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

In the G(n, m) Erdős-Rényi model, a graph with n vertices and m edges is generated uniformly at random.

Arguments: 

Returns: 

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

See also: 

#include <igraph.h>

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

    /* Initialize the library. */
    igraph_setup();

    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, IGRAPH_NO_MULTIPLE
    );

    /* 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;
}

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

In the G(n, p) Erdős-Rényi model, also known as the Gilbert model, or Bernoulli random graph, a graph with n vertices is generated such that every possible edge is included in the graph independently with probability p. This is equivalent to a maximum entropy random graph model model with a constraint on the expected edge count. Setting p = 1/2 generates all graphs on n vertices with the same probability.

The expected mean degree of the graph is approximately p n; set p = k/n when a mean degree of approximately k is desired. More precisely, the expected mean degree is p(n-1) in (undirected or directed) graphs without self-loops, p(n+1) in undirected graphs with self-loops, and p n in directed graphs with self-loops.

Arguments: 

Returns: 

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

See also: 

#include <igraph.h>

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

    /* Initialize the library. */
    igraph_setup();

    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;
}

igraph_error_t igraph_iea_game(
        igraph_t *graph,
        igraph_int_t n, igraph_int_t m,
        igraph_bool_t directed, igraph_bool_t loops);

This model generates random multigraphs on n vertices with m edges through independent edge assignment (IEA). Each of the m edges is assigned uniformly at random to an ordered vertex pair, independently of each other.

This model does not sample multigraphs uniformly. 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.

An alternative way to think of this model is that it performs uniform sampling of edge-labeled graphs, i.e. graphs in which not only vertices, but also edges carry unique identities and are distinguishable.

Arguments: 

Returns: 

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

See also: 

igraph_error_t igraph_sbm_game(
        igraph_t *graph,
        const igraph_matrix_t *pref_matrix,
        const igraph_vector_int_t *block_sizes,
        igraph_bool_t directed,
        igraph_edge_type_sw_t allowed_edge_types);

This function samples graphs from a stochastic block model, a generalization of the G(n,p) model where the connection probability p (or expected number of edges for multigraphs) is specified separately between and within a given group of vertices.

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

Reference:

Faust, K., & Wasserman, S. (1992a). Blockmodels: Interpretation and evaluation. Social Networks, 14, 5-–61. https://doi.org/10.1016/0378-8733(92)90013-W

Arguments: 

Returns: 

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_error_t igraph_hsbm_game(igraph_t *graph, igraph_int_t n,
                     igraph_int_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: 

Returns: 

See also: 

igraph_error_t igraph_hsbm_list_game(igraph_t *graph, igraph_int_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: 

Returns: 

See also: 

igraph_error_t igraph_preference_game(igraph_t *graph, igraph_int_t nodes,
                           igraph_int_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: 

Returns: 

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_error_t igraph_asymmetric_preference_game(igraph_t *graph, igraph_int_t nodes,
                                      igraph_int_t no_out_types,
                                      igraph_int_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: 

Returns: 

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_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: 

Returns: 

See also: 

igraph_error_t igraph_correlated_pair_game(igraph_t *graph1, igraph_t *graph2,
                                igraph_int_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: 

Returns: 

See also: 

igraph_error_t igraph_barabasi_game(igraph_t *graph, igraph_int_t n,
                         igraph_real_t power,
                         igraph_int_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: 

Returns: 

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

#include <igraph.h>

int main(void) {

    igraph_t g;
    igraph_vector_int_t v;
    igraph_vector_int_t v2, v3;

    /* Initialize the library. */
    igraph_setup();

    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_int_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, IGRAPH_LOOPS);
    for (igraph_int_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_int_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;
}

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

int main(void) {
    igraph_t g;
    igraph_bool_t simple;

    /* Initialize the library. */
    igraph_setup();

    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, IGRAPH_DIRECTED);
    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, IGRAPH_DIRECTED);
    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, IGRAPH_DIRECTED);
    if (simple) {
        return 9;
    }

    igraph_destroy(&g);

    return 0;
}

igraph_error_t igraph_barabasi_aging_game(igraph_t *graph,
                               igraph_int_t nodes,
                               igraph_int_t m,
                               const igraph_vector_int_t *outseq,
                               igraph_bool_t outpref,
                               igraph_real_t pa_exp,
                               igraph_real_t aging_exp,
                               igraph_int_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: 

Returns: 

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

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

Arguments: 

Returns: 

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

igraph_error_t igraph_recent_degree_aging_game(igraph_t *graph,
                                    igraph_int_t nodes,
                                    igraph_int_t m,
                                    const igraph_vector_int_t *outseq,
                                    igraph_bool_t outpref,
                                    igraph_real_t pa_exp,
                                    igraph_real_t aging_exp,
                                    igraph_int_t aging_bins,
                                    igraph_int_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: 

Returns: 

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

igraph_error_t igraph_lastcit_game(igraph_t *graph,
                        igraph_int_t nodes, igraph_int_t edges_per_node,
                        igraph_int_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 (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: 

Returns: 

See also: 

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.

igraph_error_t igraph_growing_random_game(igraph_t *graph, igraph_int_t n,
                               igraph_int_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: 

Returns: 

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

igraph_error_t igraph_callaway_traits_game(igraph_t *graph, igraph_int_t nodes,
                                igraph_int_t types, igraph_int_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: 

Returns: 

Added in version 0.2.

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

igraph_error_t igraph_establishment_game(igraph_t *graph, igraph_int_t nodes,
                              igraph_int_t types, igraph_int_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: 

Returns: 

Added in version 0.2.

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

igraph_error_t igraph_cited_type_game(igraph_t *graph, igraph_int_t nodes,
                           const igraph_vector_int_t *types,
                           const igraph_vector_t *pref,
                           igraph_int_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: 

Returns: 

See also: 

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

igraph_error_t igraph_citing_cited_type_game(igraph_t *graph, igraph_int_t nodes,
                                  const igraph_vector_int_t *types,
                                  const igraph_matrix_t *pref,
                                  igraph_int_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: 

Returns: 

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

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

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

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:

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 https://www.cs.cmu.edu/~jure/pubs/powergrowth-tkdd.pdf, our implementation is based on this.

Arguments: 

Returns: 

Time complexity: TODO.

igraph_error_t igraph_degree_sequence_game(
        igraph_t *graph,
        const igraph_vector_int_t *out_degrees,
        const igraph_vector_int_t *in_degrees,
        igraph_degseq_t method);

This function generates random graphs with a prescribed degree sequence. Several sampling methods are available, which respect different constraints (simple graph or multigraphs, connected graphs, etc.), and provide different tradeoffs between performance and unbiased sampling. See Section 2.1 of Horvát and Modes (2021) for an overview of sampling techniques for graphs with fixed degrees.

References:

Fabien Viger, and Matthieu Latapy: Efficient and Simple Generation of Random Simple Connected Graphs with Prescribed Degree Sequence, Journal of Complex Networks 4, no. 1, pp. 15–37 (2015). https://doi.org/10.1093/comnet/cnv013.

Szabolcs Horvát, and Carl D Modes: Connectedness Matters: Construction and Exact Random Sampling of Connected Networks, Journal of Physics: Complexity 2, no. 1, pp. 015008 (2021). https://doi.org/10.1088/2632-072x/abced5.

Arguments: 

Returns: 

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

See also: 

#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;

    /* Initialize the library. */
    igraph_setup();

    /* 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, IGRAPH_LOOPS)) {
        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, IGRAPH_DIRECTED) || !is_simple) {
        return 4;
    }
    if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_OUT, IGRAPH_NO_LOOPS)) {
        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, IGRAPH_LOOPS)) {
        return 7;
    }
    igraph_vector_int_print(&vec);
    if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_IN, IGRAPH_LOOPS)) {
        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, IGRAPH_DIRECTED) || !is_simple) {
        return 10;
    }
    if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_OUT, IGRAPH_LOOPS)) {
        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, IGRAPH_DIRECTED) || !is_simple) {
        return 13;
    }
    if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_OUT, IGRAPH_LOOPS)) {
        return 14;
    }
    igraph_vector_int_print(&vec);
    if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_IN, IGRAPH_LOOPS)) {
        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;
}

igraph_error_t igraph_k_regular_game(igraph_t *graph,
                          igraph_int_t no_of_nodes, igraph_int_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: 

Returns: 

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

igraph_error_t igraph_rewire(igraph_t *graph, igraph_int_t n, igraph_edge_type_sw_t allowed_edge_types);

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: 

Returns: 

Time complexity: TODO.

igraph_error_t igraph_chung_lu_game(igraph_t *graph,
                                    const igraph_vector_t *out_weights,
                                    const igraph_vector_t *in_weights,
                                    igraph_bool_t loops,
                                    igraph_chung_lu_t variant);

The Chung-Lu model is useful for generating random graphs with fixed expected degrees. This function implements both the original model of Chung and Lu, as well as some additional variants with useful properties.

In the original Chung-Lu model, each pair of vertices i and j is connected with independent probability p_ij = w_i w_j / S, where w_i is a weight associated with vertex i and S = sum_k w_k is the sum of weights. In the directed variant, vertices have both out-weights, w^out, and in-weights, w^in, with equal sums, S = sum_k w^out_k = sum_k w^in_k. The connection probability between i and j is p_ij = w^out_i w^in_j / S.

This model is commonly used to create random graphs with a fixed expected degree sequence. The expected degree of vertex i is approximately equal to the weight w_i. Specifically, if the graph is directed and self-loops are allowed, then the expected out- and in-degrees are precisely w^out and w^in. If self-loops are disallowed, then the expected out- and in-degrees are w^out (S - w^in) / S and w^in (S - w^out) / S, respectively. If the graph is undirected, then the expected degrees with and without self-loops are w (S + w) / S and w (S - w) / S, respectively.

A limitation of the original Chung-Lu model is that when some of the weights are large, the formula for p_ij yields values larger than 1. Chung and Lu's original paper excludes the use of such weights. When p_ij > 1, this function simply issues a warning and creates a connection between i and j. However, in this case the expected degrees will no longer relate to the weights in the manner stated above. Thus the original Chung-Lu model cannot produce certain (large) expected degrees.

The overcome this limitation, this function implements additional variants of the model, with modified expressions for the connection probability p_ij between vertices i and j. Let q_ij = w_i w_j / S, or q_ij = w^out_i w^in_j / S in the directed case. All model variants become equivalent in the limit of sparse graphs where q_ij approaches zero. In the original Chung-Lu model, selectable by setting variant to IGRAPH_CHUNG_LU_ORIGINAL, p_ij = min(q_ij, 1). The IGRAPH_CHUNG_LU_MAXENT variant, sometiems referred to a the generalized random graph, uses p_ij = q_ij / (1 + q_ij), and is equivalent to a maximum entropy model (i.e. exponential random graph model) with a constraint on expected degrees; see Park and Newman (2004), Section B, setting exp(-Theta_ij) = w_i w_j / S. This model is also discussed by Britton, Deijfen and Martin-Löf (2006). By virtue of being a degree-constrained maximum entropy model, it produces graphs with the same degree sequence with the same probability. A third variant can be requested with IGRAPH_CHUNG_LU_NR, and uses p_ij = 1 - exp(-q_ij). This is the underlying simple graph of a multigraph model introduced by Norros and Reittu (2006). For a discussion of these three model variants, see Section 16.4 of Bollobás, Janson, Riordan (2007), as well as Van Der Hofstad (2013).

References:

Chung F and Lu L: Connected components in a random graph with given degree sequences. Annals of Combinatorics 6, 125-145 (2002). https://doi.org/10.1007/PL00012580

Miller JC and Hagberg A: Efficient Generation of Networks with Given Expected Degrees (2011). https://doi.org/10.1007/978-3-642-21286-4_10

Park J and Newman MEJ: Statistical mechanics of networks. Physical Review E 70, 066117 (2004). https://doi.org/10.1103/PhysRevE.70.066117

Britton T, Deijfen M, Martin-Löf A: Generating Simple Random Graphs with Prescribed Degree Distribution. J Stat Phys 124, 1377–1397 (2006). https://doi.org/10.1007/s10955-006-9168-x

Norros I and Reittu H: On a conditionally Poissonian graph process. Advances in Applied Probability 38, 59–75 (2006). https://doi.org/10.1239/aap/1143936140

Bollobás B, Janson S, Riordan O: The phase transition in inhomogeneous random graphs. Random Struct Algorithms 31, 3–122 (2007). https://doi.org/10.1002/rsa.20168

Van Der Hofstad R: Critical behavior in inhomogeneous random graphs. Random Struct Algorithms 42, 480–508 (2013). https://doi.org/10.1002/rsa.20450

Arguments: 

Returns: 

See also: 

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

igraph_error_t igraph_static_fitness_game(igraph_t *graph, igraph_int_t no_of_edges,
                               const igraph_vector_t *fitness_out, const igraph_vector_t *fitness_in,
                               igraph_edge_type_sw_t allowed_edge_types);

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.

The expected degree (though not the actual degree) of each vertex will be proportional to its fitness. This is exactly true when self-loops and multi-edges are allowed, and approximately true otherwise. If you need to generate a graph with an exact degree sequence, consider igraph_degree_sequence_game() and igraph_realize_degree_sequence() instead.

To generate random undirected graphs with a given expected degree sequence, set fitness_out (and in the directed case fitness_out) to the desired expected degrees, and no_of_edges to the corresponding edge count, i.e. half the sum of expected degrees in the undirected case, and the sum of out- or in-degrees in the directed case.

This model is similar to the better-known Chung-Lu model, implemented in igraph as igraph_chung_lu_game(), but with a sharply fixed edge count.

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: 

Returns: 

See also: 

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

igraph_error_t igraph_static_power_law_game(igraph_t *graph,
                                 igraph_int_t no_of_nodes, igraph_int_t no_of_edges,
                                 igraph_real_t exponent_out, igraph_real_t exponent_in,
                                 igraph_edge_type_sw_t allowed_edge_types,
                                 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. https://doi.org/10.1103/PhysRevLett.87.278701

Chung F and Lu L: Connected components in a random graph with given degree sequences. Annals of Combinatorics 6, 125-145, 2002. https://doi.org/10.1007/PL00012580

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. https://doi.org/10.1103/PhysRevLett.103.135702

Arguments: 

Returns: 

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

igraph_error_t igraph_watts_strogatz_game(
        igraph_t *graph, igraph_int_t dim,
        igraph_int_t size, igraph_int_t nei,
        igraph_real_t p,
        igraph_edge_type_sw_t allowed_edge_types);

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. https://doi.org/10.1038/30918

Arguments: 

Returns: 

See also: 

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.

igraph_error_t igraph_rewire_edges(igraph_t *graph, igraph_real_t prob,
                                   igraph_edge_type_sw_t allowed_edge_types);

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: 

Returns: 

See also: 

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

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: 

Returns: 

See also: 

Time complexity: O(|E|).

igraph_error_t igraph_grg_game(igraph_t *graph, igraph_int_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: 

Returns: 

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

#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;

    /* Initialize the library. */
    igraph_setup();

    /* 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 */ false, &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_int_t e = IGRAPH_EIT_GET(eit);
        igraph_int_t u = IGRAPH_FROM(&graph, e);
        igraph_int_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(&graph, &weights, &avg_dist, NULL, IGRAPH_UNDIRECTED, /* unconn */ true);

    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;
}

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: 

Returns: 

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_error_t igraph_simple_interconnected_islands_game(
        igraph_t *graph,
        igraph_int_t islands_n,
        igraph_int_t islands_size,
        igraph_real_t islands_pin,
        igraph_int_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: 

Returns: 

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

igraph_error_t igraph_tree_game(igraph_t *graph, igraph_int_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: 

Returns: 

See also: 

typedef unsigned int igraph_edge_type_sw_t;

This type is used with multiple functions to specify what types of non-simple edges to allow, create or consider a graph. The constants below are treated as "switches" that can be turned on individually and combined using the bitwise-or operator. For example, IGRAPH_LOOPS_SW allows only self-loops but not multi-edges, while IGRAPH_LOOPS_SW | IGRAPH_MULTI_SW allows both.

Values: