networks. To achieve this, you have to draw the fitness scores from the desired power-law distribution. Alternatively, you may use `igraph_static_power_law_game` which generates the fitnesses for you with a given exponent.

Reference:

Goh K-I, Kahng B, Kim D: Universal behaviour of load distribution in scale-free networks. Phys Rev Lett 87(27):278701, 2001 https://doi.org/10.1103/PhysRevLett.87.278701.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`fitness_out`: | A numeric vector containing the fitness of each vertex. For directed graphs, this specifies the out-fitness of each vertex.
`fitness_in`: | If `NULL`, the generated graph will be undirected. If not `NULL`, this argument specifies the in-fitness of each vertex.
`no_of_edges`: | The number of edges in the generated graph.
`loops`: | Whether to allow loop edges in the generated graph.
`multiple`: | Whether to allow multiple edges in the generated graph.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid parameter `IGRAPH_ENOMEM`: there is not enough memory for the operation.

Time complexity: O(|V| + |E| log |E|).

### 2.11. `igraph_static_power_law_game` — Generates a non-growing random graph with expected power-law degree distributions.

    igraph_error_t igraph_static_power_law_game(igraph_t *graph,
                                     igraph_integer_t no_of_nodes, igraph_integer_t no_of_edges,
                                     igraph_real_t exponent_out, igraph_real_t exponent_in,
                                     igraph_bool_t loops, igraph_bool_t multiple,
                                     igraph_bool_t finite_size_correction);

This game generates a directed or undirected random graph where the degrees of vertices follow power-law distributions with prescribed exponents. For directed graphs, the exponents of the in- and out-degree distributions may be specified separately.

The game simply uses `igraph_static_fitness_game` with appropriately constructed fitness vectors. In particular, the fitness of vertex i is i-alpha, where alpha = 1/(gamma-1) and gamma is the exponent given in the arguments.

To remove correlations between in- and out-degrees in case of directed graphs, the in-fitness vector will be shuffled after it has been set up and before `igraph_static_fitness_game` is called.

Note that significant finite size effects may be observed for exponents smaller than 3 in the original formulation of the game. This function provides an argument that lets you remove the finite size effects by assuming that the fitness of vertex i is (i+i0-1)-alpha, where i0 is a constant chosen appropriately to ensure that the maximum degree is less than the square root of the number of edges times the average degree; see the paper of Chung and Lu, and Cho et al for more details.

References:

Goh K-I, Kahng B, Kim D: Universal behaviour of load distribution in scale-free networks. Phys Rev Lett 87(27):278701, 2001.

Chung F and Lu L: Connected components in a random graph with given degree sequences. Annals of Combinatorics 6, 125-145, 2002.

Cho YS, Kim JS, Park J, Kahng B, Kim D: Percolation transitions in scale-free networks under the Achlioptas process. Phys Rev Lett 103:135702, 2009.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`no_of_nodes`: | The number of nodes in the generated graph.
`no_of_edges`: | The number of edges in the generated graph.
`exponent_out`: | The power law exponent of the degree distribution. For directed graphs, this specifies the exponent of the out-degree distribution. It must be greater than or equal to 2. If you pass `IGRAPH_INFINITY` here, you will get back an Erdős-Rényi random network.
`exponent_in`: | If negative, the generated graph will be undirected. If greater than or equal to 2, this argument specifies the exponent of the in-degree distribution. If non-negative but less than 2, an error will be generated.
`loops`: | Whether to allow loop edges in the generated graph.
`multiple`: | Whether to allow multiple edges in the generated graph.
`finite_size_correction`: | Whether to use the proposed finite size correction of Cho et al.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid parameter `IGRAPH_ENOMEM`: there is not enough memory for the operation.

Time complexity: O(|V| + |E| log |E|).

### 2.12. `igraph_forest_fire_game` — Generates a network according to the “forest fire game”.

    igraph_error_t igraph_forest_fire_game(igraph_t *graph, igraph_integer_t nodes,
                                igraph_real_t fw_prob, igraph_real_t bw_factor,
                                igraph_integer_t pambs, igraph_bool_t directed);

The forest fire model intends to reproduce the following network characteristics, observed in real networks:

- Heavy-tailed in- and out-degree distributions.

- Community structure.

- Densification power-law. The network is densifying in time, according to a power-law rule.

- Shrinking diameter. The diameter of the network decreases in time.

The network is generated in the following way. One vertex is added at a time. This vertex connects to (cites) `ambs` vertices already present in the network, chosen uniformly random. Now, for each cited vertex `v` we do the following procedure:

1. We generate two random numbers, `x` and `y`, that are geometrically distributed with means `p/(1-p)` and `rp(1-rp)`. (`p` is `fw_prob`, `r` is `bw_factor`.) The new vertex cites `x` outgoing neighbors and `y` incoming neighbors of `v`, from those which are not yet cited by the new vertex. If there are less than `x` or `y` such vertices available then we cite all of them.

2. The same procedure is applied to all the newly cited vertices.

See also: Jure Leskovec, Jon Kleinberg and Christos Faloutsos. Graphs over time: densification laws, shrinking diameters and possible explanations. KDD '05: Proceeding of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining , 177--187, 2005.

Note however, that the version of the model in the published paper is incorrect in the sense that it cannot generate the kind of graphs the authors claim. A corrected version is available from http://cs.stanford.edu/people/jure/pubs/powergrowth-tkdd.pdf , our implementation is based on this.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`nodes`: | The number of vertices in the graph.
`fw_prob`: | The forward burning probability.
`bw_factor`: | The backward burning ratio. The backward burning probability is calculated as `bw_factor * fw_prob`.
`pambs`: | The number of ambassador vertices.
`directed`: | Whether to create a directed graph.

Returns:

`` | Error code.

Time complexity: TODO.

### 2.13. `igraph_rewire` — Randomly rewires a graph while preserving its degree sequence.

    igraph_error_t igraph_rewire(igraph_t *graph, igraph_integer_t n, igraph_rewiring_t mode);

This function generates a new graph based on the original one by randomly "rewriting" edges while preserving the original graph's degree sequence. The rewiring is done "in place", so no new graph will be allocated. If you would like to keep the original graph intact, use `igraph_copy()` beforehand. All graph attributes will be lost.

The rewiring is performed with degree-preserving edge switches: Two arbitrary edges are picked uniformly at random, namely `(a, b)` and `(c, d)`, then they are replaced by `(a, d)` and `(b, c)` if this preserves the constraints specified by `mode`.

Arguments:

`graph`: | The graph object to be rewired.
`n`: | Number of rewiring trials to perform.
`mode`: | The rewiring algorithm to be used. It can be one of the following flags:
`IGRAPH_REWIRING_SIMPLE` | This method does not create or destroy self-loops, and does not create multi-edges.
`IGRAPH_REWIRING_SIMPLE_LOOPS` | This method allows the creation or destruction of self-loops. Self-loops are created by switching edges that have a single common endpoint.

Returns:

`` | Error code:
`IGRAPH_EINVMODE` | Invalid rewiring mode.
`IGRAPH_ENOMEM` | Not enough memory for temporary data.

Time complexity: TODO.

### 2.14. `igraph_growing_random_game` — Generates a growing random graph.

    igraph_error_t igraph_growing_random_game(igraph_t *graph, igraph_integer_t n,
                                   igraph_integer_t m, igraph_bool_t directed,
                                   igraph_bool_t citation);

This function simulates a growing random graph. We start out with one vertex. In each step a new vertex is added and a number of new edges are also added. These graphs are known to be different from standard (not growing) random graphs.

Arguments:

`graph`: | Uninitialized graph object.
`n`: | The number of vertices in the graph.
`m`: | The number of edges to add in a time step (i.e. after adding a vertex).
`directed`: | Boolean, whether to generate a directed graph.
`citation`: | Boolean, if `true`, the edges always originate from the most recently added vertex and are connected to a previous vertex.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid `n` or `m` parameter.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.

### 2.15. `igraph_callaway_traits_game` — Simulates a growing network with vertex types.

    igraph_error_t igraph_callaway_traits_game(igraph_t *graph, igraph_integer_t nodes,
                                    igraph_integer_t types, igraph_integer_t edges_per_step,
                                    const igraph_vector_t *type_dist,
                                    const igraph_matrix_t *pref_matrix,
                                    igraph_bool_t directed,
                                    igraph_vector_int_t *node_type_vec);

The different types of vertices prefer to connect other types of vertices with a given probability.

The simulation goes like this: in each discrete time step a new vertex is added to the graph. The type of this vertex is generated based on `type_dist`. Then two vertices are selected uniformly randomly from the graph. The probability that they will be connected depends on the types of these vertices and is taken from `pref_matrix`. Then another two vertices are selected and this is repeated `edges_per_step` times in each time step.

References:

D. S. Callaway, J. E. Hopcroft, J. M. Kleinberg, M. E. J. Newman, and S. H. Strogatz, Are randomly grown graphs really random? Phys. Rev. E 64, 041902 (2001). https://doi.org/10.1103/PhysRevE.64.041902

Arguments:

`graph`: | Pointer to an uninitialized graph.
`nodes`: | The number of nodes in the graph.
`types`: | Number of node types.
`edges_per_step`: | The number of connections tried in each time step.
`type_dist`: | Vector giving the distribution of the vertex types. If `NULL`, the distribution is assumed to be uniform.
`pref_matrix`: | Matrix giving the connection probabilities for the vertex types.
`directed`: | Logical, whether to generate a directed graph.
`node_type_vec`: | An initialized vector or `NULL`. If not `NULL`, the type of each node will be stored here.

