For using the igraph C library
This section deals with functions which find small induced subgraphs in a graph. These were first defined for subgraphs of two and three vertices by Holland and Leinhardt, and named dyad census and triad census.
igraph_error_t igraph_dyad_census(const igraph_t *graph, igraph_real_t *mut, igraph_real_t *asym, igraph_real_t *null);
Dyad census means classifying each pair of vertices of a directed
graph into three categories: mutual (there is at least one edge from
a
to b
and also from b
to a
); asymmetric (there is at least
one edge either from a
to b
or from b
to a
, but not the other
way) and null (no edges between a
and b
in either direction).
Holland, P.W. and Leinhardt, S. (1970). A Method for Detecting Structure in Sociometric Data. American Journal of Sociology, 70, 492513.
Arguments:

The input graph. For an undirected graph, there are no asymmetric connections. 

Pointer to a real, the number of mutual dyads is stored here. 

Pointer to a real, the number of asymmetric dyads is stored here. 

Pointer to a real, the number of null dyads is stored here. 
Returns:
Error code. 
See also:
Time complexity: O(V+E), the number of vertices plus the number of edges.
igraph_error_t igraph_triad_census(const igraph_t *graph, igraph_vector_t *res);
Calculating the triad census means classifying every triple of vertices in a directed graph. A triple can be in one of 16 states:

A, B, C, the empty graph. 

A>B, C, a graph with a single directed edge. 

A<>B, C, a graph with a mutual connection between two vertices. 

A<B>C, the binary outtree. 

A>B<C, the binary intree. 

A>B>C, the directed line. 

A<>B<C. 

A<>B>C. 

A>B<C, A>C. 

A<B<C, A>C. 

A<>B<>C. 

A<B>C, A<>C. 

A>B<C, A<>C. 

A>B>C, A<>C. 

A>B<>C, A<>C. 

A<>B<>C, A<>C, the complete graph. 
See also Davis, J.A. and Leinhardt, S. (1972). The Structure of Positive Interpersonal Relations in Small Groups. In J. Berger (Ed.), Sociological Theories in Progress, Volume 2, 218251. Boston: Houghton Mifflin.
This function calls igraph_motifs_randesu()
which is an
implementation of the FANMOD motif finder tool, see igraph_motifs_randesu()
for details. Note that the order of the
triads is not the same for igraph_triad_census()
and igraph_motifs_randesu()
.
Arguments:

The input graph. A warning is given for undirected graphs, as the result is undefined for those. 

Pointer to an initialized vector, the result is stored
here in the same order as given in the list above. Note that this
order is different than the one used by 
Returns:
Error code. 
See also:
Time complexity: TODO.
igraph_error_t igraph_adjacent_triangles(const igraph_t *graph, igraph_vector_t *res, const igraph_vs_t vids);
Arguments:

The input graph. Edge directions and multiplicities are ignored. 

Initiliazed vector, the results are stored here. 

The vertices to perform the calculation for. 
Returns:
Error mode. 
See also:

Time complexity: O(d^2 n), d is the average vertex degree of the queried vertices, n is their number.
igraph_error_t igraph_list_triangles(const igraph_t *graph, igraph_vector_int_t *res);
The triangles are reported as a long list of vertex ID triplets. Use
the int
variant of igraph_matrix_view_from_vector()
to create a
matrix view into the vector where each triangle is stored in a column of the
matrix (see the example).
Arguments:

The input graph, edge directions are ignored. Multiple edges are ignored. 

Pointer to an initialized integer vector, the result is stored here, in a long list of triples of vertex IDs. Each triple is a triangle in the graph. Each triangle is listed exactly once. 
Returns:
Error code. 
See also:

