For using the igraph C library
igraph_modularity
— Calculates the modularity of a graph with respect to some clusters or vertex types.igraph_modularity_matrix
— Calculates the modularity matrix.igraph_community_optimal_modularity
— Calculate the community structure with the highest modularity valueigraph_community_to_membership
— Creates a membership vector from a community structure dendrogram.igraph_reindex_membership
— Makes the IDs in a membership vector contiguous.igraph_compare_communities
— Compares community structures using various metrics.igraph_split_join_distance
— Calculates the splitjoin distance of two community structures.
igraph_error_t igraph_modularity(const igraph_t *graph, const igraph_vector_int_t *membership, const igraph_vector_t *weights, const igraph_real_t resolution, const igraph_bool_t directed, igraph_real_t *modularity);
The modularity of a graph with respect to some clustering of the vertices (or assignment of vertex types) measures how strongly separated the different clusters are from each other compared to a random null model. It is defined as
Q = 1/(2m) sum_ij (A_ij  γ k_i k_j / (2m)) δ(c_i,c_j)
,
where m
is the number of edges, A_ij
is the adjacency matrix,
k_i
is the degree of vertex i
, c_i
is the cluster that vertex i
belongs to
(or its vertex type), δ(i,j)=1
if i=j
and 0 otherwise,
and the sum goes over all i
, j
pairs of vertices. Note that in this formula,
the diagonal of the adjacency matrix contains twice the number of selfloops.
The resolution parameter γ
allows weighting the random null model, which
might be useful when finding partitions with a high modularity. Maximizing modularity
with higher values of the resolution parameter typically results in more, smaller clusters
when finding partitions with a high modularity. Lower values typically results in
fewer, larger clusters. The original definition of modularity is retrieved
when setting γ = 1
.
Modularity can also be calculated on directed graphs. This only requires a relatively modest change,
Q = 1/m sum_ij (A_ij  γ k^out_i k^in_j / m) δ(c_i,c_j)
,
where k^out_i
is the outdegree of node i
and k^in_j
is the indegree of node j
.
Modularity on weighted graphs is also meaningful. When taking
edge weights into account, A_ij
equals the weight of the corresponding edge
(or 0 if there is no edge), k_i
is the strength (i.e. the weighted degree) of
vertex i
, with similar counterparts for a directed graph, and m
is the total
weight of all edges.
Note that the modularity is not welldefined for graphs with no edges.
igraph returns NaN
for graphs with no edges; see
https://github.com/igraph/igraph/issues/1539 for
a detailed discussion.
For the original definition of modularity, see Newman, M. E. J., and Girvan, M. (2004). Finding and evaluating community structure in networks. Physical Review E 69, 026113. https://doi.org/10.1103/PhysRevE.69.026113
For the directed definition of modularity, see Leicht, E. A., and Newman, M. E. J. (2008). Community Structure in Directed Networks. Physical Review Letters 100, 118703. https://doi.org/10.1103/PhysRevLett.100.118703
For the introduction of the resolution parameter γ
, see Reichardt, J., and
Bornholdt, S. (2006). Statistical mechanics of community detection. Physical
Review E 74, 016110. https://doi.org/10.1103/PhysRevE.74.016110
Arguments:

The input graph. 

Numeric vector of integer values which gives the type of each vertex, i.e. the cluster to which it belongs. It does not have to be consecutive, i.e. empty communities are allowed. 

Weight vector or 

The resolutin parameter 

Whether to use the directed or undirected version of modularity. Ignored for undirected graphs. 

Pointer to a real number, the result will be 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_modularity_matrix(const igraph_t *graph, const igraph_vector_t *weights, const igraph_real_t resolution, igraph_matrix_t *modmat, igraph_bool_t directed);
This function returns the modularity matrix, which is defined as
B_ij = A_ij  γ k_i k_j / (2m)
for undirected graphs, where A_ij
is the adjacency matrix, γ
is the
resolution parameter, k_i
is the degree of vertex i
, and m
is the
number of edges in the graph. When there are no edges, or the weights add up
to zero, the result is undefined.
For directed graphs the modularity matrix is changed to
B_ij = A_ij  γ k^out_i k^in_j / m
where k^out_i
is the outdegree of node i
and k^in_j
is the
indegree of node j
.
Note that selfloops in undirected graphs are multiplied by 2 in this implementation. If weights are specified, the weighted counterparts of the adjacency matrix and degrees are used.
Arguments:

The input graph. 

Edge weights, pointer to a vector. If this is a null pointer then every edge is assumed to have a weight of 1. 

The resolution parameter 

Pointer to an initialized matrix in which the modularity matrix is stored. 

For directed graphs: if the edges should be treated as undirected. For undirected graphs this is ignored. 
See also:
igraph_error_t igraph_community_optimal_modularity(const igraph_t *graph, igraph_real_t *modularity, igraph_vector_int_t *membership, const igraph_vector_t *weights);
This function calculates the optimal community structure for a graph, in terms of maximal modularity score.
The calculation is done by transforming the modularity maximization into an integer programming problem, and then calling the GLPK library to solve that. Please see Ulrik Brandes et al.: On Modularity Clustering, IEEE Transactions on Knowledge and Data Engineering 20(2):172188, 2008.
Note that modularity optimization is an NPcomplete problem, and all known algorithms for it have exponential time complexity. This means that you probably don't want to run this function on larger graphs. Graphs with up to fifty vertices should be fine, graphs with a couple of hundred vertices might be possible.
Arguments:

The input graph. It is always treated as undirected. 

Pointer to a real number, or a null pointer. If it is not a null pointer, then a optimal modularity value is returned here. 

Pointer to a vector, or a null pointer. If not a null pointer, then the membership vector of the optimal community structure is stored here. 

Vector giving the weights of the edges. If it is

Returns:
Error code.
When GLPK is not available, 
See also:

Time complexity: exponential in the number of vertices.
Example 24.1. File examples/simple/igraph_community_optimal_modularity.c
#include <igraph.h> void prepare_weights_vector(igraph_vector_t* weights, const igraph_t* graph) { igraph_integer_t i, n = igraph_ecount(graph); igraph_vector_resize(weights, n); for (i = 0; i < n; i++) { VECTOR(*weights)[i] = i % 5; } } int main(void) { igraph_t graph; igraph_vector_int_t v; igraph_integer_t edges[] = { 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 10, 0, 11, 0, 12, 0, 13, 0, 17, 0, 19, 0, 21, 0, 31, 1, 2, 1, 3, 1, 7, 1, 13, 1, 17, 1, 19, 1, 21, 1, 30, 2, 3, 2, 7, 2, 27, 2, 28, 2, 32, 2, 9, 2, 8, 2, 13, 3, 7, 3, 12, 3, 13, 4, 6, 4, 10, 5, 6, 5, 10, 5, 16, 6, 16, 8, 30, 8, 32, 8, 33, 9, 33, 13, 33, 14, 32, 14, 33, 15, 32, 15, 33, 18, 32, 18, 33, 19, 33, 20, 32, 20, 33, 22, 32, 22, 33, 23, 25, 23, 27, 23, 32, 23, 33, 23, 29, 24, 25, 24, 27, 24, 31, 25, 31, 26, 29, 26, 33, 27, 33, 28, 31, 28, 33, 29, 32, 29, 33, 30, 32, 30, 33, 31, 32, 31, 33, 32, 33 }; igraph_vector_int_t membership; igraph_vector_t weights; igraph_real_t modularity; igraph_bool_t simple; igraph_error_t retval; igraph_vector_int_view(&v, edges, sizeof(edges) / sizeof(edges[0])); igraph_create(&graph, &v, 0, IGRAPH_UNDIRECTED); igraph_vector_init(&weights, 0); igraph_is_simple(&graph, &simple); if (!simple) { return 1; } igraph_vector_int_init(&membership, 0); igraph_set_error_handler(&igraph_error_handler_printignore); /* Zachary karate club, unweighted */ retval = igraph_community_optimal_modularity(&graph, &modularity, &membership, 0); if (retval == IGRAPH_UNIMPLEMENTED) { return 77; } if (fabs(modularity  0.4197896) > 0.0000001) { return 2; } /* Zachary karate club, weighted */ prepare_weights_vector(&weights, &graph); igraph_community_optimal_modularity(&graph, &modularity, &membership, &weights); if (fabs(modularity  0.5115767) > 0.0000001) { return 4; } igraph_destroy(&graph); /* simple graph with loop edges, unweighted */ igraph_small(&graph, 6, IGRAPH_UNDIRECTED, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 0, 0, 0, 2, 2, 1); igraph_community_optimal_modularity(&graph, &modularity, &membership, 0); if (fabs(modularity  0.28125) > 0.00001) { return 3; } /* simple graph with loop edges, weighted */ prepare_weights_vector(&weights, &graph); igraph_community_optimal_modularity(&graph, &modularity, &membership, &weights); if (fabs(modularity  0.36686) > 0.00001) { return 5; } igraph_destroy(&graph); igraph_vector_int_destroy(&membership); igraph_vector_destroy(&weights); return 0; }
igraph_error_t igraph_community_to_membership(const igraph_matrix_int_t *merges, igraph_integer_t nodes, igraph_integer_t steps, igraph_vector_int_t *membership, igraph_vector_int_t *csize);
This function creates a membership vector from a community
structure dendrogram. A membership vector contains for each vertex
the id of its graph component, the graph components are numbered
from zero, see the same argument of igraph_connected_components()
for an example of a membership vector.
Many community detection algorithms return with a merges
matrix, igraph_community_walktrap()
and igraph_community_edge_betweenness()
are two examples. The matrix
contains the merge operations performed while mapping the
hierarchical structure of a network. If the matrix has n1
rows,
where n
is the number of vertices in the graph, then it contains
the hierarchical structure of the whole network and it is called a
dendrogram.
This function performs steps
merge operations as prescribed by
the merges
matrix and returns the current state of the network.
If merges
is not a complete dendrogram, it is possible to
take steps
steps if steps
is not bigger than the number
lines in merges
.
Arguments:

The twocolumn matrix containing the merge
operations. See 

The number of leaf nodes in the dendrogram. 

Integer constant, the number of steps to take. 

Pointer to an initialized vector, the membership results will be stored here, if not NULL. The vector will be resized as needed. 

Pointer to an initialized vector, or NULL. If not NULL then the sizes of the components will be stored here, the vector will be resized as needed. 
See also:

Time complexity: O(V), the number of vertices in the graph.
igraph_error_t igraph_reindex_membership(igraph_vector_int_t *membership, igraph_vector_int_t *new_to_old, igraph_integer_t *nb_clusters);
This function reindexes component IDs in a membership vector in a way that the new IDs start from zero and go up to C1, where C is the number of unique component IDs in the original vector. The supplied membership is expected to fall in the range 0, ..., n  1.
Arguments:

Numeric vector which gives the type of each vertex, i.e. the component to which it belongs. The vector will be altered inplace. 

Pointer to a vector which will contain the
old component ID for each new one, or 

Pointer to an integer for the number of
distinct clusters. If not 
Time complexity: should be O(n) for n elements.
igraph_error_t igraph_compare_communities(const igraph_vector_int_t *comm1, const igraph_vector_int_t *comm2, igraph_real_t* result, igraph_community_comparison_t method);
This function assesses the distance between two community structures using the variation of information (VI) metric of Meila (2003), the normalized mutual information (NMI) of Danon et al (2005), the splitjoin distance of van Dongen (2000), the Rand index of Rand (1971) or the adjusted Rand index of Hubert and Arabie (1985).
Some of these measures are defined based on the entropy of a discrete
random variable associated with a given clustering C
of vertices.
Let p_i
be the probability that a randomly picked vertex would be part
of cluster i
. Then the entropy of the clustering is
H(C) =  \sum_i p_i log p_i
Similarly, we can define the joint entropy of two clusterings C_1
and C_2
based on the probability p_ij
that a random vertex is part of cluster i
in the first clustering and cluster j
in the second one:
H(C_1, C_2) =  \sum_ii p_ij log p_ij
The mutual information of C_1
and C_2
is then
MI(C_1, C_2) = H(C_1) + H(C_2)  H(C_1, C_2) >= 0
.
A large mutual information indicates a high overlap between the two clusterings.
The normalized mutual information, as computed by igraph, is
NMI(C_1, C_2) = 2 MI(C_1, C_2) / (H(C_1) + H(C_2))
.
It takes its value from the interval (0, 1], with 1 achieved when the two clusterings coincide.
The variation of information is defined as
VI(C_1, C_2) = [H(C_1)  MI(C_1, C_2)] + [H(C_2)  MI(C_1, C_2)]
.
Lower values of the variation of information indicate a smaller difference between
the two clusterings, with VI = 0
achieved precisely when they coincide.
The Rand index is defined as the probability that the two clusterings agree about the cluster memberships of a randomly chosen vertex pair. All vertex pairs are considered, and the two clusterings are considered to be in agreement about the memberships of a vertex pair if either the two vertices are in the same cluster in both clusterings, or they are in different clusters in both clusterings. The Rand index is then the number of vertex pairs in agreement, divided by the total number of vertex pairs. A Rand index of zero means that the two clusterings disagree about the membership of all vertex pairs, while 1 means that the two clusterings are identical.
The adjusted Rand index is similar to the Rand index, but it takes into account that agreement between the two clusterings may also occur by chance even if the two clusterings are chosen completely randomly. The adjusted Rand index therefore subtracts the expected fraction of agreements from the value of the Rand index, and divides the result by one minus the expected fraction of agreements. The maximum value of the adjusted Rand index is still 1 (similarly to the Rand index), indicating maximum agreement, but the value may be less than zero if there is less agreement between the two clusterings than what would be expected by chance.
For an explanation of the splitjoin distance, see igraph_split_join_distance()
.
References:
Meilă M: Comparing clusterings by the variation of information. In: Schölkopf B, Warmuth MK (eds.). Learning Theory and Kernel Machines: 16th Annual Conference on Computational Learning Theory and 7th Kernel Workshop, COLT/Kernel 2003, Washington, DC, USA. Lecture Notes in Computer Science, vol. 2777, Springer, 2003. ISBN: 9783540407201. https://doi.org/10.1007/9783540451679_14
Danon L, DiazGuilera A, Duch J, Arenas A: Comparing community structure identification. J Stat Mech P09008, 2005. https://doi.org/10.1088/17425468/2005/09/P09008
van Dongen S: Performance criteria for graph clustering and Markov cluster experiments. Technical Report INSR0012, National Research Institute for Mathematics and Computer Science in the Netherlands, Amsterdam, May 2000. https://ir.cwi.nl/pub/4461
Rand WM: Objective criteria for the evaluation of clustering methods. J Am Stat Assoc 66(336):846850, 1971. https://doi.org/10.2307/2284239
Hubert L and Arabie P: Comparing partitions. Journal of Classification 2:193218, 1985. https://doi.org/10.1007/BF01908075
Arguments:

the membership vector of the first community structure 

the membership vector of the second community structure 

the result is stored here. 

the comparison method to use. 
Returns:
Error code. 
See also:
Time complexity: O(n log(n)).
igraph_error_t igraph_split_join_distance(const igraph_vector_int_t *comm1, const igraph_vector_int_t *comm2, igraph_integer_t *distance12, igraph_integer_t *distance21);
The splitjoin distance between partitions A and B is the sum of the projection distance of A from B and the projection distance of B from A. The projection distance is an asymmetric measure and it is defined as follows:
First, each set in partition A is evaluated against all sets in partition B. For each set in partition A, the best matching set in partition B is found and the overlap size is calculated. (Matching is quantified by the size of the overlap between the two sets). Then, the maximal overlap sizes for each set in A are summed together and subtracted from the number of elements in A.
The splitjoin distance will be returned in two arguments, distance12
will contain the projection distance of the first partition from the
second, while distance21
will be the projection distance of the second
partition from the first. This makes it easier to detect whether a
partition is a subpartition of the other, since in this case, the
corresponding distance will be zero.
Reference:
van Dongen S: Performance criteria for graph clustering and Markov cluster experiments. Technical Report INSR0012, National Research Institute for Mathematics and Computer Science in the Netherlands, Amsterdam, May 2000.
Arguments:

the membership vector of the first community structure 

the membership vector of the second community structure 

pointer to an 

pointer to an 
Returns:
Error code. 
See also:

Time complexity: O(n log(n)).
igraph_error_t igraph_community_spinglass(const igraph_t *graph, const igraph_vector_t *weights, igraph_real_t *modularity, igraph_real_t *temperature, igraph_vector_int_t *membership, igraph_vector_int_t *csize, igraph_integer_t spins, igraph_bool_t parupdate, igraph_real_t starttemp, igraph_real_t stoptemp, igraph_real_t coolfact, igraph_spincomm_update_t update_rule, igraph_real_t gamma, igraph_spinglass_implementation_t implementation, igraph_real_t gamma_minus);
This function implements the community structure detection algorithm proposed by Joerg Reichardt and Stefan Bornholdt. The algorithm is described in their paper: Statistical Mechanics of Community Detection, http://arxiv.org/abs/condmat/0603718 .
From version 0.6, igraph also supports an extension to the algorithm that allows negative edge weights. This is described in V. A. Traag and Jeroen Bruggeman: Community detection in networks with positive and negative links, http://arxiv.org/abs/0811.2329 .
Arguments:

The input graph, it may be directed but the direction of the edges is not used in the algorithm. 

The vector giving the edge weights, it may be 

Pointer to a real number, if not 

Pointer to a real number, if not 

Pointer to an initialized vector or 

Pointer to an initialized vector or 

Integer giving the number of spins, i.e. the maximum number of clusters. Usually it is not a program to give a high number here, the default was 25 in the original code. Even if the number of spins is high the number of clusters in the result might be small. 

A logical constant, whether to update all spins in
parallel. The default for this argument was 

Real number, the temperature at the start. The value of this argument was 1.0 in the original code. 

Real number, the algorithm stops at this temperature. The default was 0.01 in the original code. 

Real number, the cooling factor for the simulated annealing. The default was 0.99 in the original code. 

The type of the update rule. Possible values: 

Real number. The gamma parameter of the algorithm. This defines the weight of the missing and existing links in the quality function for the clustering. The default value in the original code was 1.0, which is equal weight to missing and existing edges. Smaller values make the existing links contibute more to the energy function which is minimized in the algorithm. Bigger values make the missing links more important. (If my understanding is correct.) 

Constant, chooses between the two
implementations of the spinglass algorithm that are included
in igraph. 

Real number. Parameter for the 
Returns:
Error code. 
See also:
igraph_community_spinglass_single() for calculating the community of a single vertex. 
Time complexity: TODO.
igraph_error_t igraph_community_spinglass_single(const igraph_t *graph, const igraph_vector_t *weights, igraph_integer_t vertex, igraph_vector_int_t *community, igraph_real_t *cohesion, igraph_real_t *adhesion, igraph_integer_t *inner_links, igraph_integer_t *outer_links, igraph_integer_t spins, igraph_spincomm_update_t update_rule, igraph_real_t gamma);
This function implements the community structure detection algorithm proposed by Joerg Reichardt and Stefan Bornholdt. It is described in their paper: Statistical Mechanics of Community Detection, http://arxiv.org/abs/condmat/0603718 .
This function calculates the community of a single vertex without calculating all the communities in the graph.
Arguments:

The input graph, it may be directed but the direction of the edges is not used in the algorithm. 

Pointer to a vector with the weights of the edges.
Alternatively 

The vertex ID of the vertex of which ths community is calculated. 

Pointer to an initialized vector, the result, the IDs of the vertices in the community of the input vertex will be stored here. The vector will be resized as needed. 

Pointer to a real variable, if not 

Pointer to a real variable, if not 

Pointer to an integer, if not 

Pointer to an integer, if not 

The number of spins to use, this can be higher than the actual number of clusters in the network, in which case some clusters will contain zero vertices. 

The type of the update rule. Possible values: 

Real number. The gamma parameter of the algorithm. This defined the weight of the missing and existing links in the quality function for the clustering. The default value in the original code was 1.0, which is equal weight to missing and existing edges. Smaller values make the existing links contibute more to the energy function which is minimized in the algorithm. Bigger values make the missing links more important. (If my understanding is correct.) 
Returns:
Error code. 
See also:
igraph_community_spinglass() for the traditional version of the algorithm. 
Time complexity: TODO.
igraph_community_leading_eigenvector
— Leading eigenvector community finding (proper version).igraph_community_leading_eigenvector_callback_t
— Callback for the leading eigenvector community finding method.igraph_le_community_to_membership
— Vertex membership from the leading eigenvector community structureThe function documented in these section implements the “leading eigenvector” method developed by Mark Newman and published in MEJ Newman: Finding community structure using the eigenvectors of matrices, Phys Rev E 74:036104 (2006).
The heart of the method is the definition of the modularity matrix, B, which is B=AP, A being the adjacency matrix of the (undirected) network, and P contains the probability that certain edges are present according to the “configuration model” In other words, a Pij element of P is the probability that there is an edge between vertices i and j in a random network in which the degrees of all vertices are the same as in the input graph.
The leading eigenvector method works by calculating the eigenvector of the modularity matrix for the largest positive eigenvalue and then separating vertices into two community based on the sign of the corresponding element in the eigenvector. If all elements in the eigenvector are of the same sign that means that the network has no underlying community structure. Check Newman's paper to understand why this is a good method for detecting community structure.
The leading eigenvector community structure detection method is
implemented in igraph_community_leading_eigenvector()
. After
the initial split, the following splits are done in a way to
optimize modularity regarding to the original network. Note that
any further refinement, for example using KernighanLin, as
proposed in Section V.A of Newman (2006), is not implemented here.
Example 24.2. File examples/simple/igraph_community_leading_eigenvector.c
#include <igraph.h> int main(void) { igraph_t g; igraph_matrix_int_t merges; igraph_vector_int_t membership; igraph_vector_t x; igraph_arpack_options_t options; /* Zachary Karate club */ igraph_small(&g, 0, IGRAPH_UNDIRECTED, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 10, 0, 11, 0, 12, 0, 13, 0, 17, 0, 19, 0, 21, 0, 31, 1, 2, 1, 3, 1, 7, 1, 13, 1, 17, 1, 19, 1, 21, 1, 30, 2, 3, 2, 7, 2, 8, 2, 9, 2, 13, 2, 27, 2, 28, 2, 32, 3, 7, 3, 12, 3, 13, 4, 6, 4, 10, 5, 6, 5, 10, 5, 16, 6, 16, 8, 30, 8, 32, 8, 33, 9, 33, 13, 33, 14, 32, 14, 33, 15, 32, 15, 33, 18, 32, 18, 33, 19, 33, 20, 32, 20, 33, 22, 32, 22, 33, 23, 25, 23, 27, 23, 29, 23, 32, 23, 33, 24, 25, 24, 27, 24, 31, 25, 31, 26, 29, 26, 33, 27, 33, 28, 31, 28, 33, 29, 32, 29, 33, 30, 32, 30, 33, 31, 32, 31, 33, 32, 33, 1); igraph_matrix_int_init(&merges, 0, 0); igraph_vector_int_init(&membership, 0); igraph_vector_init(&x, 0); igraph_arpack_options_init(&options); igraph_community_leading_eigenvector(&g, /*weights=*/ 0, &merges, &membership, 1, &options, /*modularity=*/ 0, /*start=*/ 0, /*eigenvalues=*/ 0, /*eigenvectors=*/ 0, /*history=*/ 0, /*callback=*/ 0, /*callback_extra=*/ 0); igraph_matrix_int_print(&merges); igraph_vector_int_print(&membership); printf("\n"); /* Make all the steps */ igraph_community_leading_eigenvector(&g, /*weights=*/ 0, &merges, &membership, igraph_vcount(&g), &options, /*modularity=*/ 0, /*start=*/ 0, /*eigenvalues=*/ 0, /*eigenvectors=*/ 0, /*history=*/ 0, /*callback=*/ 0, /*callback_extra=*/ 0); igraph_matrix_int_print(&merges); igraph_vector_int_print(&membership); igraph_vector_destroy(&x); igraph_vector_int_destroy(&membership); igraph_matrix_int_destroy(&merges); igraph_destroy(&g); return 0; }
igraph_error_t igraph_community_leading_eigenvector( const igraph_t *graph, const igraph_vector_t *weights, igraph_matrix_int_t *merges, igraph_vector_int_t *membership, igraph_integer_t steps, igraph_arpack_options_t *options, igraph_real_t *modularity, igraph_bool_t start, igraph_vector_t *eigenvalues, igraph_vector_list_t *eigenvectors, igraph_vector_t *history, igraph_community_leading_eigenvector_callback_t *callback, void *callback_extra);
Newman's leading eigenvector method for detecting community structure. This is the proper implementation of the recursive, divisive algorithm: each split is done by maximizing the modularity regarding the original network, see MEJ Newman: Finding community structure in networks using the eigenvectors of matrices, Phys Rev E 74:036104 (2006).
Arguments:

The undirected input graph. 


The weights of the edges, or a null pointer for unweighted graphs. 


The result of the algorithm, a matrix containing the
information about the splits performed. The matrix is built in
the opposite way however, it is like the result of an
agglomerative algorithm. If at the end of the algorithm (after



The membership of the vertices after all the
splits were performed will be stored here. The vector must be
initialized before calling and will be resized as needed.
This argument is ignored if it is 


The maximum number of steps to perform. It might happen that some component (or the whole network) has no underlying community structure and no further steps can be done. If you want as many steps as possible then supply the number of vertices in the network here. 


The options for ARPACK. Supply 


If not a null pointer, then it must be a pointer to a real number and the modularity score of the final division is stored here. 


Boolean, whether to use the community structure given
in the 


Pointer to an initialized vector or a null pointer. If not a null pointer, then the eigenvalues calculated along the community structure detection are stored here. The nonpositive eigenvalues, that do not result a split, are stored as well. 


If not a null pointer, then the eigenvectors
that are calculated in each step of the algorithm are stored here,
in a list of vectors. Each eigenvector is stored in an



Pointer to an initialized vector or a null pointer. If not a null pointer, then a trace of the algorithm is stored here, encoded numerically. The various operations:



A null pointer or a function of type 


Extra argument to pass to the callback function. 
Returns:
Error code. 
See also:

Time complexity: O(E+V^2*steps), V is the number of vertices, E the number of edges, “steps” the number of splits performed.
typedef igraph_error_t igraph_community_leading_eigenvector_callback_t( const igraph_vector_int_t *membership, igraph_integer_t comm, igraph_real_t eigenvalue, const igraph_vector_t *eigenvector, igraph_arpack_function_t *arpack_multiplier, void *arpack_extra, void *extra);
The leading eigenvector community finding implementation in igraph
is able to call a callback function, after each eigenvalue
calculation. This callback function must be of igraph_community_leading_eigenvector_callback_t
type.
The following arguments are passed to the callback:
Arguments:

The actual membership vector, before recording the potential change implied by the newly found eigenvalue. 

The id of the community that the algorithm tried to split in the last iteration. The community IDs are indexed from zero here! 

The eigenvalue the algorithm has just found. 

The eigenvector corresponding to the eigenvalue the algorithm just found. 

A function that was passed to 

The extra argument that was passed to the ARPACK solver. 

Extra argument that as passed to 
See also:
igraph_error_t igraph_le_community_to_membership(const igraph_matrix_int_t *merges, igraph_integer_t steps, igraph_vector_int_t *membership, igraph_vector_int_t *csize);
This function creates a membership vector from the
result of igraph_community_leading_eigenvector()
,
It takes membership
and performs steps
merges, according to the supplied
merges
matrix.
Arguments:

The twocolumn matrix containing the merge
operations. See 

The number of steps to make according to 

Initially the starting membership vector,
on output the resulting membership vector, after performing 

Optionally the sizes of the communities is stored here, if this is not a null pointer, but an initialized vector. 
Returns:
Error code. 
Time complexity: O(V), the number of vertices.
igraph_error_t igraph_community_walktrap(const igraph_t *graph, const igraph_vector_t *weights, igraph_integer_t steps, igraph_matrix_int_t *merges, igraph_vector_t *modularity, igraph_vector_int_t *membership);
finding algorithm, see Pascal Pons, Matthieu Latapy: Computing communities in large networks using random walks, https://arxiv.org/abs/physics/0512106
Currently the original C++ implementation is used in igraph, see https://wwwcomplexnetworks.lip6.fr/~latapy/PP/walktrap.html We are grateful to Matthieu Latapy and Pascal Pons for providing this source code.
In contrast to the original implementation, isolated vertices are allowed in the graph and they are assumed to have a single incident loop edge with weight 1.
Arguments:

The input graph, edge directions are ignored. 

Numeric vector giving the weights of the edges. If it is a NULL pointer then all edges will have equal weights. The weights are expected to be positive. 

Integer constant, the length of the random walks. Typically, good results are obtained with values between 38 with 45 being a reasonable default. 

Pointer to a matrix, the merges performed by the
algorithm will be stored here (if not NULL). Each merge is a
row in a twocolumn matrix and contains the IDs of the merged
clusters. Clusters are numbered from zero and cluster numbers
smaller than the number of nodes in the network belong to the
individual vertices as singleton clusters. In each step a new
cluster is created from two other clusters and its id will be
one larger than the largest cluster id so far. This means that
before the first merge we have 

Pointer to a vector. If not NULL then the modularity score of the current clustering is stored here after each merge operation. 

Pointer to a vector. If not a NULL pointer, then
the membership vector corresponding to the maximal modularity
score is stored here. If it is not a NULL pointer, then neither

Returns:
Error code. 
See also:
Time complexity: O(EV^2) in the worst case, O(V^2 logV) typically, V is the number of vertices, E is the number of edges.
Example 24.3. File examples/simple/walktrap.c
#include <igraph.h> int main(void) { igraph_t g; igraph_matrix_int_t merges; igraph_vector_t modularity; igraph_integer_t no_of_nodes; igraph_integer_t i; igraph_small(&g, 5, IGRAPH_UNDIRECTED, 0, 1, 0, 2, 0, 3, 0, 4, 1, 2, 1, 3, 1, 4, 2, 3, 2, 4, 3, 4, 5, 6, 5, 7, 5, 8, 5, 9, 6, 7, 6, 8, 6, 9, 7, 8, 7, 9, 8, 9, 0, 5, 4, 9, 1); igraph_vector_init(&modularity, 0); igraph_matrix_int_init(&merges, 0, 0); igraph_community_walktrap(&g, NULL /* no weights */, 4 /* steps */, &merges, &modularity, /* membership=*/ NULL); no_of_nodes = igraph_vcount(&g); printf("Merges:\n"); for (i = 0; i < igraph_matrix_int_nrow(&merges); i++) { printf("%2.1" IGRAPH_PRId " + %2." IGRAPH_PRId " > %2." IGRAPH_PRId " (modularity %4.2f)\n", MATRIX(merges, i, 0), MATRIX(merges, i, 1), no_of_nodes + i, VECTOR(modularity)[i]); } igraph_destroy(&g); igraph_matrix_int_destroy(&merges); igraph_vector_destroy(&modularity); return 0; }
igraph_error_t igraph_community_edge_betweenness(const igraph_t *graph, igraph_vector_int_t *result, igraph_vector_t *edge_betweenness, igraph_matrix_int_t *merges, igraph_vector_int_t *bridges, igraph_vector_t *modularity, igraph_vector_int_t *membership, igraph_bool_t directed, const igraph_vector_t *weights);
Community structure detection based on the betweenness of the edges in the network. The algorithm was invented by M. Girvan and M. Newman, see: M. Girvan and M. E. J. Newman: Community structure in social and biological networks, Proc. Nat. Acad. Sci. USA 99, 78217826 (2002). https://doi.org/10.1073/pnas.122653799
The idea is that the betweenness of the edges connecting two communities is typically high, as many of the shortest paths between nodes in separate communities go through them. So we gradually remove the edge with highest betweenness from the network, and recalculate edge betweenness after every removal. This way sooner or later the network splits into two components, then after a while one of these components splits again into two smaller components, and so on until all edges are removed. This is a divisive hierarchical approach, the result of which is a dendrogram.
In directed graphs, when directed
is set to true, the directed version
of betweenness and modularity are used, however, only splits into
weakly connected components are detected.
Arguments:

The input graph. 

Pointer to an initialized vector, the result will be
stored here, the IDs of the removed edges in the order of their
removal. It will be resized as needed. It may be 

Pointer to an initialized vector or


Pointer to an initialized matrix or 

Pointer to an initialized vector of 

If not a null pointer, then the modularity values
of the different divisions are stored here, in the order
corresponding to the merge matrix. The modularity values will
take weights into account if 

If not a null pointer, then the membership vector, corresponding to the highest modularity value, is stored here. 

Logical constant. Controls whether to calculate directed betweenness (i.e. directed paths) for directed graphs, and whether to use the directed version of modularity. It is ignored for undirected graphs. 

An optional vector containing edge weights. If null, the unweighted edge betweenness scores will be calculated and used. If not null, the weighted edge betweenness scores will be calculated and used. 
Returns:
Error code. 
See also:
Time complexity: O(VE^2), as the betweenness calculation requires O(VE) and we do it E1 times.
Example 24.4. File examples/simple/igraph_community_edge_betweenness.c
#include <igraph.h> int igraph_vector_between(const igraph_vector_t* v, const igraph_vector_t* lo, const igraph_vector_t* hi) { return igraph_vector_all_le(lo, v) && igraph_vector_all_ge(hi, v); } void test_unweighted(void) { igraph_t g; igraph_vector_int_t edges; igraph_vector_t eb; igraph_integer_t i; igraph_integer_t no_of_edges; /* Zachary Karate club */ igraph_small(&g, 0, IGRAPH_UNDIRECTED, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 10, 0, 11, 0, 12, 0, 13, 0, 17, 0, 19, 0, 21, 0, 31, 1, 2, 1, 3, 1, 7, 1, 13, 1, 17, 1, 19, 1, 21, 1, 30, 2, 3, 2, 7, 2, 8, 2, 9, 2, 13, 2, 27, 2, 28, 2, 32, 3, 7, 3, 12, 3, 13, 4, 6, 4, 10, 5, 6, 5, 10, 5, 16, 6, 16, 8, 30, 8, 32, 8, 33, 9, 33, 13, 33, 14, 32, 14, 33, 15, 32, 15, 33, 18, 32, 18, 33, 19, 33, 20, 32, 20, 33, 22, 32, 22, 33, 23, 25, 23, 27, 23, 29, 23, 32, 23, 33, 24, 25, 24, 27, 24, 31, 25, 31, 26, 29, 26, 33, 27, 33, 28, 31, 28, 33, 29, 32, 29, 33, 30, 32, 30, 33, 31, 32, 31, 33, 32, 33, 1); igraph_vector_int_init(&edges, 0); igraph_vector_init(&eb, 0); igraph_community_edge_betweenness(&g, &edges, &eb, 0 /*merges */, 0 /*bridges */, /*modularity=*/ 0, /*membership=*/ 0, IGRAPH_UNDIRECTED, /*weights=*/ 0); no_of_edges = igraph_ecount(&g); for (i = 0; i < no_of_edges; i++) { printf("%" IGRAPH_PRId " ", VECTOR(edges)[i]); } printf("\n"); for (i = 0; i < no_of_edges; i++) { printf("%.2f ", VECTOR(eb)[i]); } printf("\n"); /* Try it once again without storage space for edges */ igraph_community_edge_betweenness(&g, 0, &eb, 0 /*merges */, 0 /*bridges */, /*modularity=*/ 0, /*membership=*/ 0, IGRAPH_UNDIRECTED, /*weights=*/ 0); for (i = 0; i < no_of_edges; i++) { printf("%.2f ", VECTOR(eb)[i]); } printf("\n"); igraph_vector_destroy(&eb); igraph_vector_int_destroy(&edges); igraph_destroy(&g); } #define EPS 1e4 void test_weighted(void) { igraph_t g; igraph_vector_int_t edges; igraph_vector_t eb, weights; igraph_real_t weights_array[] = { 4, 1, 3, 2, 5, 8, 6, 7 }; igraph_integer_t edges_array1[] = { 2, 3, 0, 1, 4, 7, 5, 6 }; igraph_integer_t edges_array2[] = { 2, 3, 6, 5, 0, 1, 4, 7 }; igraph_real_t eb_array1_lo[] = { 4, 5, 3 + 1 / 3.0  EPS, 4, 2.5, 4, 1, 1 }; igraph_real_t eb_array1_hi[] = { 4, 5, 3 + 1 / 3.0 + EPS, 4, 2.5, 4, 1, 1 }; igraph_real_t eb_array2_lo[] = { 4, 5, 3 + 1 / 3.0  EPS, 6, 1.5, 2, 1, 1 }; igraph_real_t eb_array2_hi[] = { 4, 5, 3 + 1 / 3.0 + EPS, 6, 1.5, 2, 1, 1 }; igraph_vector_int_t edges_sol1, edges_sol2; igraph_vector_t eb_sol1_lo, eb_sol1_hi, eb_sol2_lo, eb_sol2_hi; igraph_vector_int_view(&edges_sol1, edges_array1, sizeof(edges_array1) / sizeof(edges_array1[0])); igraph_vector_int_view(&edges_sol2, edges_array2, sizeof(edges_array2) / sizeof(edges_array2[0])); igraph_vector_view(&eb_sol1_lo, eb_array1_lo, sizeof(eb_array1_lo) / sizeof(eb_array1_lo[0])); igraph_vector_view(&eb_sol2_lo, eb_array2_lo, sizeof(eb_array2_lo) / sizeof(eb_array2_lo[0])); igraph_vector_view(&eb_sol1_hi, eb_array1_hi, sizeof(eb_array1_hi) / sizeof(eb_array1_hi[0])); igraph_vector_view(&eb_sol2_hi, eb_array2_hi, sizeof(eb_array2_hi) / sizeof(eb_array2_hi[0])); /* Small graph as follows: ABCA, ADEA, BD, CE */ igraph_small(&g, 0, IGRAPH_UNDIRECTED, 0, 1, 0, 2, 0, 3, 0, 4, 1, 2, 1, 3, 2, 4, 3, 4, 1); igraph_vector_view(&weights, weights_array, igraph_ecount(&g)); igraph_vector_int_init(&edges, 0); igraph_vector_init(&eb, 0); igraph_community_edge_betweenness(&g, &edges, &eb, 0 /*merges */, 0 /*bridges */, /*modularity=*/ 0, /*membership=*/ 0, IGRAPH_UNDIRECTED, &weights); if (!igraph_vector_int_all_e(&edges_sol1, &edges) && !igraph_vector_int_all_e(&edges_sol2, &edges)) { printf("Error, edges vector was: \n"); igraph_vector_int_print(&edges); exit(2); } if (!igraph_vector_between(&eb, &eb_sol1_lo, &eb_sol1_hi) && !igraph_vector_between(&eb, &eb_sol2_lo, &eb_sol2_hi)) { printf("Error, eb vector was: \n"); igraph_vector_print(&eb); exit(2); } /* Try it once again without storage space for edges */ igraph_community_edge_betweenness(&g, 0, &eb, 0 /*merges */, 0 /*bridges */, /*modularity=*/ 0, /*membership=*/ 0, IGRAPH_UNDIRECTED, &weights); if (!igraph_vector_between(&eb, &eb_sol1_lo, &eb_sol1_hi) && !igraph_vector_between(&eb, &eb_sol2_lo, &eb_sol2_hi)) { printf("Error, eb vector was: \n"); igraph_vector_print(&eb); exit(2); } igraph_vector_destroy(&eb); igraph_vector_int_destroy(&edges); igraph_destroy(&g); } void test_zero_edge_graph(void) { igraph_t g; igraph_vector_t eb; igraph_vector_int_t res; igraph_full(&g, 1, 0, 0); igraph_vector_int_init(&res, igraph_ecount(&g)); igraph_vector_init(&eb, igraph_ecount(&g)); igraph_community_edge_betweenness(&g, &res, // result &eb, // edge_betweenness result NULL, // merges result NULL, // bridges NULL, // modularity NULL, // membership IGRAPH_UNDIRECTED, // directed NULL // weights ); igraph_vector_destroy(&eb); printf("No crash\n"); igraph_vector_int_destroy(&res); igraph_destroy(&g); } int main(void) { test_unweighted(); test_weighted(); test_zero_edge_graph(); return 0; }
igraph_error_t igraph_community_eb_get_merges(const igraph_t *graph, const igraph_bool_t directed, const igraph_vector_int_t *edges, const igraph_vector_t *weights, igraph_matrix_int_t *res, igraph_vector_int_t *bridges, igraph_vector_t *modularity, igraph_vector_int_t *membership);
This function is handy if you have a sequence of edges which are
gradually removed from the network and you would like to know how
the network falls apart into separate components. The edge sequence
may come from the igraph_community_edge_betweenness()
function, but this is not necessary. Note that igraph_community_edge_betweenness()
can also calculate the
dendrogram, via its merges
argument. Merges happen when the
edge removal process is run backwards and two components become
connected.
Arguments:

The input graph. 

Vector containing the edges to be removed from the network, all edges are expected to appear exactly once in the vector. 

Whether to use the directed or undirected version of modularity. Will be ignored for undirected graphs. 

An optional vector containing edge weights. If null,
the unweighted modularity scores will be calculated. If not null,
the weighted modularity scores will be calculated. Ignored if both


Pointer to an initialized matrix, if not 

Pointer to an initialized vector of 

If not a null pointer, then the modularity values for the different divisions, corresponding to the merges matrix, will be stored here. 

If not a null pointer, then the membership vector for the best division (in terms of modularity) will be stored here. 
Returns:
Error code. 
See also:
Time complexity: O(E+VlogV), V is the number of vertices, E is the number of edges.
igraph_error_t igraph_community_fastgreedy(const igraph_t *graph, const igraph_vector_t *weights, igraph_matrix_int_t *merges, igraph_vector_t *modularity, igraph_vector_int_t *membership);
This function implements the fast greedy modularity optimization algorithm for finding community structure, see A Clauset, MEJ Newman, C Moore: Finding community structure in very large networks, http://www.arxiv.org/abs/condmat/0408187 for the details.
Some improvements proposed in K Wakita, T Tsurumi: Finding community structure in megascale social networks, http://www.arxiv.org/abs/cs.CY/0702048v1 have also been implemented.
Arguments:

The input graph. It must be a graph without multiple edges. This is checked and an error message is given for graphs with multiple edges. 

Potentially a numeric vector containing edge weights. Supply a null pointer here for unweighted graphs. The weights are expected to be nonnegative. 

Pointer to an initialized matrix or 

Pointer to an initialized vector or 

Pointer to a vector. If not a null pointer, then the membership vector corresponding to the best split (in terms of modularity) is stored here. 
Returns:
Error code. 
See also:

Time complexity: O(EVlogV) in the worst case, O(E+Vlog^2V) typically, V is the number of vertices, E is the number of edges.
Example 24.5. File examples/simple/igraph_community_fastgreedy.c
#include <igraph.h> void show_results(igraph_t *g, igraph_vector_t *mod, igraph_matrix_int_t *merges, igraph_vector_int_t *membership, FILE* f) { igraph_integer_t i = 0; igraph_vector_int_t our_membership; igraph_vector_int_init(&our_membership, 0); if (mod != 0) { i = igraph_vector_which_max(mod); fprintf(f, "Modularity: "); igraph_real_fprintf(f, VECTOR(*mod)[i]); fprintf(f, "\n"); } else { fprintf(f, "Modularity: \n"); } if (membership != 0) { igraph_vector_int_update(&our_membership, membership); } else if (merges != 0) { igraph_community_to_membership(merges, igraph_vcount(g), i, &our_membership, 0); } printf("Membership: "); for (i = 0; i < igraph_vector_int_size(&our_membership); i++) { printf("%" IGRAPH_PRId " ", VECTOR(our_membership)[i]); } printf("\n"); igraph_vector_int_destroy(&our_membership); } int main(void) { igraph_t g; igraph_vector_t modularity, weights; igraph_vector_int_t membership; igraph_matrix_int_t merges; igraph_vector_init(&modularity, 0); igraph_matrix_int_init(&merges, 0, 0); igraph_vector_init(&weights, 0); igraph_vector_int_init(&membership, 0); /* Simple unweighted graph */ igraph_small(&g, 10, IGRAPH_UNDIRECTED, 0, 1, 0, 2, 0, 3, 0, 4, 1, 2, 1, 3, 1, 4, 2, 3, 2, 4, 3, 4, 5, 6, 5, 7, 5, 8, 5, 9, 6, 7, 6, 8, 6, 9, 7, 8, 7, 9, 8, 9, 0, 5, 1); igraph_community_fastgreedy(&g, 0, &merges, &modularity, /*membership=*/ 0); show_results(&g, &modularity, &merges, 0, stdout); /* Same simple graph, with uniform edge weights */ igraph_vector_resize(&weights, igraph_ecount(&g)); igraph_vector_fill(&weights, 2); igraph_community_fastgreedy(&g, &weights, &merges, &modularity, /*membership=*/ 0); show_results(&g, &modularity, &merges, 0, stdout); igraph_destroy(&g); /* Simple nonuniform weighted graph, with and without weights */ igraph_small(&g, 6, IGRAPH_UNDIRECTED, 0, 1, 1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 3, 5, 4, 5, 1); igraph_vector_resize(&weights, 8); igraph_vector_fill(&weights, 1); VECTOR(weights)[0] = 10; VECTOR(weights)[1] = 10; igraph_community_fastgreedy(&g, 0, &merges, &modularity, /*membership=*/ 0); show_results(&g, &modularity, &merges, 0, stdout); igraph_community_fastgreedy(&g, &weights, &merges, &modularity, /*membership=*/ 0); show_results(&g, &modularity, &merges, 0, stdout); igraph_destroy(&g); /* Zachary Karate club */ igraph_small(&g, 0, IGRAPH_UNDIRECTED, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 10, 0, 11, 0, 12, 0, 13, 0, 17, 0, 19, 0, 21, 0, 31, 1, 2, 1, 3, 1, 7, 1, 13, 1, 17, 1, 19, 1, 21, 1, 30, 2, 3, 2, 7, 2, 8, 2, 9, 2, 13, 2, 27, 2, 28, 2, 32, 3, 7, 3, 12, 3, 13, 4, 6, 4, 10, 5, 6, 5, 10, 5, 16, 6, 16, 8, 30, 8, 32, 8, 33, 9, 33, 13, 33, 14, 32, 14, 33, 15, 32, 15, 33, 18, 32, 18, 33, 19, 33, 20, 32, 20, 33, 22, 32, 22, 33, 23, 25, 23, 27, 23, 29, 23, 32, 23, 33, 24, 25, 24, 27, 24, 31, 25, 31, 26, 29, 26, 33, 27, 33, 28, 31, 28, 33, 29, 32, 29, 33, 30, 32, 30, 33, 31, 32, 31, 33, 32, 33, 1); igraph_community_fastgreedy(&g, 0, &merges, &modularity, /*membership=*/ 0); show_results(&g, &modularity, &merges, 0, stdout); igraph_destroy(&g); /* Simple disconnected graph with isolates */ igraph_small(&g, 9, IGRAPH_UNDIRECTED, 0, 1, 0, 2, 0, 3, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 4, 7, 5, 6, 5, 7, 6, 7, 1); igraph_community_fastgreedy(&g, 0, &merges, &modularity, /*membership=*/ 0); show_results(&g, &modularity, &merges, 0, stdout); igraph_destroy(&g); /* Disjoint union of two rings */ igraph_small(&g, 20, IGRAPH_UNDIRECTED, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 0, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 10, 19, 1); igraph_community_fastgreedy(&g, 0, &merges, &modularity, /*membership=*/ 0); show_results(&g, &modularity, &merges, 0, stdout); igraph_destroy(&g); /* Completely empty graph */ igraph_small(&g, 10, IGRAPH_UNDIRECTED, 1); igraph_community_fastgreedy(&g, 0, &merges, &modularity, /*membership=*/ 0); show_results(&g, &modularity, &merges, 0, stdout); igraph_destroy(&g); /* Ring graph with loop edges */ igraph_small(&g, 6, IGRAPH_UNDIRECTED, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 0, 0, 0, 2, 2, 1); igraph_community_fastgreedy(&g, 0, &merges, &modularity, /*membership=*/ 0); show_results(&g, &modularity, &merges, 0, stdout); igraph_destroy(&g); /* Regression test  graph with two vertices and two edges */ igraph_small(&g, 2, IGRAPH_UNDIRECTED, 0, 0, 1, 1, 1); igraph_community_fastgreedy(&g, 0, &merges, &modularity, /*membership=*/ 0); show_results(&g, &modularity, &merges, 0, stdout); igraph_destroy(&g); /* Regression test  asking for optimal membership vector but not * providing a modularity vector */ igraph_small(&g, 10, IGRAPH_UNDIRECTED, 0, 1, 0, 2, 0, 3, 0, 4, 1, 2, 1, 3, 1, 4, 2, 3, 2, 4, 3, 4, 5, 6, 5, 7, 5, 8, 5, 9, 6, 7, 6, 8, 6, 9, 7, 8, 7, 9, 8, 9, 0, 5, 1); igraph_community_fastgreedy(&g, 0, &merges, 0, &membership); show_results(&g, 0, &merges, &membership, stdout); igraph_destroy(&g); /* Regression test  asking for optimal membership vector but not * providing a merge matrix */ igraph_small(&g, 10, IGRAPH_UNDIRECTED, 0, 1, 0, 2, 0, 3, 0, 4, 1, 2, 1, 3, 1, 4, 2, 3, 2, 4, 3, 4, 5, 6, 5, 7, 5, 8, 5, 9, 6, 7, 6, 8, 6, 9, 7, 8, 7, 9, 8, 9, 0, 5, 1); igraph_community_fastgreedy(&g, 0, 0, &modularity, &membership); show_results(&g, &modularity, 0, &membership, stdout); /* Regression test  asking for optimal membership vector but not * providing a merge matrix or a modularity vector */ igraph_community_fastgreedy(&g, 0, 0, 0, &membership); show_results(&g, 0, 0, &membership, stdout); igraph_destroy(&g); igraph_vector_int_destroy(&membership); igraph_vector_destroy(&modularity); igraph_vector_destroy(&weights); igraph_matrix_int_destroy(&merges); return 0; }
igraph_error_t igraph_community_multilevel(const igraph_t *graph, const igraph_vector_t *weights, const igraph_real_t resolution, igraph_vector_int_t *membership, igraph_matrix_int_t *memberships, igraph_vector_t *modularity);
This function implements the multilevel modularity optimization algorithm for finding community structure, see Blondel, V. D., Guillaume, J.L., Lambiotte, R., & Lefebvre, E. (2008). Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment, 10008(10), 6. https://doi.org/10.1088/17425468/2008/10/P10008 for the details (preprint: http://arxiv.org/abs/0803.0476). The algorithm is sometimes known as the "Louvain" algorithm.
The algorithm is based on the modularity measure and a hierarchical approach. Initially, each vertex is assigned to a community on its own. In every step, vertices are reassigned to communities in a local, greedy way: in a random order, each vertex is moved to the community with which it achieves the highest contribution to modularity. When no vertices can be reassigned, each community is considered a vertex on its own, and the process starts again with the merged communities. The process stops when there is only a single vertex left or when the modularity cannot be increased any more in a step.
The resolution parameter gamma
allows finding communities at different
resolutions. Higher values of the resolution parameter typically result in
more, smaller communities. Lower values typically result in fewer, larger
communities. The original definition of modularity is retrieved when setting
gamma=1
. Note that the returned modularity value is calculated using
the indicated resolution parameter. See igraph_modularity()
for more details.
This function was contributed by Tom Gregorovic.
Arguments:

The input graph. It must be an undirected graph. 

Numeric vector containing edge weights. If 

Resolution parameter. Must be greater than or equal to 0. Lower values favor fewer, larger communities; higher values favor more, smaller communities. Set it to 1 to use the classical definition of modularity. 

The membership vector, the result is returned here. For each vertex it gives the ID of its community. The vector must be initialized and it will be resized accordingly. 

Numeric matrix that will contain the membership vector after
each level, if not 

Numeric vector that will contain the modularity score
after each level, if not 
Returns:
Error code. 
Time complexity: in average near linear on sparse graphs.
Example 24.6. File examples/simple/igraph_community_multilevel.c
#include <igraph.h> void show_results(igraph_t *g, igraph_vector_int_t *membership, igraph_matrix_int_t *memberships, igraph_vector_t *modularity, FILE* f) { igraph_integer_t i, j, no_of_nodes = igraph_vcount(g); j = igraph_vector_which_max(modularity); for (i = 0; i < igraph_vector_int_size(membership); i++) { if (VECTOR(*membership)[i] != MATRIX(*memberships, j, i)) { fprintf(f, "WARNING: best membership vector element %" IGRAPH_PRId " does not match the best one in the membership matrix\n", i); } } fprintf(f, "Modularities:\n"); igraph_vector_print(modularity); for (i = 0; i < igraph_matrix_int_nrow(memberships); i++) { for (j = 0; j < no_of_nodes; j++) { fprintf(f, "%" IGRAPH_PRId " ", MATRIX(*memberships, i, j)); } fprintf(f, "\n"); } fprintf(f, "\n"); } int main(void) { igraph_t g; igraph_vector_t modularity; igraph_vector_int_t edges; igraph_vector_int_t membership; igraph_matrix_int_t memberships; igraph_integer_t i, j, k; igraph_vector_init(&modularity, 0); igraph_vector_int_init(&membership, 0); igraph_matrix_int_init(&memberships, 0, 0); igraph_rng_seed(igraph_rng_default(), 42); /* Unweighted test graph from the paper of Blondel et al */ igraph_small(&g, 16, IGRAPH_UNDIRECTED, 0, 2, 0, 3, 0, 4, 0, 5, 1, 2, 1, 4, 1, 7, 2, 4, 2, 5, 2, 6, 3, 7, 4, 10, 5, 7, 5, 11, 6, 7, 6, 11, 8, 9, 8, 10, 8, 11, 8, 14, 8, 15, 9, 12, 9, 14, 10, 11, 10, 12, 10, 13, 10, 14, 11, 13, 1); igraph_community_multilevel(&g, 0, 1, &membership, &memberships, &modularity); show_results(&g, &membership, &memberships, &modularity, stdout); /* Higher resolution */ igraph_community_multilevel(&g, 0, 1.5, &membership, &memberships, &modularity); show_results(&g, &membership, &memberships, &modularity, stdout); igraph_destroy(&g); /* Ring of 30 cliques */ igraph_vector_int_init(&edges, 0); for (i = 0; i < 30; i++) { for (j = 0; j < 5; j++) { for (k = j + 1; k < 5; k++) { igraph_vector_int_push_back(&edges, i * 5 + j); igraph_vector_int_push_back(&edges, i * 5 + k); } } } for (i = 0; i < 30; i++) { igraph_vector_int_push_back(&edges, i * 5 % 150); igraph_vector_int_push_back(&edges, (i * 5 + 6) % 150); } igraph_create(&g, &edges, 150, 0); igraph_community_multilevel(&g, 0, 1, &membership, &memberships, &modularity); show_results(&g, &membership, &memberships, &modularity, stdout); igraph_destroy(&g); igraph_vector_destroy(&modularity); igraph_vector_int_destroy(&membership); igraph_vector_int_destroy(&edges); igraph_matrix_int_destroy(&memberships); return 0; }
igraph_error_t igraph_community_leiden(const igraph_t *graph, const igraph_vector_t *edge_weights, const igraph_vector_t *node_weights, const igraph_real_t resolution_parameter, const igraph_real_t beta, const igraph_bool_t start, const igraph_integer_t n_iterations, igraph_vector_int_t *membership, igraph_integer_t *nb_clusters, igraph_real_t *quality);
This function implements the Leiden algorithm for finding community structure, see Traag, V. A., Waltman, L., & van Eck, N. J. (2019). From Louvain to Leiden: guaranteeing wellconnected communities. Scientific reports, 9(1), 5233. http://dx.doi.org/10.1038/s4159801941695z
It is similar to the multilevel algorithm, often called the Louvain algorithm, but it is faster and yields higher quality solutions. It can optimize both modularity and the Constant Potts Model, which does not suffer from the resolutionlimit (see preprint http://arxiv.org/abs/1104.3083).
The Leiden algorithm consists of three phases: (1) local moving of nodes, (2) refinement of the partition and (3) aggregation of the network based on the refined partition, using the nonrefined partition to create an initial partition for the aggregate network. In the local move procedure in the Leiden algorithm, only nodes whose neighborhood has changed are visited. Only moves that strictly improve the quality function are made. The refinement is done by restarting from a singleton partition within each cluster and gradually merging the subclusters. When aggregating, a single cluster may then be represented by several nodes (which are the subclusters identified in the refinement).
The Leiden algorithm provides several guarantees. The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. At each iteration all clusters are guaranteed to be connected and wellseparated. After an iteration in which nothing has changed, all nodes and some parts are guaranteed to be locally optimally assigned. Note that even if a single iteration did not result in any change, it is still possible that a subsequent iteration might find some improvement. Each iteration explores different subsets of nodes to consider for moving from one cluster to another. Finally, asymptotically, all subsets of all clusters are guaranteed to be locally optimally assigned. For more details, please see Traag, Waltman & van Eck (2019).
The objective function being optimized is
1 / 2m sum_ij (A_ij  gamma n_i n_j)d(s_i, s_j)
where m is the total edge weight, A_ij is the weight of edge (i, j), gamma is
the socalled resolution parameter, n_i is the node weight of node i, s_i is
the cluster of node i and d(x, y) = 1 if and only if x = y and 0 otherwise.
By setting n_i = k_i, the degree of node i, and dividing gamma by 2m, you
effectively obtain an expression for modularity. Hence, the standard
modularity will be optimized when you supply the degrees as node_weights
and by supplying as a resolution parameter 1.0/(2*m), with m the number of
edges.
Arguments:

The input graph. It must be an undirected graph. 

Numeric vector containing edge weights. If 

Numeric vector containing node weights. If 

The resolution parameter used, which is represented by gamma in the objective function mentioned in the documentation. 

The randomness used in the refinement step when merging. A small
amount of randomness ( 

Start from membership vector. If this is true, the optimization will start from the provided membership vector. If this is false, the optimization will start from a singleton partition. 

Iterate the core Leiden algorithm for the indicated number of times. If this is a negative number, it will continue iterating until an iteration did not change the clustering. 

The membership vector. This is both used as the initial
membership from which optimisation starts and is updated in place. It
must hence be properly initialized. When finding clusters from scratch it
is typically started using a singleton clustering. This can be achieved
using 

The number of clusters contained in 

The quality of the partition, in terms of the objective
function as included in the documentation. If 
Returns:
Error code. 
Time complexity: near linear on sparse graphs.
Example 24.7. File examples/simple/igraph_community_leiden.c
#include <igraph.h> int main(void) { igraph_t graph; igraph_vector_int_t membership; igraph_vector_int_t degree; igraph_vector_t weights; igraph_integer_t nb_clusters, i; igraph_real_t quality; /* Set default seed to get reproducible results */ igraph_rng_seed(igraph_rng_default(), 0); /* Simple unweighted graph */ igraph_small(&graph, 10, IGRAPH_UNDIRECTED, 0, 1, 0, 2, 0, 3, 0, 4, 1, 2, 1, 3, 1, 4, 2, 3, 2, 4, 3, 4, 5, 6, 5, 7, 5, 8, 5, 9, 6, 7, 6, 8, 6, 9, 7, 8, 7, 9, 8, 9, 0, 5, 1); /* Perform Leiden algorithm using CPM for 1 iteration */ igraph_vector_int_init(&membership, igraph_vcount(&graph)); igraph_community_leiden(&graph, NULL, NULL, 0.05, 0.01, 0, 1, &membership, &nb_clusters, &quality); printf("Leiden found %" IGRAPH_PRId " clusters using CPM (resolution parameter 0.05), quality is %.4f.\n", nb_clusters, quality); printf("Membership: "); igraph_vector_int_print(&membership); printf("\n"); /* Start from existing membership for 10 iterations to improve it further */ igraph_community_leiden(&graph, NULL, NULL, 0.05, 0.01, 1, 10, &membership, &nb_clusters, &quality); printf("Iterated Leiden, using CPM (resolution parameter 0.05), quality is %.4f.\n", quality); printf("Membership: "); igraph_vector_int_print(&membership); printf("\n"); /* Initialize degree vector to use for optimizing modularity */ igraph_vector_int_init(°ree, igraph_vcount(&graph)); igraph_degree(&graph, °ree, igraph_vss_all(), IGRAPH_ALL, 1); igraph_vector_init(&weights, igraph_vector_int_size(°ree)); for (i = 0; i < igraph_vector_int_size(°ree); i++) { VECTOR(weights)[i] = VECTOR(degree)[i]; } /* Perform Leiden algorithm using modularity until stable iteration */ igraph_community_leiden(&graph, NULL, &weights, 1.0 / (2 * igraph_ecount(&graph)), 0.01, 0, 1, &membership, &nb_clusters, &quality); printf("Leiden found %" IGRAPH_PRId " clusters using modularity, quality is %.4f.\n", nb_clusters, quality); printf("Membership: "); igraph_vector_int_print(&membership); printf("\n"); igraph_vector_destroy(&weights); igraph_vector_int_destroy(°ree); igraph_vector_int_destroy(&membership); igraph_destroy(&graph); return 0; }
igraph_error_t igraph_community_fluid_communities(const igraph_t *graph, igraph_integer_t no_of_communities, igraph_vector_int_t *membership);
The algorithm is based on the simple idea of several fluids interacting in a nonhomogeneous environment (the graph topology), expanding and contracting based on their interaction and density. Weighted graphs are not supported.
This function implements the community detection method described in: Parés F, Gasulla DG, et. al. (2018) Fluid Communities: A Competitive, Scalable and Diverse Community Detection Algorithm. In: Complex Networks & Their Applications VI: Proceedings of Complex Networks 2017 (The Sixth International Conference on Complex Networks and Their Applications), Springer, vol 689, p 229. https://doi.org/10.1007/9783319721507_19
Arguments:

The input graph. The graph must be simple and connected. Edge directions will be ignored. 

The number of communities to be found. Must be greater than 0 and fewer than number of vertices in the graph. 

The result vector mapping vertices to the communities they are assigned to. 

If not a null pointer, then it must be a pointer to a real number. The modularity score of the detected community structure is stored here. 
Returns:
Error code. 
Time complexity: O(E)
igraph_error_t igraph_community_label_propagation(const igraph_t *graph, igraph_vector_int_t *membership, igraph_neimode_t mode, const igraph_vector_t *weights, const igraph_vector_int_t *initial, const igraph_vector_bool_t *fixed);
This function implements the label propagationbased community detection algorithm described by Raghavan, Albert and Kumara. This version extends the original method by the ability to take edge weights into consideration and also by allowing some labels to be fixed.
Weights are taken into account as follows: when the new label of node
i
is determined, the algorithm iterates over all edges incident on
node i
and calculate the total weight of edges leading to other
nodes with label 0, 1, 2, ..., k
 1 (where k
is the number of possible
labels). The new label of node i
will then be the label whose edges
(among the ones incident on node i
) have the highest total weight.
For directed graphs, it is important to know that labels can circulate freely only within the strongly connected components of the graph and may propagate in only one direction (or not at all) between strongly connected components. You should treat directed edges as directed only if you are aware of the consequences.
References:
Raghavan, U.N. and Albert, R. and Kumara, S.: Near linear time algorithm to detect community structures in largescale networks. Phys Rev E 76, 036106 (2007). https://doi.org/10.1103/PhysRevE.76.036106
Šubelj, L.: Label propagation for clustering. Chapter in "Advances in Network Clustering and Blockmodeling" edited by P. Doreian, V. Batagelj & A. Ferligoj (Wiley, New York, 2018). https://doi.org/10.1002/9781119483298.ch5 https://arxiv.org/abs/1709.05634
Arguments:

The input graph. Note that the algorithm wsa originally
defined for undirected graphs. You are advised to set 

The membership vector, the result is returned here. For each vertex it gives the ID of its community (label). 

Whether to consider edge directions for the label propagation,
and if so, which direction the labels should propagate. Ignored for
undirected graphs. 

The weight vector, it should contain a positive weight for all the edges. 

The initial state. If 

Boolean vector denoting which labels are fixed. Of course this makes sense only if you provided an initial state, otherwise this element will be ignored. Note that vertices without labels cannot be fixed. The fixed status will be ignord for these with a warning. Also note that label numbers by themselves have no meaning, and igraph may renumber labels. However, comembership constraints will be respected: two vertices can be fixed to be in the same or in different communities. 

If not a null pointer, then it must be a pointer
to a real number. The modularity score of the detected community
structure is stored here. Note that igraph will calculate the
directed modularity if the input graph is directed, even if
you set 
Returns:
Error code. 
Time complexity: O(m+n)
Example 24.8. File examples/simple/igraph_community_label_propagation.c
#include <igraph.h> #include <stdio.h> int main(void) { igraph_t graph; igraph_vector_int_t membership; igraph_real_t modularity; igraph_famous(&graph, "Zachary"); /* We use Zachary's karate club network. */ /* Label propagation is a stochastic method; the result will depend on the random seed. */ igraph_rng_seed(igraph_rng_default(), 123); /* All igraph functions that returns their result in an igraph_vector_t must be given an already initialized vector. */ igraph_vector_int_init(&membership, 0); igraph_community_label_propagation( &graph, &membership, /* mode = */ IGRAPH_ALL, /* weights= */ NULL, /* initial= */ NULL, /* fixed= */ NULL ); /* Also calculate the modularity of the partition */ igraph_modularity( &graph, &membership, /* weights= */ NULL, /* resolution = */ 1, /* directed= */ 0, &modularity); printf("%" IGRAPH_PRId " communities found; modularity score is %g.\n", igraph_vector_int_max(&membership) + 1, modularity); printf("Communities membership: "); igraph_vector_int_print(&membership); /* Destroy data structures at the end. */ igraph_vector_int_destroy(&membership); igraph_destroy(&graph); return 0; }
igraph_error_t igraph_community_infomap(const igraph_t * graph, const igraph_vector_t *e_weights, const igraph_vector_t *v_weights, igraph_integer_t nb_trials, igraph_vector_int_t *membership, igraph_real_t *codelength);
Implementation of the InfoMap community detection algorithm of Martin Rosvall and Carl T. Bergstrom.
For more details, see the visualization of the math and the map generator at https://www.mapequation.org . The original paper describing the algorithm is: M. Rosvall and C. T. Bergstrom, Maps of information flow reveal community structure in complex networks, PNAS 105, 1118 (2008) (http://dx.doi.org/10.1073/pnas.0706851105, http://arxiv.org/abs/0707.0609). A more detailed paper about the algorithm is: M. Rosvall, D. Axelsson, and C. T. Bergstrom, The map equation, Eur. Phys. J. Special Topics 178, 13 (2009). (http://dx.doi.org/10.1140/epjst/e2010011791, http://arxiv.org/abs/0906.1405)
The original C++ implementation of Martin Rosvall is used, see http://www.tp.umu.se/~rosvall/downloads/infomap_undir.tgz . Integration in igraph was done by Emmanuel Navarro (who is grateful to Martin Rosvall and Carl T. Bergstrom for providing this source code).
Note that the graph must not contain isolated vertices.
If you want to specify a random seed (as in the original
implementation) you can use igraph_rng_seed()
.
Arguments:

The input graph. 

Numeric vector giving the weights of the edges. The random walker will favour edges with high weights over edges with low weights; the probability of picking a particular outbound edge from a node is directly proportional to its weight. If it is a NULL pointer then all edges will have equal weights. The weights are expected to be positive. 

Numeric vector giving the weights of the vertices. Vertices with higher weights are favoured by the random walker when it needs to "teleport" to a new node after getting stuck in a sink node (i.e. a node with no outbound edges). The probability of picking a vertex when the random walker teleports is directly proportional to the weight of the vertex. If this argument is a NULL pointer then all vertices will have equal weights. Weights are expected to be positive. 

The number of attempts to partition the network (can be any integer value equal or larger than 1). 

Pointer to a vector. The membership vector is stored here. 

Pointer to a real. If not NULL the code length of the partition is stored here. 
Returns:
Error code. 
See also:
Time complexity: TODO.
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