Returns:

`` | Error code.

Added in version 0.2.

Time complexity: O(|V|*k*log(|V|)), |V| is the number of vertices, k is `edges_per_step`.

### 2.16. `igraph_establishment_game` — Generates a graph with a simple growing model with vertex types.

    igraph_error_t igraph_establishment_game(igraph_t *graph, igraph_integer_t nodes,
                                  igraph_integer_t types, igraph_integer_t k,
                                  const igraph_vector_t *type_dist,
                                  const igraph_matrix_t *pref_matrix,
                                  igraph_bool_t directed,
                                  igraph_vector_int_t *node_type_vec);

The simulation goes like this: a single vertex is added at each time step. This new vertex tries to connect to `k` vertices in the graph. The probability that such a connection is realized depends on the types of the vertices involved.

Arguments:

`graph`: | Pointer to an uninitialized graph.
`nodes`: | The number of vertices in the graph.
`types`: | The number of vertex types.
`k`: | The number of connections tried in each time step.
`type_dist`: | Vector giving the distribution of vertex types. If `NULL`, the distribution is assumed to be uniform.
`pref_matrix`: | Matrix giving the connection probabilities for different vertex types.
`directed`: | Logical, whether to generate a directed graph.
`node_type_vec`: | An initialized vector or `NULL`. If not `NULL`, the type of each node will be stored here.

Returns:

`` | Error code.

Added in version 0.2.

Time complexity: O(|V|*k*log(|V|)), |V| is the number of vertices and k is the `k` parameter.

### 2.17. `igraph_preference_game` — Generates a graph with vertex types and connection preferences.

    igraph_error_t igraph_preference_game(igraph_t *graph, igraph_integer_t nodes,
                               igraph_integer_t types,
                               const igraph_vector_t *type_dist,
                               igraph_bool_t fixed_sizes,
                               const igraph_matrix_t *pref_matrix,
                               igraph_vector_int_t *node_type_vec,
                               igraph_bool_t directed,
                               igraph_bool_t loops);

This is practically the nongrowing variant of `igraph_establishment_game()`. A given number of vertices are generated. Every vertex is assigned to a vertex type according to the given type probabilities. Finally, every vertex pair is evaluated and an edge is created between them with a probability depending on the types of the vertices involved.

In other words, this function generates a graph according to a block-model. Vertices are divided into groups (or blocks), and the probability the two vertices are connected depends on their groups only.

Arguments:

`graph`: | Pointer to an uninitialized graph.
`nodes`: | The number of vertices in the graph.
`types`: | The number of vertex types.
`type_dist`: | Vector giving the distribution of vertex types. If `NULL`, all vertex types will have equal probability. See also the `fixed_sizes` argument.
`fixed_sizes`: | Boolean. If true, then the number of vertices with a given vertex type is fixed and the `type_dist` argument gives these numbers for each vertex type. If true, and `type_dist` is `NULL`, then the function tries to make vertex groups of the same size. If this is not possible, then some groups will have an extra vertex.
`pref_matrix`: | Matrix giving the connection probabilities for different vertex types. This should be symmetric if the requested graph is undirected.
`node_type_vec`: | A vector where the individual generated vertex types will be stored. If `NULL`, the vertex types won't be saved.
`directed`: | Logical, whether to generate a directed graph. If undirected graphs are requested, only the lower left triangle of the preference matrix is considered.
`loops`: | Logical, whether loop edges are allowed.

Returns:

`` | Error code.

Added in version 0.3.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.

See also:

`` | `igraph_asymmetric_preference_game()`, `igraph_establishment_game()`, `igraph_callaway_traits_game()`

### 2.18. `igraph_asymmetric_preference_game` — Generates a graph with asymmetric vertex types and connection preferences.

    igraph_error_t igraph_asymmetric_preference_game(igraph_t *graph, igraph_integer_t nodes,
                                          igraph_integer_t no_out_types,
                                          igraph_integer_t no_in_types,
                                          const igraph_matrix_t *type_dist_matrix,
                                          const igraph_matrix_t *pref_matrix,
                                          igraph_vector_int_t *node_type_out_vec,
                                          igraph_vector_int_t *node_type_in_vec,
                                          igraph_bool_t loops);

This is the asymmetric variant of `igraph_preference_game()`. A given number of vertices are generated. Every vertex is assigned to an "outgoing" and an "incoming " vertex type according to the given joint type probabilities. Finally, every vertex pair is evaluated and a directed edge is created between them with a probability depending on the "outgoing" type of the source vertex and the "incoming" type of the target vertex.

