For using the igraph C library
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.
igraph_create — Creates a graph with the specified edges.igraph_small — Shorthand to create a small graph, giving the edges as arguments.igraph_adjacency — Creates a graph from an adjacency matrix.igraph_weighted_adjacency — Creates a graph from a weighted adjacency matrix.igraph_adjlist — Creates a graph from an adjacency list.igraph_star — Creates a star graph, every vertex connects only to the center.igraph_lattice — Arbitrary dimensional square lattices.igraph_ring — Creates a ring graph, a one dimensional lattice.igraph_tree — Creates a tree in which almost all vertices have the same number of children.igraph_full — Creates a full graph (directed or undirected, with or without loops).igraph_full_citation — Creates a full citation graphigraph_realize_degree_sequence — Generates a graph with the given degree sequence.igraph_famous — Create a famous graph by simply providing its nameigraph_lcf — Creates a graph from LCF notation.igraph_lcf_vector — Creates a graph from LCF notation.igraph_from_prufer — Generates a tree from a Prüfer sequence.igraph_atlas — Create a small graph from the “Graph Atlas”.igraph_de_bruijn — Generate a de Bruijn graph.igraph_kautz — Generate a Kautz graph.igraph_extended_chordal_ring — Create an extended chordal ring
int igraph_create(igraph_t *graph, const igraph_vector_t *edges,
igraph_integer_t n, igraph_bool_t directed);
Arguments:
|
An uninitialized graph object. |
|
The edges to add, the first two elements are the first edge, etc. |
|
The number of vertices in the graph, if smaller or equal
to the highest vertex id in the |
|
Boolean, whether to create a directed graph or
not. If yes, then the first edge points from the first
vertex id in |
Returns:
Error code:
|
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
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2006-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard street, Cambridge, MA 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> int main() { igraph_t g; igraph_vector_t v1, v2; int ret; /* simple use */ igraph_vector_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_init(&v2, 0); igraph_get_edgelist(&g, &v2, 0); igraph_vector_sort(&v1); igraph_vector_sort(&v2); if (!igraph_vector_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_sort(&v1); igraph_vector_sort(&v2); if (!igraph_vector_all_e(&v1, &v2)) { return 3; } igraph_destroy(&g); /* error: IGRAPH_EINVEVECTOR */ igraph_set_error_handler(igraph_error_handler_ignore); igraph_vector_resize(&v1, 9); VECTOR(v1)[8] = 0; ret = igraph_create(&g, &v1, 0, 0); if (ret != IGRAPH_EINVEVECTOR) { return 4; } /* error: IGRAPH_EINVVID */ igraph_vector_resize(&v1, 8); VECTOR(v1)[7] = -1; ret = igraph_create(&g, &v1, 10, 1); if (ret != IGRAPH_EINVVID) { return 5; } igraph_vector_destroy(&v1); igraph_vector_destroy(&v2); return 0; }
int igraph_small(igraph_t *graph, igraph_integer_t n, igraph_bool_t directed,
...);
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.
Note that only graphs which have vertices less than the highest value of the 'int' type can be created this way. If you give larger values then the result is undefined.
Arguments:
|
Pointer to an uninitialized graph object. The result will be stored here. |
||||
|
The number of vertices in the graph; a nonnegative integer. |
||||
|
Logical constant; gives whether the graph should be directed. Supported values are:
|
||||
|
The additional arguments giving the edges of the graph. Don't forget to supply an additional '-1' after the last (meaningful) 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
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2006-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard st, Cambridge MA, 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> int main() { 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; }
int igraph_adjacency(igraph_t *graph, igraph_matrix_t *adjmatrix,
igraph_adjacency_t mode);
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:
|
Pointer to an uninitialized graph object. |
||||||||||||||
|
The adjacency matrix. How it is interpreted
depends on the |
||||||||||||||
|
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
|
Returns:
Error code,
|
Time complexity: O(|V||V|), |V| is the number of vertices in the graph.
Example 9.3. File examples/simple/igraph_adjacency.c
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2006-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard street, Cambridge, MA 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> int main() { return 0; }
int igraph_weighted_adjacency(igraph_t *graph, igraph_matrix_t *adjmatrix,
igraph_adjacency_t mode, const char* attr,
igraph_bool_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:
|
Pointer to an uninitialized graph object. |
||||||||||||||
|
The weighted adjacency matrix. How it is interpreted
depends on the |
||||||||||||||
|
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
|
||||||||||||||
|
the name of the attribute that will store the edge weights.
If |
||||||||||||||
|
Logical scalar, whether to ignore the diagonal elements in the adjacency matrix. |
Returns:
Error code,
|
Time complexity: O(|V||V|), |V| is the number of vertices in the graph.