Time complexity: O(d^2 n), d is the average degree, n is the number of vertices.
Example 19.1. File examples/simple/igraph_list_triangles.c
#include <igraph.h> int main(void) { igraph_t g; igraph_vector_int_t v; igraph_matrix_int_t result; igraph_full(&g, 5, 0, IGRAPH_NO_LOOPS); printf("Triangles in a full graph of 5 vertices:\n"); igraph_vector_int_init(&v, 0); igraph_list_triangles(&g, &v); igraph_matrix_int_view_from_vector(&result, &v, /* nrow = */ 3); igraph_matrix_int_print(&result); igraph_vector_int_destroy(&v); igraph_destroy(&g); return 0; }
igraph_motifs_randesu
— Count the number of motifs in a graph.igraph_motifs_randesu_no
— Count the total number of motifs in a graph.igraph_motifs_randesu_estimate
— Estimate the total number of motifs in a graph.igraph_motifs_randesu_callback
— Finds motifs in a graph and calls a function for each of them.igraph_motifs_handler_t
— Callback type for igraph_motifs_randesu_callback
igraph_error_t igraph_motifs_randesu(const igraph_t *graph, igraph_vector_t *hist, igraph_integer_t size, const igraph_vector_t *cut_prob);
Motifs are small weakly connected induced subgraphs of a given structure in a graph. It is argued that the motif profile (i.e. the number of different motifs in the graph) is characteristic for different types of networks and network function is related to the motifs in the graph.
This function is able to find directed motifs of sizes three and four and undirected motifs of sizes three to six (i.e. the number of different subgraphs with three to six vertices in the network).
In a big network the total number of motifs can be very large, so
it takes a lot of time to find all of them. In this case, a sampling
method can be used. This function is capable of doing sampling via the
cut_prob
argument. This argument gives the probability that
a branch of the motif search tree will not be explored. See
S. Wernicke and F. Rasche: FANMOD: a tool for fast network motif
detection, Bioinformatics 22(9), 11521153, 2006 for details.
https://doi.org/10.1093/bioinformatics/btl038
Set the cut_prob
argument to a zero vector for finding all
motifs.
Directed motifs will be counted in directed graphs and undirected motifs in undirected graphs.
Arguments:

The graph to find the motifs in. 

The result of the computation, it gives the number of
motifs found for each isomorphism class. See


The size of the motifs to search for. For directed graphs, only 3 and 4 are implemented, for undirected, 3 to 6. The limitation is not in the motif finding code, but the graph isomorphism code. 

Vector of probabilities for cutting the search tree
at a given level. The first element is the first level, etc.
Supply all zeros here (of length 
Returns:
Error code. 
See also:

Time complexity: TODO.
Example 19.2. File examples/simple/igraph_motifs_randesu.c
#include <igraph.h> /* This is a callback function suitable for use with igraph_motifs_randesu_callback(). * It prints each motif it is calld with. */ igraph_error_t print_motif(const igraph_t *graph, igraph_vector_int_t *vids, igraph_integer_t isoclass, void* extra) { printf("Found isoclass %2" IGRAPH_PRId ": ", isoclass); igraph_vector_int_print(vids); return IGRAPH_SUCCESS; /* Return 'IGRAPH_SUCCESS': do not interrupt the search. */ } int main(void) { igraph_t graph; igraph_vector_t hist; igraph_real_t zeros[] = { 0.0, 0.0, 0.0, 0.0, 0.0 }; igraph_vector_t cut_prob; /* Compute the 4motif distritbuion in Zachary's karate club network. */ igraph_famous(&graph, "Zachary"); igraph_vector_init(&hist, 0); igraph_motifs_randesu(&graph, &hist, 4, igraph_vector_view(&cut_prob, zeros, 4)); /* Compute the total number of motifs (connected 4vertex subgraphs) * so that we can print the normalized distribution. */ igraph_real_t sum = 0.0; igraph_integer_t n = igraph_vector_size(&hist); for (igraph_integer_t i=0; i < n; i++) { if (!isnan(VECTOR(hist)[i])) { sum += VECTOR(hist)[i]; } } printf("4motif distribution:\n"); for (igraph_integer_t i=0; i < n; i++) { /* Print NaN values in a platformindependent manner: */ igraph_real_printf(VECTOR(hist)[i] / sum); printf(" "); } printf("\n\n"); igraph_vector_destroy(&hist); igraph_destroy(&graph); /* Identify the vertices of each threemotif in a small Kautz graph. */ igraph_kautz(&graph, 2, 1); igraph_motifs_randesu_callback(&graph, 3, igraph_vector_view(&cut_prob, zeros, 3), &print_motif, NULL); igraph_destroy(&graph); return 0; }
igraph_error_t igraph_motifs_randesu_no(const igraph_t *graph, igraph_integer_t *no, igraph_integer_t size, const igraph_vector_t *cut_prob);
This function counts the total number of motifs in a graph, i.e. the number of of (weakly) connected triplets or quadruplets, without assigning isomorphism classes to them.
Arguments:

The graph object to study. 

Pointer to an integer type, the result will be stored here. 

The size of the motifs to count. 

Vector giving the probabilities that a branch of the search tree will be cut at a given level. 
Returns:
Error code. 
See also:
Time complexity: TODO.
igraph_error_t igraph_motifs_randesu_estimate(const igraph_t *graph, igraph_integer_t *est, igraph_integer_t size, const igraph_vector_t *cut_prob, igraph_integer_t sample_size, const igraph_vector_int_t *parsample);
This function estimates the total number of weakly connected induced
subgraphs, called motifs, of a fixed number of vertices. For
example, an undirected complete graph on n
vertices
will have one motif of size n
, and n
motifs
of size
n  1
. As another example, one triangle
and a separate vertex will have zero motifs of size four.
This function is useful for large graphs for which it is not feasible to count all the different motifs, because there are very many of them.
The total number of motifs is estimated by taking a sample of
vertices and counts all motifs in which these vertices are
included. (There is also a cut_prob
parameter which gives the
probabilities to cut a branch of the search tree.)
Directed motifs will be counted in directed graphs and undirected motifs in undirected graphs.
Arguments:

The graph object to study. 

Pointer to an integer type, the result will be stored here. 

The size of the motifs to look for. 

Vector giving the probabilities to cut a branch of
the search tree and omit counting the motifs in that branch.
It contains a probability for each level. Supply 

The number of vertices to use as the
sample. This parameter is only used if the 

Either pointer to an initialized vector or a null
pointer. If a vector then the vertex IDs in the vector are
used as a sample. If a null pointer then the 
Returns:
Error code. 
See also:
Time complexity: TODO.
igraph_error_t igraph_motifs_randesu_callback(const igraph_t *graph, igraph_integer_t size, const igraph_vector_t *cut_prob, igraph_motifs_handler_t *callback, void* extra);
Similarly to igraph_motifs_randesu()
, this function is able to find
directed motifs of sizes three and four and undirected motifs of sizes
three to six (i.e. the number of different subgraphs with three to six
vertices in the network). However, instead of
counting them, the function will call a callback function for each motif
found to allow further tests or postprocessing.
The cut_prob
argument also allows sampling the motifs, just like for
igraph_motifs_randesu()
. Set the cut_prob
argument to a zero vector
for finding all motifs.
Arguments:

The graph to find the motifs in. 

The size of the motifs to search for. Only three and four are implemented currently. The limitation is not in the motif finding code, but the graph isomorphism code. 

Vector of probabilities for cutting the search tree
at a given level. The first element is the first level, etc.
Supply all zeros here (of length 

A pointer to a function of type 

Extra argument to pass to the callback function. 
Returns:
Error code. 
Time complexity: TODO.
Example 19.3. File examples/simple/igraph_motifs_randesu.c
#include <igraph.h> /* This is a callback function suitable for use with igraph_motifs_randesu_callback(). * It prints each motif it is calld with. */ igraph_error_t print_motif(const igraph_t *graph, igraph_vector_int_t *vids, igraph_integer_t isoclass, void* extra) { printf("Found isoclass %2" IGRAPH_PRId ": ", isoclass); igraph_vector_int_print(vids); return IGRAPH_SUCCESS; /* Return 'IGRAPH_SUCCESS': do not interrupt the search. */ } int main(void) { igraph_t graph; igraph_vector_t hist; igraph_real_t zeros[] = { 0.0, 0.0, 0.0, 0.0, 0.0 }; igraph_vector_t cut_prob; /* Compute the 4motif distritbuion in Zachary's karate club network. */ igraph_famous(&graph, "Zachary"); igraph_vector_init(&hist, 0); igraph_motifs_randesu(&graph, &hist, 4, igraph_vector_view(&cut_prob, zeros, 4)); /* Compute the total number of motifs (connected 4vertex subgraphs) * so that we can print the normalized distribution. */ igraph_real_t sum = 0.0; igraph_integer_t n = igraph_vector_size(&hist); for (igraph_integer_t i=0; i < n; i++) { if (!isnan(VECTOR(hist)[i])) { sum += VECTOR(hist)[i]; } } printf("4motif distribution:\n"); for (igraph_integer_t i=0; i < n; i++) { /* Print NaN values in a platformindependent manner: */ igraph_real_printf(VECTOR(hist)[i] / sum); printf(" "); } printf("\n\n"); igraph_vector_destroy(&hist); igraph_destroy(&graph); /* Identify the vertices of each threemotif in a small Kautz graph. */ igraph_kautz(&graph, 2, 1); igraph_motifs_randesu_callback(&graph, 3, igraph_vector_view(&cut_prob, zeros, 3), &print_motif, NULL); igraph_destroy(&graph); return 0; }
typedef igraph_error_t igraph_motifs_handler_t(const igraph_t *graph, igraph_vector_int_t *vids, igraph_integer_t isoclass, void* extra);
igraph_motifs_randesu_callback()
calls a specified callback
function whenever a new motif is found during a motif search. This
callback function must be of type igraph_motifs_handler_t
. It has
the following arguments:
Arguments:

The graph that that algorithm is working on. Of course this must not be modified. 

The IDs of the vertices in the motif that has just been found. This vector is owned by the motif search algorithm, so do not modify or destroy it; make a copy of it if you need it later. 

The isomorphism class of the motif that has just been
found. Use 

The extra argument that was passed to 
Returns:

See also:
← Chapter 18. Graph coloring  Chapter 20. Generating layouts for graph drawing → 