Arguments:

`graph`: | Pointer to an uninitialized graph.
`nodes`: | The number of vertices in the graph.
`no_out_types`: | The number of vertex out-types.
`no_in_types`: | The number of vertex in-types.
`type_dist_matrix`: | Matrix of size `out_types * in_types`, giving the joint distribution of vertex types. If `NULL`, incoming and outgoing vertex types are independent and uniformly distributed.
`pref_matrix`: | Matrix of size `out_types * in_types`, giving the connection probabilities for different vertex types.
`node_type_out_vec`: | A vector where the individual generated "outgoing" vertex types will be stored. If `NULL`, the vertex types won't be saved.
`node_type_in_vec`: | A vector where the individual generated "incoming" vertex types will be stored. If `NULL`, the vertex types won't be saved.
`loops`: | Logical, whether loop edges are allowed.

Returns:

`` | Error code.

Added in version 0.3.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.

See also:

`` | `igraph_preference_game()`

### 2.19. `igraph_recent_degree_game` — Stochastic graph generator based on the number of incident edges a node has gained recently.

    igraph_error_t igraph_recent_degree_game(igraph_t *graph, igraph_integer_t nodes,
                                  igraph_real_t power,
                                  igraph_integer_t time_window,
                                  igraph_integer_t m,
                                  const igraph_vector_int_t *outseq,
                                  igraph_bool_t outpref,
                                  igraph_real_t zero_appeal,
                                  igraph_bool_t directed);

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`nodes`: | The number of vertices in the graph, this is the same as the number of time steps.
`power`: | The exponent, the probability that a node gains a new edge is proportional to the number of edges it has gained recently (in the last `window` time steps) to `power`.
`time_window`: | Integer constant, the size of the time window to use to count the number of recent edges.
`m`: | Integer constant, the number of edges to add per time step if the `outseq` parameter is a null pointer or a zero-length vector.
`outseq`: | The number of edges to add in each time step. This argument is ignored if it is a null pointer or a zero length vector. In this case the constant `m` parameter is used.
`outpref`: | Logical constant, if true the edges originated by a vertex also count as recent incident edges. For most applications it is reasonable to set it to false.
`zero_appeal`: | Constant giving the attractiveness of the vertices which haven't gained any edge recently.
`directed`: | Logical constant, whether to generate a directed graph.

Returns:

`` | Error code.

Time complexity: O(|V|*log(|V|)+|E|), |V| is the number of vertices, |E| is the number of edges in the graph.

### 2.20. `igraph_barabasi_aging_game` — Preferential attachment with aging of vertices.

    igraph_error_t igraph_barabasi_aging_game(igraph_t *graph,
                                   igraph_integer_t nodes,
                                   igraph_integer_t m,
                                   const igraph_vector_int_t *outseq,
                                   igraph_bool_t outpref,
                                   igraph_real_t pa_exp,
                                   igraph_real_t aging_exp,
                                   igraph_integer_t aging_bins,
                                   igraph_real_t zero_deg_appeal,
                                   igraph_real_t zero_age_appeal,
                                   igraph_real_t deg_coef,
                                   igraph_real_t age_coef,
                                   igraph_bool_t directed);

This game starts with one vertex (if `nodes` > 0). In each step a new node is added, and it is connected to `m` existing nodes. Existing nodes to connect to are chosen with probability dependent on their (in-)degree (`k`) and age (`l`). The degree-dependent part is `deg_coef * k^pa_exp + zero_deg_appeal`, while the age-dependent part is `age_coef * l^aging_exp + zero_age_appeal`, which are multiplied to obtain the final weight.

The age `l` is based on the number of vertices in the network and the `aging_bins` argument: the age of a node is incremented by 1 after each `floor(nodes / aging_bins) + 1` time steps.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`nodes`: | The number of vertices in the graph.
`m`: | The number of edges to add in each time step. Ignored if `outseq` is a non-zero length vector.
`outseq`: | The number of edges to add in each time step. If it is `NULL` or a zero-length vector then it is ignored and the `m` argument is used instead.
`outpref`: | Logical constant, whether the edges initiated by a vertex contribute to the probability to gain a new edge.
`pa_exp`: | The exponent of the preferential attachment, a small positive number usually, the value 1 yields the classic linear preferential attachment.
`aging_exp`: | The exponent of the aging, this is a negative number usually.
`aging_bins`: | Integer constant, the number of age bins to use.
`zero_deg_appeal`: | The degree dependent part of the attractiveness of the zero degree vertices.
`zero_age_appeal`: | The age dependent part of the attractiveness of the vertices of age zero. This parameter is usually zero.
`deg_coef`: | The coefficient for the degree.
`age_coef`: | The coefficient for the age.
`directed`: | Logical constant, whether to generate a directed graph.

Returns:

`` | Error code.

Time complexity: O((|V|+|V|/aging_bins)*log(|V|)+|E|). |V| is the number of vertices, |E| the number of edges.

### 2.21. `igraph_recent_degree_aging_game` — Preferential attachment based on the number of edges gained recently, with aging of vertices.

    igraph_error_t igraph_recent_degree_aging_game(igraph_t *graph,
                                        igraph_integer_t nodes,
                                        igraph_integer_t m,
                                        const igraph_vector_int_t *outseq,
                                        igraph_bool_t outpref,
                                        igraph_real_t pa_exp,
                                        igraph_real_t aging_exp,
                                        igraph_integer_t aging_bins,
                                        igraph_integer_t time_window,
                                        igraph_real_t zero_appeal,
                                        igraph_bool_t directed);

This game is very similar to `igraph_barabasi_aging_game()`, except that instead of the total number of incident edges the number of edges gained in the last `time_window` time steps are counted.

The degree dependent part of the attractiveness is given by k to the power of `pa_exp` plus `zero_appeal`; the age dependent part is l to the power to `aging_exp`. k is the number of edges gained in the last `time_window` time steps, l is the age of the vertex.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`nodes`: | The number of vertices in the graph.
`m`: | The number of edges to add in each time step. If the `outseq` argument is not a null vector or a zero-length vector then it is ignored.
`outseq`: | Vector giving the number of edges to add in each time step. If it is a null pointer or a zero-length vector then it is ignored and the `m` argument is used.
`outpref`: | Logical constant, if true the edges initiated by a vertex are also counted. Normally it is false.
`pa_exp`: | The exponent for the preferential attachment.
`aging_exp`: | The exponent for the aging, normally it is negative: old vertices gain edges with less probability.
`aging_bins`: | Integer constant, the number of age bins to use.
`time_window`: | The time window to use to count the number of incident edges for the vertices.
`zero_appeal`: | The degree dependent part of the attractiveness for zero degree vertices.
`directed`: | Logical constant, whether to create a directed graph.

Returns:

`` | Error code.

Time complexity: O((|V|+|V|/aging_bins)*log(|V|)+|E|). |V| is the number of vertices, |E| the number of edges.

### 2.22. `igraph_lastcit_game` — Simulates a citation network, based on time passed since the last citation.

    igraph_error_t igraph_lastcit_game(igraph_t *graph,
                            igraph_integer_t nodes, igraph_integer_t edges_per_node,
                            igraph_integer_t agebins,
                            const igraph_vector_t *preference,
                            igraph_bool_t directed);

This is a quite special stochastic graph generator, it models an evolving graph. In each time step a single vertex is added to the network and it cites a number of other vertices (as specified by the `edges_per_step` argument). The cited vertices are selected based on the last time they were cited. Time is measured by the addition of vertices and it is binned into `agebins` bins. So if the current time step is `t` and the last citation to a given `i` vertex was made in time step `t0`, then \c (t-t0)/binwidth is calculated where binwidth is `nodes`/agebins+1, in the last expression '/' denotes integer division, so the fraction part is omitted.

The `preference` argument specifies the preferences for the citation lags, i.e. its first elements contains the attractivity of the very recently cited vertices, etc. The last element is special, it contains the attractivity of the vertices which were never cited. This element should be bigger than zero.

Note that this function generates networks with multiple edges if `edges_per_step` is bigger than one, call `igraph_simplify()` on the result to get rid of these edges.

Arguments:

`graph`: | Pointer to an uninitialized graph object, the result will be stored here.
`nodes`: | The number of vertices in the network.
`edges_per_node`: | The number of edges to add in each time step.
`agebins`: | The number of age bins to use.
`preference`: | Pointer to an initialized vector of length `agebins`+1. This contains the "attractivity" of the various age bins, the last element is the attractivity of the vertices which were never cited, and it should be greater than zero. It is a good idea to have all positive values in this vector. Preferences cannot be negative.
`directed`: | Logical constant, whether to create directed networks.

Returns:

`` | Error code.

See also:

`` | `igraph_barabasi_aging_game()`.

Time complexity: O(|V|*a+|E|*log|V|), |V| is the number of vertices, |E| is the total number of edges, a is the `agebins` parameter.

### 2.23. `igraph_cited_type_game` — Simulates a citation based on vertex types.

    igraph_error_t igraph_cited_type_game(igraph_t *graph, igraph_integer_t nodes,
                               const igraph_vector_int_t *types,
                               const igraph_vector_t *pref,
                               igraph_integer_t edges_per_step,
                               igraph_bool_t directed);

Function to create a network based on some vertex categories. This function creates a citation network: in each step a single vertex and `edges_per_step` citing edges are added. Nodes with different categories may have different probabilities to get cited, as given by the `pref` vector.

Note that this function might generate networks with multiple edges if `edges_per_step` is greater than one. You might want to call `igraph_simplify()` on the result to remove multiple edges.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`nodes`: | The number of vertices in the network.
`types`: | Numeric vector giving the categories of the vertices, so it should contain `nodes` non-negative integer numbers. Types are numbered from zero.
`pref`: | The attractivity of the different vertex categories in a vector. Its length should be the maximum element in `types` plus one (types are numbered from zero).
`edges_per_step`: | Integer constant, the number of edges to add in each time step.
`directed`: | Logical constant, whether to create a directed network.

Returns:

`` | Error code.

See also:

`` | `igraph_citing_cited_type_game()` for a bit more general game.

Time complexity: O((|V|+|E|)log|V|), |V| and |E| are number of vertices and edges, respectively.

### 2.24. `igraph_citing_cited_type_game` — Simulates a citation network based on vertex types.

    igraph_error_t igraph_citing_cited_type_game(igraph_t *graph, igraph_integer_t nodes,
                                      const igraph_vector_int_t *types,
                                      const igraph_matrix_t *pref,
                                      igraph_integer_t edges_per_step,
                                      igraph_bool_t directed);

This game is similar to `igraph_cited_type_game()` but here the category of the citing vertex is also considered.

An evolving citation network is modeled here, a single vertex and its `edges_per_step` citation are added in each time step. The odds the a given vertex is cited by the new vertex depends on the category of both the citing and the cited vertex and is given in the `pref` matrix. The categories of the citing vertex correspond to the rows, the categories of the cited vertex to the columns of this matrix. I.e. the element in row `i` and column `j` gives the probability that a `j` vertex is cited, if the category of the citing vertex is `i`.

Note that this function might generate networks with multiple edges if `edges_per_step` is greater than one. You might want to call `igraph_simplify()` on the result to remove multiple edges.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`nodes`: | The number of vertices in the network.
`types`: | A numeric vector of length `nodes`, containing the categories of the vertices. The categories are numbered from zero.
`pref`: | The preference matrix, a square matrix is required, both the number of rows and columns should be the maximum element in `types` plus one (types are numbered from zero).
`edges_per_step`: | Integer constant, the number of edges to add in each time step.
`directed`: | Logical constant, whether to create a directed network.

Returns:

`` | Error code.

Time complexity: O((|V|+|E|)log|V|), |V| and |E| are number of vertices and edges, respectively.

### 2.25. `igraph_sbm_game` — Sample from a stochastic block model.

    igraph_error_t igraph_sbm_game(igraph_t *graph, igraph_integer_t n,
                        const igraph_matrix_t *pref_matrix,
                        const igraph_vector_int_t *block_sizes,
                        igraph_bool_t directed, igraph_bool_t loops);

This function samples graphs from a stochastic block model by (doing the equivalent of) Bernoulli trials for each potential edge with the probabilities given by the Bernoulli rate matrix, `pref_matrix`. See Faust, K., & Wasserman, S. (1992a). Blockmodels: Interpretation and evaluation. Social Networks, 14, 5-–61.

The order of the vertex IDs in the generated graph corresponds to the `block_sizes` argument.

Arguments:

`graph`: | The output graph. This should be a pointer to an uninitialized graph.
`n`: | Number of vertices.
`pref_matrix`: | The matrix giving the Bernoulli rates. This is a KxK matrix, where K is the number of groups. The probability of creating an edge between vertices from groups i and j is given by element (i,j).
`block_sizes`: | An integer vector giving the number of vertices in each group.
`directed`: | Boolean, whether to create a directed graph. If this argument is false, then `pref_matrix` must be symmetric.
`loops`: | Boolean, whether to create self-loops.

Returns:

`` | Error code.

Time complexity: O(|V|+|E|+K^2), where |V| is the number of vertices, |E| is the number of edges, and K is the number of groups.

See also:

`` | `igraph_erdos_renyi_game_gnp()` for a simple Bernoulli graph.

### 2.26. `igraph_hsbm_game` — Hierarchical stochastic block model.

    igraph_error_t igraph_hsbm_game(igraph_t *graph, igraph_integer_t n,
                         igraph_integer_t m, const igraph_vector_t *rho,
                         const igraph_matrix_t *C, igraph_real_t p);

The function generates a random graph according to the hierarchical stochastic block model.

Arguments:

`graph`: | The generated graph is stored here.
`n`: | The number of vertices in the graph.
`m`: | The number of vertices per block. n/m must be integer.
`rho`: | The fraction of vertices per cluster, within a block. Must sum up to 1, and rho * m must be integer for all elements of rho.
`C`: | A square, symmetric numeric matrix, the Bernoulli rates for the clusters within a block. Its size must mach the size of the `rho` vector.
`p`: | The Bernoulli rate of connections between vertices in different blocks.

Returns:

`` | Error code.

See also:

`` | `igraph_sbm_game()` for the classic stochastic block model, `igraph_hsbm_list_game()` for a more general version.

### 2.27. `igraph_hsbm_list_game` — Hierarchical stochastic block model, more general version.

    igraph_error_t igraph_hsbm_list_game(igraph_t *graph, igraph_integer_t n,
                              const igraph_vector_int_t *mlist,
                              const igraph_vector_list_t *rholist,
                              const igraph_matrix_list_t *Clist,
                              igraph_real_t p);

The function generates a random graph according to the hierarchical stochastic block model.

Arguments:

`graph`: | The generated graph is stored here.
`n`: | The number of vertices in the graph.
`mlist`: | An integer vector of block sizes.
`rholist`: | A list of rho vectors (`igraph_vector_t` objects), one for each block.
`Clist`: | A list of square matrices (`igraph_matrix_t` objects), one for each block, specifying the Bernoulli rates of connections within the block.
`p`: | The Bernoulli rate of connections between vertices in different blocks.

Returns:

`` | Error code.

See also:

`` | `igraph_sbm_game()` for the classic stochastic block model, `igraph_hsbm_game()` for a simpler general version.

### 2.28. `igraph_dot_product_game` — Generates a random dot product graph.

    igraph_error_t igraph_dot_product_game(igraph_t *graph, const igraph_matrix_t *vecs,
                                igraph_bool_t directed);

In this model, each vertex is represented by a latent position vector. Probability of an edge between two vertices are given by the dot product of their latent position vectors.

See also Christine Leigh Myers Nickel: Random dot product graphs, a model for social networks. Dissertation, Johns Hopkins University, Maryland, USA, 2006.

Arguments:

`graph`: | The output graph is stored here.
`vecs`: | A matrix in which each latent position vector is a column. The dot product of the latent position vectors should be in the [0,1] interval, otherwise a warning is given. For negative dot products, no edges are added; dot products that are larger than one always add an edge.
`directed`: | Should the generated graph be directed?

Returns:

`` | Error code.

Time complexity: O(n*n*m), where n is the number of vertices, and m is the length of the latent vectors.

See also:

`` | `igraph_sample_dirichlet()`, `igraph_sample_sphere_volume()`, `igraph_sample_sphere_surface()` for functions to generate the latent vectors.

### 2.29. `igraph_tree_game` — Generates a random tree with the given number of nodes.

    igraph_error_t igraph_tree_game(igraph_t *graph, igraph_integer_t n, igraph_bool_t directed, igraph_random_tree_t method);

This function samples uniformly from the set of labelled trees, i.e. it generates each labelled tree with the same probability.

Note that for `n=0`, the null graph is returned, which is not considered to be a tree by `igraph_is_tree()`.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`n`: | The number of nodes in the tree.
`directed`: | Whether to create a directed tree. The edges are oriented away from the root.
`method`: | The algorithm to use to generate the tree. Possible values:
`IGRAPH_RANDOM_TREE_PRUFER` | This algorithm samples Prüfer sequences uniformly, then converts them to trees. Directed trees are not currently supported.
`IGRAPH_RANDOM_LERW` | This algorithm effectively performs a loop-erased random walk on the complete graph to uniformly sample its spanning trees (Wilson's algorithm).

Returns:

`` | Error code: `IGRAPH_ENOMEM`: there is not enough memory to perform the operation. `IGRAPH_EINVAL`: invalid tree size

See also:

`` | `igraph_from_prufer()`

### 2.30. `igraph_correlated_game` — Generates a random graph correlated to an existing graph.

    igraph_error_t igraph_correlated_game(const igraph_t *old_graph, igraph_t *new_graph,
                               igraph_real_t corr, igraph_real_t p,
                               const igraph_vector_int_t *permutation);

Sample a new graph by perturbing the adjacency matrix of a given simple graph and shuffling its vertices.

Arguments:

`old_graph`: | The original graph, it must be simple.
`new_graph`: | The new graph will be stored here.
`corr`: | A scalar in the unit interval [0,1], the target Pearson correlation between the adjacency matrices of the original and the generated graph (the adjacency matrix being used as a vector).
`p`: | A numeric scalar, the probability of an edge between two vertices, it must in the open (0,1) interval. Typically, the density of `old_graph`.
`permutation`: | A permutation to apply to the vertices of the generated graph. It can also be a null pointer, in which case the vertices will not be permuted.

Returns:

`` | Error code

See also:

`` | `igraph_correlated_pair_game()` for generating a pair of correlated random graphs in one go.

### 2.31. `igraph_correlated_pair_game` — Generates pairs of correlated random graphs.

    igraph_error_t igraph_correlated_pair_game(igraph_t *graph1, igraph_t *graph2,
                                    igraph_integer_t n, igraph_real_t corr, igraph_real_t p,
                                    igraph_bool_t directed,
                                    const igraph_vector_int_t *permutation);

Sample two random graphs, with given correlation.

Arguments:

`graph1`: | The first graph will be stored here.
`graph2`: | The second graph will be stored here.
`n`: | The number of vertices in both graphs.
`corr`: | A scalar in the unit interval, the target Pearson correlation between the adjacency matrices of the original the generated graph (the adjacency matrix being used as a vector).
`p`: | A numeric scalar, the probability of an edge between two vertices, it must in the open (0,1) interval.
`directed`: | Whether to generate directed graphs.
`permutation`: | A permutation to apply to the vertices of the second graph. It can also be a null pointer, in which case the vertices will not be permuted.

Returns:

`` | Error code

See also:

`` | `igraph_correlated_game()` for generating a correlated pair to a given graph.

### 2.32. `igraph_simple_interconnected_islands_game` — Generates a random graph made of several interconnected islands, each island being a random graph.

    igraph_error_t igraph_simple_interconnected_islands_game(
            igraph_t *graph,
            igraph_integer_t islands_n,
            igraph_integer_t islands_size,
            igraph_real_t islands_pin,
            igraph_integer_t n_inter);

All islands are of the same size. Within an island, each edge is generated with the same probability. A fixed number of additional edges are then generated for each unordered pair of islands to connect them. The generated graph is guaranteed to be simple.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`islands_n`: | The number of islands in the graph.
`islands_size`: | The size of islands in the graph.
`islands_pin`: | The probability to create each possible edge within islands.
`n_inter`: | The number of edges to create between two islands. It may be larger than `islands_size` squared, but in this case it is assumed to be `islands_size` squared.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid parameter `IGRAPH_ENOMEM`: there is not enough memory for the operation.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.

## 3. Deprecated functions

### 3.1. `igraph_erdos_renyi_game` — Generates a random (Erdős-Rényi) graph.

    igraph_error_t igraph_erdos_renyi_game(igraph_t *graph, igraph_erdos_renyi_t type,
                                igraph_integer_t n, igraph_real_t p_or_m,
                                igraph_bool_t directed, igraph_bool_t loops);

This function is deprecated; use `igraph_erdos_renyi_game_gnm()` or `igraph_erdos_renyi_game_gnp()` instead.

Arguments:

`graph`: | Pointer to an uninitialized graph object.
`type`: | The type of the random graph, possible values:
`IGRAPH_ERDOS_RENYI_GNM` | G(n,m) graph, m edges are selected uniformly randomly in a graph with n vertices.
`IGRAPH_ERDOS_RENYI_GNP` | G(n,p) graph, every possible edge is included in the graph with probability p.
`n`: | The number of vertices in the graph.
`p_or_m`: | This is the p parameter for G(n,p) graphs and the m parameter for G(n,m) graphs.
`directed`: | Logical, whether to generate a directed graph.
`loops`: | Logical, whether to generate loops (self) edges.

Returns:

`` | Error code: `IGRAPH_EINVAL`: invalid `type`, `n`, `p` or `m` parameter. `IGRAPH_ENOMEM`: there is not enough memory for the operation.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.

See also:

`` | `igraph_barabasi_game()`, `igraph_growing_random_game()`, `igraph_erdos_renyi_game_gnm()`, `igraph_erdos_renyi_game_gnp()`

### 3.2. `igraph_lattice` — Arbitrary dimensional square lattices (deprecated).

    igraph_error_t igraph_lattice(igraph_t *graph, const igraph_vector_int_t *dimvector,
                       igraph_integer_t nei, igraph_bool_t directed, igraph_bool_t mutual,
                       igraph_bool_t circular);

### Warning

Deprecated since version 0.10.0. Please do not use this function in new code; use `igraph_square_lattice()` instead.

### 3.3. `igraph_tree` — Creates a k-ary tree in which almost all vertices have k children (deprecated alias).

    igraph_error_t igraph_tree(igraph_t *graph, igraph_integer_t n, igraph_integer_t children,
                    igraph_tree_mode_t type);

### Warning

Deprecated since version 0.10.0. Please do not use this function in new code; use `igraph_kary_tree()` instead.