Example 9.4. File examples/simple/igraph_weighted_adjacency.c
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2006-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard st, Cambridge MA, 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> #include <stdarg.h> void print(igraph_t *g) { igraph_vector_t el; long int i, j, n; char ch = igraph_is_directed(g) ? '>' : '-'; igraph_vector_init(&el, 0); igraph_get_edgelist(g, &el, 0); n = igraph_ecount(g); for (i = 0, j = 0; i < n; i++, j += 2) { printf("%ld --%c %ld: %ld\n", (long)VECTOR(el)[j], ch, (long)VECTOR(el)[j + 1], (long)EAN(g, "weight", i)); } printf("\n"); igraph_vector_destroy(&el); } int main() { igraph_t g; igraph_matrix_t mat; int m[4][4] = { { 0, 1, 2, 0 }, { 2, 0, 0, 1 }, { 0, 0, 1, 0 }, { 0, 1, 0, 0 } }; long int i, j; 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_set_attribute_table(&igraph_cattribute_table); /* [ 0 1 2 0 ] [ 2 0 0 1 ] [ 0 0 1 0 ] [ 0 1 0 0 ] */ igraph_weighted_adjacency(&g, &mat, IGRAPH_ADJ_DIRECTED, 0, /*loops=*/ 1); print(&g); igraph_destroy(&g); /* [ 0 1 2 0 ] [ - 0 0 1 ] [ - - 1 0 ] [ - - - 0 ] */ igraph_weighted_adjacency(&g, &mat, IGRAPH_ADJ_UPPER, 0, /*loops=*/ 1); print(&g); igraph_destroy(&g); /* [ 0 - - - ] [ 2 0 - - ] [ 0 0 1 - ] [ 0 1 0 0 ] */ igraph_weighted_adjacency(&g, &mat, IGRAPH_ADJ_LOWER, 0, /*loops=*/ 1); print(&g); igraph_destroy(&g); /* [ 0 1 0 0 ] [ 1 0 0 1 ] [ 0 0 1 0 ] [ 0 1 0 0 ] */ igraph_weighted_adjacency(&g, &mat, IGRAPH_ADJ_MIN, 0, /*loops=*/ 1); print(&g); igraph_destroy(&g); /* [ 0 2 2 0 ] [ 2 0 0 1 ] [ 2 0 1 0 ] [ 0 1 0 0 ] */ igraph_weighted_adjacency(&g, &mat, IGRAPH_ADJ_MAX, 0, /*loops=*/ 1); print(&g); igraph_destroy(&g); /* [ 0 3 2 0 ] [ 3 0 0 2 ] [ 2 0 1 0 ] [ 0 2 0 0 ] */ igraph_weighted_adjacency(&g, &mat, IGRAPH_ADJ_PLUS, 0, /*loops=*/ 1); print(&g); igraph_destroy(&g); igraph_matrix_destroy(&mat); if (IGRAPH_FINALLY_STACK_SIZE() != 0) { return 1; } return 0; }
int 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:
|
Pointer to an uninitialized graph object. |
|
The adjacency list. |
|
Whether or not to create a directed graph. |
|
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:
|
Time complexity: O(|V|+|E|).
int igraph_star(igraph_t *graph, igraph_integer_t n, igraph_star_mode_t mode,
igraph_integer_t center);
Arguments:
|
Pointer to an uninitialized graph object, this will be the result. |
||||||||
|
Integer constant, the number of vertices in the graph. |
||||||||
|
Constant, gives the type of the star graph to create. Possible values:
|
||||||||
|
Id of the vertex which will be the center of the graph. |
Returns:
|
Error code:
|
Time complexity: O(|V|), the number of vertices in the graph.
See also:
|
Example 9.5. File examples/simple/igraph_star.c
/* IGraph library. Copyright (C) 2020 The igraph development team <igraph@igraph.org> This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see <https://www.gnu.org/licenses/>. */ #include <igraph.h> #include <stdio.h> int main() { 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; }
int igraph_lattice(igraph_t *graph, const igraph_vector_t *dimvector,
igraph_integer_t nei, igraph_bool_t directed, igraph_bool_t mutual,
igraph_bool_t circular);
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_0, i_1, i_2, ..., i_d)
in a lattice of size (n_0, n_1, ..., n_d) will be
i_0 + n_0 * i_1 + n_0 * n_1 * i_2 + ... .
Arguments:
|
An uninitialized graph object. |
|
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. |
|
Integer value giving the distance (number of steps) within which two vertices will be connected. |
|
Boolean, whether to create a directed graph.
If the |
|
Boolean, if the graph is directed this gives whether to create all connections as mutual. |
|
Boolean, defines whether the generated lattice is periodic. |
Returns:
Error code:
|
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.
int igraph_ring(igraph_t *graph, igraph_integer_t n, igraph_bool_t directed,
igraph_bool_t mutual, igraph_bool_t circular);
An undirected (circular) ring on n vertices is commonly known in graph theory as the cycle graph C_n.
Arguments:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the ring. |
|
Logical, whether to create a directed ring. |
|
Logical, whether to create mutual edges in a directed ring. It is ignored for undirected graphs. |
|
Logical, if false, the ring will be open (this is not a real ring actually). |
Returns:
Error code:
|
Time complexity: O(|V|), the number of vertices in the graph.
See also:
|
Example 9.6. File examples/simple/igraph_ring.c
/* IGraph library. Copyright (C) 2020 The igraph development team <igraph@igraph.org> This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see <https://www.gnu.org/licenses/>. */ #include <igraph.h> #include <stdio.h> int main() { 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; }
int igraph_tree(igraph_t *graph, igraph_integer_t n, igraph_integer_t children,
igraph_tree_mode_t type);
Arguments:
|
Pointer to an uninitialized graph object. |
||||||
|
Integer, the number of vertices in the graph. |
||||||
|
Integer, the number of children of a vertex in the tree. |
||||||
|
Constant, gives whether to create a directed tree, and if this is the case, also its orientation. Possible values:
|
Returns:
Error code:
|
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.
See also:
|
Example 9.7. File examples/simple/igraph_tree.c
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2006-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard st, Cambridge MA, 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> int main() { igraph_t graph; igraph_bool_t res; /* Create a directed binary tree on 15 nodes, with edges pointing towards the root. */ igraph_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; }
int 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. A full graph in igraph should be
distinguished from the concept of complete graphs as used in graph theory.
If n is a positive integer, then the complete graph K_n on n vertices is
the undirected simple graph with the following property. For any distinct
pair (u,v) of vertices in K_n, uv (or equivalently vu) is an edge of K_n.
In igraph, a full graph on n vertices can be K_n, a directed version of
K_n, or K_n with at least one loop edge. In any case, if F is a full graph
on n vertices as generated by igraph, then K_n is a subgraph of the
undirected version of F.
Arguments:
|
Pointer to an uninitialized graph object. |
|
Integer, the number of vertices in the graph. |
|
Logical, whether to create a directed graph. |
|
Logical, whether to include self-edges (loops). |
Returns:
Error code:
|
Time complexity: O(|V|+|E|), |V| is the number of vertices, |E| the number of edges in the graph. Of course this is the same as O(|E|)=O(|V||V|) here.
See also:
|
Example 9.8. File examples/simple/igraph_full.c
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2006-2020 The igraph development team <igraph@igraph.org> This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see <https://www.gnu.org/licenses/>. */ #include <igraph.h> #include <stdio.h> int main() { igraph_t graph; long int 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 %ld vertices has %ld edges.\n", (long int) igraph_vcount(&graph), (long int) 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 %ld vertices has %ld edges.\n", (long int) igraph_vcount(&graph), (long int) 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 %ld vertices with self-loops has %ld edges.\n", (long int) igraph_vcount(&graph), (long int) 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 %ld vertices with self-loops has %ld edges.\n", (long int) igraph_vcount(&graph), (long int) igraph_ecount(&graph)); igraph_destroy(&graph); return 0; }
int 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 zero then an undirected graph is
created, and it is just a full graph.
Arguments:
|
Pointer to an uninitialized graph object, the result is stored here. |
|
The number of vertices. |
|
Whether to created a directed graph. If zero an undirected graph is created. |
Returns:
Error code. |
Time complexity: O(|V|^2), as we have many edges.
int igraph_realize_degree_sequence(
igraph_t *graph,
const igraph_vector_t *outdeg, const igraph_vector_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/2098746
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, Connectivity matters: Construction and exact random sampling of connected graphs (2020). https://arxiv.org/abs/2009.03747
Arguments:
|
Pointer to an uninitialized graph object. |
||||||||
|
The degree sequence of an undirected graph
(if |
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|
The in-degree sequence of a directed graph.
Pass |
||||||||
|
The types of edges to allow in the graph. For directed graphs,
only
|
||||||||
|
The method to generate the graph. Possible values:
|
Returns:
|
Error code:
|
See also:
|
int 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, eg. igraph_full() for creating a full graph.
The following graphs are supported:
|
The bull graph, 5 vertices, 5 edges, resembles the head of a bull if drawn properly. |
|
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. |
|
A non-Hamiltonian cubic symmetric graph with 28 vertices and 42 edges. |
|
The Platonic graph of the cube. A convex regular polyhedron with 8 vertices and 12 edges. |
|
A graph with 4 vertices and 5 edges, resembles a schematic diamond if drawn properly. |
|
Another Platonic solid with 20 vertices and 30 edges. |
|
The semisymmetric graph with minimum number of vertices, 20 and 40 edges. A semisymmetric graph is regular, edge transitive and not vertex transitive. |
|
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. |
|
The Frucht Graph is the smallest cubical graph whose automorphism group consists only of the identity element. It has 12 vertices and 18 edges. |
|
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. |
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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. |
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The Herschel graph is the smallest nonhamiltonian polyhedral graph. It is the unique such graph on 11 nodes, and has 18 edges. |
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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. |
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The same as the house graph with an X in the square. 5 vertices and 8 edges. |
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A Platonic solid with 12 vertices and 30 edges. |
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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. |
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The graph is a 4-arc transitive cubic graph, it has 30 vertices and 45 edges. |
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The McGee graph is the unique 3-regular 7-cage graph, it has 24 vertices and 36 edges. |
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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. |
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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. |
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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. |
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Platonic solid with 6 vertices and 12 edges. |
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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. |
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The unique (4,5)-cage graph, i.e. a 4-regular graph of girth 5. It has 19 vertices and 38 edges. |
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A smallest nontrivial graph whose automorphism group is cyclic. It has 9 vertices and 15 edges. |
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Platonic solid with 4 vertices and 6 edges. |
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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. |
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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. |
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Returns a 12-vertex, triangle-free graph with chromatic number 3 that is uniquely 3-colorable. |
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An identity graph with 25 vertices and 31 edges. An identity graph has a single graph automorphism, the trivial one. |
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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:
|
Pointer to an uninitialized graph object. |
|
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:
|
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.
int 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 http://mathworld.wolfram.com/LCFNotation.html for details.
Arguments:
|
Pointer to an uninitialized graph object. |
|
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 |
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.
Example 9.9. File examples/simple/igraph_lcf.c
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2007-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard st, Cambridge MA, 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> int main() { 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; }
int igraph_lcf_vector(igraph_t *graph, igraph_integer_t n,
const igraph_vector_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:
|
Pointer to an uninitialized graph object. |
|
Integer constant giving the number of vertices. |
|
A vector giving the shifts. |
|
An integer constant giving the number of repeats for the shifts. |
Returns:
Error code. |
See also:
Time complexity: O(|V|+|E|), linear in the number of vertices plus the number of edges.
int 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:
|
Pointer to an uninitialized graph object. |
|
The Prüfer sequence |
Returns:
|
Error code:
|
See also:
int igraph_atlas(igraph_t *graph, int number);
The number of the graph is given as a parameter. The graphs are listed:
in increasing order of number of nodes;
for a fixed number of nodes, in increasing order of the number of edges;
for fixed numbers of nodes and edges, in increasing order of the degree sequence, for example 111223 < 112222;
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:
|
Pointer to an uninitialized graph object. |
|
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 9.10. File examples/simple/igraph_atlas.c
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2006-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard street, Cambridge, MA 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> int main() { igraph_t g; int ret; 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); igraph_set_error_handler(igraph_error_handler_ignore); ret = igraph_atlas(&g, -1); if (ret != IGRAPH_EINVAL) { return 1; } ret = igraph_atlas(&g, 1253); if (ret != IGRAPH_EINVAL) { return 2; } return 0; }
int 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, eg. Wikipedia for details.
Arguments:
|
Pointer to an uninitialized graph object, the result will be stored here. |
|
Integer, the number of letters in the alphabet. |
|
Integer, the length of the strings. |
Returns:
Error code. |
See also:
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.
int 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 eg. Wikipedia for details.
Vincent Matossian wrote the first version of this function in R, thanks.
Arguments:
|
Pointer to an uninitialized graph object, the result will be stored here. |
|
Integer, |
|
Integer, |
Returns:
Error code. |
See also:
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.
int igraph_extended_chordal_ring(
igraph_t *graph, igraph_integer_t nodes, const igraph_matrix_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:
|
Pointer to an uninitialized graph object, the result will be stored here. |
|
Integer constant, the number of vertices in the graph. It must be at least 3. |
|
The matrix specifying the extra edges. The number of columns should divide the number of total vertices. The elements are allowed to be negative. |
|
Whether the graph should be directed. |
Returns:
Error code. |
See also:
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.
igraph_grg_game — Generates a geometric random graph.igraph_barabasi_game — Generates a graph based on the Barabási-Albert model.igraph_erdos_renyi_game — Generates a random (Erdős-Rényi) graph.igraph_watts_strogatz_game — The Watts-Strogatz small-world modeligraph_rewire_edges — Rewire the edges of a graph with constant probabilityigraph_rewire_directed_edges — Rewire the chosen endpoint of directed edgesigraph_degree_sequence_game — Generates a random graph with a given degree sequenceigraph_k_regular_game — Generates a random graph where each vertex has the same degree.igraph_static_fitness_game — Generates a non-growing random graph with edge probabilitiesigraph_static_power_law_game — Generates a non-growing random graph with expected power-law degree distributions.igraph_forest_fire_game — Generates a network according to the “forest fire game”igraph_rewire — Randomly rewires a graph while preserving the degree distribution.igraph_growing_random_game — Generates a growing random graph.igraph_callaway_traits_game — Simulates a growing network with vertex types.igraph_establishment_game — Generates a graph with a simple growing model with vertex types.igraph_preference_game — Generates a graph with vertex types and connection preferencesigraph_asymmetric_preference_game — Generates a graph with asymmetric vertex types and connection preferencesigraph_recent_degree_game — Stochastic graph generator based on the number of incident edges a node has gained recently.igraph_barabasi_aging_game — Preferential attachment with aging of verticesigraph_recent_degree_aging_game — Preferential attachment based on the number of edges gained recently, with aging of vertices.igraph_lastcit_game — Simulates a citation network, based on time passed since the last citation.igraph_cited_type_game — Simulates a citation based on vertex types.igraph_citing_cited_type_game — Simulates a citation network based on vertex types.igraph_sbm_game — Sample from a stochastic block modeligraph_hsbm_game — Hierarchical stochastic block modeligraph_hsbm_list_game — Hierarchical stochastic block model, more general versionigraph_dot_product_game — Generates a random dot product graph.igraph_tree_game — Generates a random tree with the given number of nodesigraph_correlated_game — Generates a random graph correlated to an existing graph.igraph_correlated_pair_game — Generates pairs of correlated random graphs.igraph_simple_interconnected_islands_game — Generates a random graph made of several interconnected islands, each island being a random graph.Games are randomized graph generators. Randomization means that they generate a different graph every time you call them.
int 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 (=vertices)
randomly on the unit square and then connecting all those pairs
which are less than radius apart in Euclidean distance.
Original code contributed by Keith Briggs, thanks Keith.
Arguments:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the graph. |
|
The radius within which the vertices will be connected. |
|
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. |
|
An initialized vector or |
|
An initialized vector or |
Returns:
Error code. |
Time complexity: TODO, less than O(|V|^2+|E|).
Example 9.11. File examples/simple/igraph_grg_game.c
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2006-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard st, Cambridge MA, 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> #include <math.h> int main() { igraph_t g; /* long int i; */ /* struct tms time; */ /* clock_t current_time,start_time; */ /* long int runs=100, n=10000; */ /* igraph_real_t r=0.01; */ /* Empty graph */ igraph_grg_game(&g, 100, 0, 0, 0, 0); if (igraph_ecount(&g) != 0) { return 1; } igraph_destroy(&g); /* Full graph */ igraph_grg_game(&g, 10, sqrt(2.0) / 2, 1, 0, 0); if (igraph_ecount(&g) != igraph_vcount(&g) * (igraph_vcount(&g) - 1) / 2) { return 2; } igraph_destroy(&g); /* Measure running time */ /* tps=sysconf(_SC_CLK_TCK); // clock ticks per second */ /* times(&time); start_time=time.tms_utime; */ /* for (i=0; i<runs; i++) { */ /* igraph_grg_game2(&g, n, r, 1); */ /* igraph_destroy(&g); */ /* } */ /* times(&time); current_time=time.tms_utime; */ /* fprintf(stdout," sorted: time=%.3fs\n",(current_time-start_time)/(double)tps); */ /* tps=sysconf(_SC_CLK_TCK); // clock ticks per second */ /* times(&time); start_time=time.tms_utime; */ /* for (i=0; i<runs; i++) { */ /* igraph_grg_game(&g, n, r, 1); */ /* igraph_destroy(&g); */ /* } */ /* times(&time); current_time=time.tms_utime; */ /* fprintf(stdout,"non-sorted: time=%.3fs\n", */ /* (current_time-start_time)/(double)tps); */ return 0; }
int igraph_barabasi_game(igraph_t *graph, igraph_integer_t n,
igraph_real_t power,
igraph_integer_t m,
const igraph_vector_t *outseq,
igraph_bool_t outpref,
igraph_real_t A,
igraph_bool_t directed,
igraph_barabasi_algorithm_t algo,
const igraph_t *start_from);
Arguments:
|
An uninitialized graph object. |
||||||
|
The number of vertices in the graph. |
||||||
|
Power of the preferential attachment. The probability
that a vertex is cited is proportional to d^power+A, where
d is its degree (see also the |
||||||
|
The number of outgoing edges generated for each
vertex. (Only if |
||||||
|
Gives the (out-)degrees of the vertices. If this is
constant, this can be a NULL pointer or an empty (but
initialized!) vector, in this case |
||||||
|
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. |
||||||
|
The probability that a vertex is cited is proportional to
d^power+A, where d is its degree (see also the |
||||||
|
Boolean, whether to generate a directed graph. |
||||||
|
The algorithm to use to generate the network. Possible values:
|
||||||
|
Either a null pointer, or a graph. In the former
case, the starting configuration is a clique of size |
Returns:
Error code:
|
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.
Example 9.12. File examples/simple/igraph_barabasi_game.c
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2006-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard street, Cambridge, MA 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> int main() { igraph_t g; igraph_vector_t v, v2; int i, ret; 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_init(&v, 0); igraph_get_edgelist(&g, &v, 0); for (i = 0; i < igraph_ecount(&g); i++) { if (VECTOR(v)[2 * i] <= VECTOR(v)[2 * i + 1]) { return 4; } } igraph_destroy(&g); /* out-degree sequence */ igraph_vector_resize(&v, 10); VECTOR(v)[0] = 0; VECTOR(v)[1] = 1; VECTOR(v)[2] = 3; VECTOR(v)[3] = 3; VECTOR(v)[4] = 4; VECTOR(v)[5] = 5; VECTOR(v)[6] = 6; VECTOR(v)[7] = 7; VECTOR(v)[8] = 8; VECTOR(v)[9] = 9; igraph_barabasi_game(&g, 10, /*power=*/ 1, 0, &v, 0, /*A=*/ 1, 1, IGRAPH_BARABASI_BAG, /*start_from=*/ 0); if (igraph_ecount(&g) != igraph_vector_sum(&v)) { return 5; } igraph_vector_init(&v2, 0); igraph_degree(&g, &v2, igraph_vss_all(), IGRAPH_OUT, 1); for (i = 0; i < igraph_vcount(&g); i++) { if (VECTOR(v)[i] != VECTOR(v2)[i]) { igraph_vector_print(&v); printf("\n"); igraph_vector_print(&v2); return 6; } } igraph_vector_destroy(&v); igraph_vector_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_init(&v, 0); igraph_get_edgelist(&g, &v, 0); for (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_destroy(&v); igraph_destroy(&g); /* Error tests */ igraph_set_error_handler(igraph_error_handler_ignore); ret = igraph_barabasi_game(&g, -10, /*power=*/ 1, 1, 0, 0, /*A=*/ 1, 0, IGRAPH_BARABASI_BAG, /*start_from=*/ 0); if (ret != IGRAPH_EINVAL) { return 9; } ret = igraph_barabasi_game(&g, 10, /*power=*/ 1, -2, 0, 0, /*A=*/ 1, 0, IGRAPH_BARABASI_BAG, /*start_from=*/ 0); if (ret != IGRAPH_EINVAL) { return 10; } igraph_vector_init(&v, 9); ret = igraph_barabasi_game(&g, 10, /*power=*/ 1, 0, &v, 0, /*A=*/ 1, 0, IGRAPH_BARABASI_BAG, /*start_from=*/ 0); if (ret != IGRAPH_EINVAL) { return 11; } igraph_vector_destroy(&v); return 0; }
Example 9.13. File examples/simple/igraph_barabasi_game2.c
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2010-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard street, Cambridge, MA 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> #include <stdio.h> int main() { 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; }
int 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);
Arguments:
|
Pointer to an uninitialized graph object. |
||||
|
The type of the random graph, possible values:
|
||||
|
The number of vertices in the graph. |
||||
|
This is the p parameter for G(n,p) graphs and the m parameter for G(n,m) graphs. |
||||
|
Logical, whether to generate a directed graph. |
||||
|
Logical, whether to generate loops (self) edges. |
Returns:
Error code:
|
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.
See also:
Example 9.14. File examples/simple/igraph_erdos_renyi_game.c
#include <igraph.h> int main() { igraph_t graph; igraph_vector_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( &graph, IGRAPH_ERDOS_RENYI_GNM, /* n= */ 100, /* m= */ 100, IGRAPH_UNDIRECTED, IGRAPH_NO_LOOPS); /* Compute the fraction of vertices contained within the largest connected component */ igraph_vector_init(&component_sizes, 0); igraph_clusters(&graph, NULL, &component_sizes, NULL, IGRAPH_STRONG); printf("Fraction of vertices in giant component: %g\n", (double) igraph_vector_max(&component_sizes) / igraph_vcount(&graph)); /* Clean up data structures when no longer needed */ igraph_vector_destroy(&component_sizes); igraph_destroy(&graph); return 0; }
int 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 a graph according to the Watts-Strogatz model of small-world networks. The graph is obtained by creating a circular undirected lattice and then rewire the edges randomly with a constant probability.
See also: Duncan J Watts and Steven H Strogatz: Collective dynamics of “small world” networks, Nature 393, 440-442, 1998.
Arguments:
|
The graph to initialize. |
|
The dimension of the lattice. |
|
The size of the lattice along each dimension. |
|
The size of the neighborhood for each vertex. This is
the same as the |
|
The rewiring probability. A real number between zero and one (inclusive). |
|
Logical, whether to generate loop edges. |
|
Logical, whether to allow multiple edges in the generated graph. |
Returns:
Error code. |
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.
int 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:
|
The input graph, this will be rewired, it can be directed or undirected. |
|
The rewiring probability a constant between zero and one (inclusive). |
|
Boolean, whether loop edges are allowed in the new graph, or not. |
|
Boolean, whether multiple edges are allowed in the new graph. |
Returns:
Error code. |
See also:
|
Time complexity: O(|V|+|E|).
int 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.
Arguments:
|
The input graph, this will be rewired, it can be
directed or undirected. If it is directed, |
||||||
|
The rewiring probability, a constant between zero and one (inclusive). |
||||||
|
Boolean, whether loop edges are allowed in the new graph, or not. |
||||||
|
The endpoints of directed edges to rewire. It is ignored for undirected graphs. Possible values:
|
Returns:
Error code. |
See also:
Time complexity: O(|E|).
int igraph_degree_sequence_game(igraph_t *graph, const igraph_vector_t *out_deg,
const igraph_vector_t *in_deg,
igraph_degseq_t method);
Arguments:
|
Pointer to an uninitialized graph object. |
||||||||
|
The degree sequence for an undirected graph (if
|
||||||||
|
It is either a zero-length vector or
|
||||||||
|
The method to generate the graph. Possible values:
|
Returns:
Error code:
|
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:
Example 9.15. File examples/simple/igraph_degree_sequence_game.c
/* -*- mode: C -*- */ /* IGraph library. Copyright (C) 2006-2012 Gabor Csardi <csardi.gabor@gmail.com> 334 Harvard street, Cambridge, MA 02139 USA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include <igraph.h> int main() { igraph_t g; igraph_vector_t outdeg, indeg, vec; igraph_bool_t is_simple; igraph_vector_init_real(&outdeg, 10, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0); igraph_vector_init_real(&indeg, 10, 4.0, 4.0, 2.0, 2.0, 4.0, 4.0, 2.0, 2.0, 3.0, 3.0); igraph_vector_init(&vec, 0); /* checking the simple method, undirected graphs */ igraph_degree_sequence_game(&g, &outdeg, 0, IGRAPH_DEGSEQ_SIMPLE); 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_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_print(&vec); igraph_destroy(&g); /* checking the simple method, directed graphs */ igraph_degree_sequence_game(&g, &outdeg, &indeg, IGRAPH_DEGSEQ_SIMPLE); 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_print(&vec); if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_IN, 1)) { return 8; } igraph_vector_print(&vec); igraph_destroy(&g); /* checking the no multiple edges method, undirected graphs */ igraph_degree_sequence_game(&g, &outdeg, 0, IGRAPH_DEGSEQ_SIMPLE_NO_MULTIPLE); 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_print(&vec); igraph_destroy(&g); /* checking the no multiple edges method, directed graphs */ igraph_degree_sequence_game(&g, &outdeg, &indeg, IGRAPH_DEGSEQ_SIMPLE_NO_MULTIPLE); 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_print(&vec); if (igraph_degree(&g, &vec, igraph_vss_all(), IGRAPH_IN, 1)) { return 15; } igraph_vector_print(&vec); igraph_destroy(&g); igraph_vector_destroy(&vec); igraph_vector_destroy(&outdeg); igraph_vector_destroy(&indeg); return 0; }
int 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 SIMPLE_NO_MULTIPLE 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:
|
Pointer to an uninitialized graph object. |
|
The number of nodes in the generated graph. |
|
The degree of each vertex in an undirected graph, or the out-degree and in-degree of each vertex in a directed graph. |
|
Whether the generated graph will be directed. |
|
Whether to allow multiple edges in the generated graph. |
Returns:
Error code:
|
Time complexity: O(|V|+|E|) if multiple is true, otherwise not known.
int 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);
proportional to node fitness scores. 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.
Arguments:
|
Pointer to an uninitialized graph object. |
|
A numeric vector containing the fitness of each vertex. For directed graphs, this specifies the out-fitness of each vertex. |
|
If |
|
The number of edges in the generated graph. |
|
Whether to allow loop edges in the generated graph. |
|
Whether to allow multiple edges in the generated graph. |
Returns:
Error code:
|
Time complexity: O(|V| + |E| log |E|).
int 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:
|
Pointer to an uninitialized graph object. |
|
The number of nodes in the generated graph. |
|
The number of edges in the generated graph. |
|
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 |
|
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. |
|
Whether to allow loop edges in the generated graph. |
|
Whether to allow multiple edges in the generated graph. |
|
Whether to use the proposed finite size correction of Cho et al. |
Returns:
Error code:
|
Time complexity: O(|V| + |E| log |E|).
int 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-degree distribution.
Heavy-tailed out-degree distribution.
Communities.
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:
We generate two random number, 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.
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:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the graph. |
|
The forward burning probability. |
|
The backward burning ratio. The backward burning
probability is calculated as |
|
The number of ambassador vertices. |
|
Whether to create a directed graph. |
Returns:
Error code. |
Time complexity: TODO.
int 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
rewiring edges while preserving the original graph's degree distribution.
Please note that 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.
Arguments:
|
The graph object to be rewired. |
||||
|
Number of rewiring trials to perform. |
||||
|
The rewiring algorithm to be used. It can be one of the following flags:
|
Returns:
|
Error code:
|
Time complexity: TODO.
int 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:
|
Uninitialized graph object. |
|
The number of vertices in the graph. |
|
The number of edges to add in a time step (i.e. after adding a vertex). |
|
Boolean, whether to generate a directed graph. |
|
Boolean, if |
Returns:
Error code:
|
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.
int 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_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.
Arguments:
|
Pointer to an uninitialized graph. |
|
The number of nodes in the graph. |
|
Number of node types. |
|
The number of connections tried in each time step. |
|
Vector giving the distribution of the vertex types.
If |
|
Matrix giving the connection probabilities for the vertex types. |
|
Logical, whether to generate a directed graph. |
|
An initialized vector or |
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.
int 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_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:
|
Pointer to an uninitialized graph. |
|
The number of vertices in the graph. |
|
The number of vertex types. |
|
The number of connections tried in each time step. |
|
Vector giving the distribution of vertex types.
If |
|
Matrix giving the connection probabilities for different vertex types. |
|
Logical, whether to generate a directed graph. |
|
An initialized vector or |
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.
int 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_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:
|
Pointer to an uninitialized graph. |
|
The number of vertices in the graph. |
|
The number of vertex types. |
|
Vector giving the distribution of vertex types. If
|
|
Boolean. If true, then the number of vertices with a
given vertex type is fixed and the |
|
Matrix giving the connection probabilities for different vertex types. This should be symmetric if the requested graph is undirected. |
|
A vector where the individual generated vertex types
will be stored. If |
|
Logical, whether to generate a directed graph. If undirected graphs are requested, only the lower left triangle of the preference matrix is considered. |
|
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:
int igraph_asymmetric_preference_game(igraph_t *graph, igraph_integer_t nodes,
igraph_integer_t out_types,
igraph_integer_t in_types,
const igraph_matrix_t *type_dist_matrix,
const igraph_matrix_t *pref_matrix,
igraph_vector_t *node_type_out_vec,
igraph_vector_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:
|
Pointer to an uninitialized graph. |
|
The number of vertices in the graph. |
|
The number of vertex out-types. |
|
The number of vertex in-types. |
|
Matrix of size |
|
Matrix of size |
|
A vector where the individual generated "outgoing"
vertex types will be stored. If |
|
A vector where the individual generated "incoming"
vertex types will be stored. If |
|
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:
int 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_t *outseq,
igraph_bool_t outpref,
igraph_real_t zero_appeal,
igraph_bool_t directed);
Arguments:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the graph, this is the same as the number of time steps. |
|
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 |
|
Integer constant, the size of the time window to use to count the number of recent edges. |
|
Integer constant, the number of edges to add per time
step if the |
|
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 |
|
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. |
|
Constant giving the attractiveness of the vertices which haven't gained any edge recently. |
|
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.
int igraph_barabasi_aging_game(igraph_t *graph,
igraph_integer_t nodes,
igraph_integer_t m,
const igraph_vector_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 summed 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:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the graph. |
|
The number of edges to add in each time step.
Ignored if |
|
The number of edges to add in each time step. If it
is |
|
Logical constant, whether the edges initiated by a vertex contribute to the probability to gain a new edge. |
|
The exponent of the preferential attachment, a small positive number usually, the value 1 yields the classic linear preferential attachment. |
|
The exponent of the aging, this is a negative number usually. |
|
Integer constant, the number of age bins to use. |
|
The degree dependent part of the attractiveness of the zero degree vertices. |
|
The age dependent part of the attractiveness of the vertices of age zero. This parameter is usually zero. |
|
The coefficient for the degree. |
|
The coefficient for the age. |
|
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.
int igraph_recent_degree_aging_game(igraph_t *graph,
igraph_integer_t nodes,
igraph_integer_t m,
const igraph_vector_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:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the graph. |
|
The number of edges to add in each time step. If the |
|
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 |
|
Logical constant, if true the edges initiated by a vertex are also counted. Normally it is false. |
|
The exponent for the preferential attachment. |
|
The exponent for the aging, normally it is negative: old vertices gain edges with less probability. |
|
Integer constant, the number of age bins to use. |
|
The time window to use to count the number of incident edges for the vertices. |
|
The degree dependent part of the attractiveness for zero degree vertices. |
|
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.
int 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:
|
Pointer to an uninitialized graph object, the result will be stored here. |
|
The number of vertices in the network. |
|
The number of edges to add in each time step. |
|
The number of age bins to use. |
|
Pointer to an initialized vector of length
|
|
Logical constant, whether to create directed networks. |
Returns:
Error code. |
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.
int igraph_cited_type_game(igraph_t *graph, igraph_integer_t nodes,
const igraph_vector_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:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the network. |
|
Numeric vector giving the categories of the vertices,
so it should contain |
|
The attractivity of the different vertex categories in
a vector. Its length should be the maximum element in |
|
Integer constant, the number of edges to add in each time step. |
|
Logical constant, whether to create a directed network. |
Returns:
Error code. |
See also:
|
Time complexity: O((|V|+|E|)log|V|), |V| and |E| are number of vertices and edges, respectively.
int igraph_citing_cited_type_game(igraph_t *graph, igraph_integer_t nodes,
const igraph_vector_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:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the network. |
|
A numeric matrix of length |
|
The preference matrix, a square matrix is required,
both the number of rows and columns should be the maximum
element in |
|
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.
int 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:
|
The output graph. |
|
Number of vertices. |
|
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). |
|
An integer vector giving the number of vertices in each group. |
|
Boolean, whether to create a directed graph. If
this argument is false, then |
|
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:
|
int 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:
|
The generated graph is stored here. |
|
The number of vertices in the graph. |
|
The number of vertices per block. n/m must be integer. |
|
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. |
|
A square, symmetric numeric matrix, the Bernoulli rates for the clusters within a block. Its size must mach the size of the \code{rho} vector. |
|
The Bernoulli rate of connections between vertices in different blocks. |
Returns:
Error code. |
See also:
|
int igraph_hsbm_list_game(igraph_t *graph, igraph_integer_t n,
const igraph_vector_int_t *mlist,
const igraph_vector_ptr_t *rholist,
const igraph_vector_ptr_t *Clist,
igraph_real_t p);
The function generates a random graph according to the hierarchical stochastic block model.
Arguments:
|
The generated graph is stored here. |
|
The number of vertices in the graph. |
|
An integer vector of block sizes. |
|
A list of rho vectors ( |
|
A list of square matrices ( |
|
The Bernoulli rate of connections between vertices in different blocks. |
Returns:
Error code. |
See also:
|
int 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:
|
The output graph is stored here. |
|
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. |
|
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:
|
int 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.
Arguments:
|
Pointer to an uninitialized graph object. |
||||
|
The number of nodes in the tree. |
||||
|
Whether to create a directed tree. The edges are oriented away from the root. |
||||
|
The algorithm to use to generate the tree. Possible values:
|
Returns:
Error code:
|
See also:
int igraph_correlated_game(const igraph_t *old_graph, igraph_t *new_graph,
igraph_real_t corr, igraph_real_t p,
const igraph_vector_t *permutation);
Sample a new graph by perturbing the adjacency matrix of a given graph and shuffling its vertices.
Arguments:
|
The original graph. |
|
The new graph will be stored here. |
|
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). |
|
A numeric scalar, the probability of an edge between two vertices, it must in the open (0,1) interval. |
|
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:
|
int 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_t *permutation);
Sample two random graphs, with given correlation.
Arguments:
|
The first graph will be stored here. |
|
The second graph will be stored here. |
|
The number of vertices in both graphs. |
|
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). |
|
A numeric scalar, the probability of an edge between two vertices, it must in the open (0,1) interval. |
|
Whether to generate directed graphs. |
|
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:
|
int 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);
Arguments:
|
Pointer to an uninitialized graph object. |
|
The number of islands in the graph. |
|
The size of islands in the graph. |
|
The probability to create each possible edge into each island. |
|
The number of edges to create between two islands. |
Returns:
Error code:
|
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.
| ← Chapter 8. Random numbers | Chapter 10. Games on Graphs → |