# igraph Reference Manual

For using the igraph C library

# Chapter 17. Graph Motifs, Dyad Census and Triad Census

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.

## 1. `igraph_dyad_census` — Calculating the dyad census as defined by Holland and Leinhardt

```int igraph_dyad_census(const igraph_t *graph, igraph_integer_t *mut,
igraph_integer_t *asym, igraph_integer_t *null);
```

Dyad census means classifying each pair of vertices of a directed graph into three categories: mutual, there is an edge from `a` to `b` and also from `b` to `a`; asymmetric, there is an edge either from `a` to `b` or from `b` to `a` but not the other way and null, no edges between `a` and `b`.

Holland, P.W. and Leinhardt, S. (1970). A Method for Detecting Structure in Sociometric Data. American Journal of Sociology, 70, 492-513.

Arguments:

 `graph`: The input graph, a warning is given if undirected as the results are undefined for undirected graphs. `mut`: Pointer to an integer, the number of mutual dyads is stored here. `asym`: Pointer to an integer, the number of asymmetric dyads is stored here. `null`: Pointer to an integer, 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.

## 2. `igraph_triad_census` — Triad census, as defined by Davis and Leinhardt

```int 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:

 `003 ` A, B, C, the empty graph. `012 ` A->B, C, a graph with a single directed edge. `102 ` A<->B, C, a graph with a mutual connection between two vertices. `021D ` A<-B->C, the binary out-tree. `021U ` A->B<-C, the binary in-tree. `021C ` A->B->C, the directed line. `111D ` A<->B<-C. `111U ` A<->B->C. `030T ` A->B<-C, A->C. `030C ` A<-B<-C, A->C. `201 ` A<->B<->C. `120D ` A<-B->C, A<->C. `120U ` A->B<-C, A<->C. `120C ` A->B->C, A<->C. `210 ` A->B<->C, A<->C. `300 ` 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, 218-251. 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:

 `graph`: The input graph. A warning is given for undirected graphs, as the result is undefined for those. `res`: 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 `igraph_motifs_randesu()`.

Returns:

 Error code.

See also:

Time complexity: TODO.

## 3. Graph motifs

### 3.1. `igraph_motifs_randesu` — Count the number of motifs in a graph

```int igraph_motifs_randesu(const igraph_t *graph, igraph_vector_t *hist,
int size, const igraph_vector_t *cut_prob);
```

Motifs are small connected subgraphs of a given structure in a graph. It is argued that the motif profile (ie. 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 the different motifs of size three and four (ie. the number of different subgraphs with three and four 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, 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), 1152--1153, 2006 for details.

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:

 `graph`: The graph to find the motifs in. `hist`: The result of the computation, it gives the number of motifs found for each isomorphism class. See `igraph_isoclass()` for help about isomorphism classes. Note that this function does not count isomorphism classes that are not connected and will report NaN (more precisely `IGRAPH_NAN`) for them. `size`: 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. `cut_prob`: 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 `size`) to find all motifs in a graph.

Returns:

 Error code.

See also:

 `igraph_motifs_randesu_estimate()` for estimating the number of motifs in a graph, this can help to set the `cut_prob` parameter; `igraph_motifs_randesu_no()` to calculate the total number of motifs of a given size in a graph; `igraph_motifs_randesu_callback()` for calling a callback function for every motif found.

Time complexity: TODO.

Example 17.1.  File `examples/simple/igraph_motifs_randesu.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>

void print_vector(igraph_vector_t *v) {
long int i, n=igraph_vector_size(v);
igraph_real_t sum=0.0;
for (i=0; i<n; i++) {
if (!igraph_is_nan(VECTOR(*v)[i])) { sum += VECTOR(*v)[i]; }
}
for (i=0; i<n; i++) {
igraph_real_printf(VECTOR(*v)[i]/sum);
printf(" ");
}
printf("\n");
}

igraph_bool_t print_motif(const igraph_t *graph, igraph_vector_t *vids,
int isoclass, void* extra) {
printf("Class %d: ", isoclass);
igraph_vector_print(vids);
return 0;
}

int main() {

igraph_t g;
igraph_vector_t hist;
igraph_vector_t cp;

igraph_vector_init_real(&cp, 8, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0);

igraph_ring(&g, 1000, IGRAPH_DIRECTED, 1, 1);
igraph_vector_init(&hist, 0);
igraph_motifs_randesu(&g, &hist, 3, &cp);
print_vector(&hist);
igraph_destroy(&g);
igraph_vector_destroy(&hist);

igraph_famous(&g, "bull");
igraph_motifs_randesu_callback(&g, 3, &cp, &print_motif, 0);
igraph_motifs_randesu_callback(&g, 4, &cp, &print_motif, 0);
igraph_destroy(&g);

igraph_vector_destroy(&cp);
return 0;
}
```

### 3.2. `igraph_motifs_randesu_no` — Count the total number of motifs in a graph

```int igraph_motifs_randesu_no(const igraph_t *graph, igraph_integer_t *no,
int size, const igraph_vector_t *cut_prob);
```

This function counts the total number of motifs in a graph without assigning isomorphism classes to them.

Directed motifs will be counted in directed graphs and undirected motifs in undirected graphs.

Arguments:

 `graph`: The graph object to study. `no`: Pointer to an integer type, the result will be stored here. `size`: The size of the motifs to count. `cut_prob`: 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.

### 3.3. `igraph_motifs_randesu_estimate` — Estimate the total number of motifs in a graph

```int igraph_motifs_randesu_estimate(const igraph_t *graph, igraph_integer_t *est,
int size, const igraph_vector_t *cut_prob,
igraph_integer_t sample_size,
const igraph_vector_t *parsample);
```

This function is useful for large graphs for which it is not feasible to count all the different motifs, because there is 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:

 `graph`: The graph object to study. `est`: Pointer to an integer type, the result will be stored here. `size`: The size of the motif to look for. `cut_prob`: 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 `size` zeros here to count all the motifs in the sample. `sample_size`: The number of vertices to use as the sample. This parameter is only used if the `parsample` argument is a null pointer. `parsample`: 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 `sample_size` argument is used to create a sample of vertices drawn with uniform probability.

Returns:

 Error code.

See also:

Time complexity: TODO.

### 3.4. `igraph_motifs_randesu_callback` — Finds motifs in a graph and calls a function for each of them

```int igraph_motifs_randesu_callback(const igraph_t *graph, int 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 the different motifs of size three and four (ie. the number of different subgraphs with three and four 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 post-processing.

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:

 `graph`: The graph to find the motifs in. `size`: 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. `cut_prob`: 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 `size`) to find all motifs in a graph. `callback`: A pointer to a function of type `igraph_motifs_handler_t`. This function will be called whenever a new motif is found. `extra`: Extra argument to pass to the callback function.

Returns:

 Error code.

Time complexity: TODO.

Example 17.2.  File `examples/simple/igraph_motifs_randesu.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>

void print_vector(igraph_vector_t *v) {
long int i, n=igraph_vector_size(v);
igraph_real_t sum=0.0;
for (i=0; i<n; i++) {
if (!igraph_is_nan(VECTOR(*v)[i])) { sum += VECTOR(*v)[i]; }
}
for (i=0; i<n; i++) {
igraph_real_printf(VECTOR(*v)[i]/sum);
printf(" ");
}
printf("\n");
}

igraph_bool_t print_motif(const igraph_t *graph, igraph_vector_t *vids,
int isoclass, void* extra) {
printf("Class %d: ", isoclass);
igraph_vector_print(vids);
return 0;
}

int main() {

igraph_t g;
igraph_vector_t hist;
igraph_vector_t cp;

igraph_vector_init_real(&cp, 8, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0);

igraph_ring(&g, 1000, IGRAPH_DIRECTED, 1, 1);
igraph_vector_init(&hist, 0);
igraph_motifs_randesu(&g, &hist, 3, &cp);
print_vector(&hist);
igraph_destroy(&g);
igraph_vector_destroy(&hist);

igraph_famous(&g, "bull");
igraph_motifs_randesu_callback(&g, 3, &cp, &print_motif, 0);
igraph_motifs_randesu_callback(&g, 4, &cp, &print_motif, 0);
igraph_destroy(&g);

igraph_vector_destroy(&cp);
return 0;
}
```

### 3.5. `igraph_motifs_handler_t` — Callback type for `igraph_motifs_randesu_callback`

```typedef igraph_bool_t igraph_motifs_handler_t(const igraph_t *graph,
igraph_vector_t *vids,
int 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:

 `graph`: The graph that that algorithm is working on. Of course this must not be modified. `vids`: 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. `isoclass`: The isomorphism class of the motif that has just been found. Use `igraph_isoclass` or `igraph_isoclass_subgraph` to find out which isomorphism class belongs to a given motif. `extra`: The extra argument that was passed to `igraph_motifs_randesu_callback()`.

Returns:

 A logical value, if TRUE (=non-zero), that is interpreted as a request to stop the motif search and return to the caller.

See also: