For using the igraph C library
Community detection is concerned with clustering the vertices of networks into tightly connected subgraphs called "communities". The following references provide a good introduction to the topic of community detection:
S. Fortunato: "Community Detection in Graphs". Physics Reports 486, no. 3–5 (2010): 75–174. https://doi.org/16/j.physrep.2009.11.002.
S. Fortunato and D. Hric: "Community Detection in Networks: A User Guide". Physics Reports 659 (2016): 1–44. https://doi.org/10.1016/j.physrep.2016.09.002.
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 value.igraph_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 split-join 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 self-loops.
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 out-degree of node i and k^in_j is the in-degree 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 well-defined 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 resolution 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 out-degree of node i and k^in_j is the
in-degree of node j.
Note that self-loops 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):172-188, 2008 https://doi.org/10.1109/TKDE.2007.190689.
Note that exact modularity optimization is an NP-complete 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 may be undirected or directed. |
|
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 n-1 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 two-column 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 C-1, 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 in-place. |
|
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 split-join 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.
igraph uses natural units for the variation of information, i.e. it uses the
natural logarithm when computing entropies.
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 split-join 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: 978-3-540-40720-1. https://doi.org/10.1007/978-3-540-45167-9_14
Danon L, Diaz-Guilera A, Duch J, Arenas A: Comparing community structure identification. J Stat Mech P09008, 2005. https://doi.org/10.1088/1742-5468/2005/09/P09008
van Dongen S: Performance criteria for graph clustering and Markov cluster experiments. Technical Report INS-R0012, 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):846-850, 1971. https://doi.org/10.2307/2284239
Hubert L and Arabie P: Comparing partitions. Journal of Classification 2:193-218, 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 split-join 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 split-join 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 INS-R0012, 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/cond-mat/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 ignored by 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. 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. It is not implemented in the |
|
Real number, the temperature at the start. A reasonable default is 1.0. |
|
Real number, the algorithm stops at this temperature. A reasonable default is 0.01. |
|
Real number, the cooling factor for the simulated annealing. A reasonable default is 0.99. |
|
The type of the update rule. Possible values: |
|
Real number. The gamma parameter of the algorithm, acting as a resolution parameter. Smaller values typically lead to larger clusters, larger values typically lead to smaller clusters. |
|
Constant, chooses between the two
implementations of the spin-glass algorithm that are included
in igraph. |
|
Real number. Parameter for the |
Returns:
Error code. |
See also:
|
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_real_t *inner_links,
igraph_real_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/cond-mat/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 structure.The 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 = A - P, 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 P_ij 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. See
igraph_modularity_matrix() for more details.
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 Kernighan-Lin, 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> #include <stdio.h> int main(void) { igraph_t g; igraph_matrix_int_t merges; igraph_vector_int_t membership; igraph_vector_t x; /* 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_community_leading_eigenvector(&g, /*weights=*/ NULL, &merges, &membership, /*stpes=*/ 1, /*options=*/ NULL, /*modularity=*/ NULL, /*start=*/ NULL, /*eigenvalues=*/ NULL, /*eigenvectors=*/ NULL, /*history=*/ NULL, /*callback=*/ NULL, /*callback_extra=*/ NULL); igraph_matrix_int_print(&merges); igraph_vector_int_print(&membership); printf("\n"); /* Make all the steps */ igraph_community_leading_eigenvector(&g, /*weights=*/ NULL, &merges, &membership, /*steps=*/ igraph_vcount(&g), /*options=*/ NULL, /*modularity=*/ NULL, /*start=*/ NULL, /*eigenvalues=*/ NULL, /*eigenvectors=*/ NULL, /*history=*/ NULL, /*callback=*/ NULL, /*callback_extra=*/ NULL); 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). https://doi.org/10.1103/PhysRevE.74.036104
Arguments:
|
The input graph. Edge directions will be ignored. |
||||||||
|
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. Unlike with most other hierarchicaly
community detection functions in igraph, the integers in this matrix
represent community indices, not vertex indices. 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 non-positive 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 two-column 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 are 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);
This function is the implementation of the Walktrap community 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://www-complexnetworks.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 3-8 with 4-5 being a reasonable default. |
|
Pointer to a matrix, the merges performed by the
algorithm will be stored here (if not |
|
Pointer to a vector. If not |
|
Pointer to a vector. If not a |
Returns:
Error code. |
See also:
Time complexity: O(|E||V|^2) in the worst case, O(|V|^2 log|V|) 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 *removed_edges,
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,
const igraph_vector_t *lengths);
Community structure detection based on the betweenness of the edges in the network. This method is also known as the Girvan-Newman algorithm.
The idea behind this method is that the betweenness of the edges connecting two communities is typically high, as many of the shortest paths between vertices in separate communities pass through them. The algorithm successively removes edges with the highest betweenness, recalculating betweenness values after each removal. This way eventually the network splits into two components, then one of these components splits again, and so on, until all edges are removed. The resulting hierarhical partitioning of the vertices can be encoded as 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.
When edge weights are given, the ratio of betweenness and weight values is used to choose which edges to remove first, as described in M. E. J. Newman: Analysis of Weighted Networks (2004), Section C. Thus, edges with large weights are treated as strong connections, and will be removed later than weak connections having similar betweenness. Weights are also used for calculating modularity.
If lengths are given, they will be considered for shortest path length calculations while computing betweenness values.
Note: In igraph 0.10, this function interpreted weights in a different, erroneous way, and issued a warning when weights were used. Please see https://github.com/igraph/igraph/issues/2229 for additional details.
References:
M. Girvan and M. E. J. Newman, Community Structure in Social and Biological Networks, PNAS 99, 7821 (2002). https://doi.org/10.1073/pnas.122653799
M. E. J. Newman, Analysis of Weighted Networks, Phys. Rev. E 70, 9 (2004). https://doi.org/10.1103/PhysRevE.70.056131
Arguments:
|
The input graph. |
|
Pointer to an initialized integer vector, which will
be resized as needed. The IDs of the removed edges in the order of their
removal will be stored here. This vector is suitable as input to
|
|
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 not |
|
An optional vector containing edge lengths. If not |
Returns:
Error code. |
See also:
Time complexity: O(|V||E|^2), as the betweenness calculation requires O(|V||E|) and we do it |E|-1 times.
Example 24.4. File examples/simple/igraph_community_edge_betweenness.c
#include <igraph.h> int main(void) { igraph_t graph; igraph_vector_int_t edges; igraph_vector_t eb, weights; igraph_matrix_int_t merges; igraph_vector_int_t bridges; igraph_small(&graph, 5, IGRAPH_UNDIRECTED, 0, 1, 1, 2, 0, 2, 0, 3, 1, 3, 1, 4, -1); igraph_vector_init_range(&weights, 1, igraph_ecount(&graph)+1); igraph_vector_init(&eb, 0); igraph_vector_int_init(&edges, 0); igraph_matrix_int_init(&merges, 0, 0); igraph_vector_int_init(&bridges, 0); igraph_community_edge_betweenness(&graph, &edges, &eb, &merges, &bridges, /*modularity*/ NULL, /*membership*/ NULL, IGRAPH_UNDIRECTED, &weights, NULL); printf("edges:\n"); igraph_vector_int_print(&edges); printf("edge betweenness:\n"); igraph_vector_print(&eb); printf("merges:\n"); igraph_matrix_int_print(&merges); printf("bridges:\n"); igraph_vector_int_print(&bridges); igraph_destroy(&graph); igraph_vector_int_destroy(&edges); igraph_vector_destroy(&eb); igraph_vector_destroy(&weights); igraph_matrix_int_destroy(&merges); igraph_vector_int_destroy(&bridges); 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|+|V|log|V|), |V| is the number of vertices, |E| is the number of edges.
igraph_community_fastgreedy — Finding community structure by greedy optimization of modularity.igraph_community_multilevel — Finding community structure by multi-level optimization of modularity (Louvain).igraph_community_leiden — Finding community structure using the Leiden algorithm.
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/cond-mat/0408187 for the details.
Some improvements proposed in K Wakita, T Tsurumi: Finding community structure in mega-scale 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 non-negative. |
|
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(|E||V|log|V|) in the worst case, O(|E|+|V|log^2|V|) 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> int main(void) { igraph_t graph; igraph_vector_t weights; igraph_matrix_int_t merges; igraph_matrix_int_init(&merges, 0, 0); igraph_vector_init_int(&weights, 8, 10, 10, 1, 1, 1, 1, 1, 1); igraph_small(&graph, 6, IGRAPH_UNDIRECTED, 0,1, 1,2, 2,3, 2,4, 2,5, 3,4, 3,5, 4,5, -1); igraph_community_fastgreedy(&graph, &weights, &merges, /*modularity*/ NULL, /*membership=*/ NULL); igraph_matrix_int_print(&merges); igraph_destroy(&graph); 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 a multi-level modularity optimization algorithm for finding community structure, 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 re-assigned 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 γ 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
γ=1. Note that the returned modularity value is calculated using
the indicated resolution parameter. See igraph_modularity() for more details.
The original version of this function was contributed by Tom Gregorovic.
Reference:
Blondel, V. D., Guillaume, J.-L., Lambiotte, R., & Lefebvre, E.: Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment, 10008(10), 6 (2008). https://doi.org/10.1088/1742-5468/2008/10/P10008
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.
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 resolution-limit (see Tragg, Van Dooren & Nesterov).
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 non-refined 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 well-separated. 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 - γ n_i n_j) δ(s_i, s_j)
where m is the total edge weight, A_ij is the weight of edge
(i, j), γ is the so-called resolution parameter, n_i
is the node weight of node i, s_i is the cluster of node
i and δ(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 γ by 2m, we 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/(2m), with m the number of edges.
References:
V. A. Traag, L. Waltman, N. J. van Eck: From Louvain to Leiden: guaranteeing well-connected communities. Scientific Reports, 9(1), 5233 (2019). http://dx.doi.org/10.1038/s41598-019-41695-z
V. A. Traag, P. Van Dooren, and Y. Nesterov: Narrow scope for resolution-limit-free community detection. Phys. Rev. E 84, 016114 (2011). https://doi.org/10.1103/PhysRevE.84.016114
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> #include <stdio.h> int main(void) { igraph_t graph; igraph_vector_int_t membership; igraph_vector_t weights; igraph_integer_t nb_clusters; 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"); /* Use degrees as vertex weights for optimizing modularity */ igraph_vector_init(&weights, igraph_vcount(&graph)); igraph_strength(&graph, &weights, igraph_vss_all(), IGRAPH_ALL, /*loops*/ true, NULL); /* 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(&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 non-homogeneous 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/10.1007/978-3-319-72150-7_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,
igraph_lpa_variant_t lpa_variant);
This function implements the label propagation-based community detection algorithm described by Raghavan, Albert and Kumara (2007). This version extends the original method by the ability to take edge weights into consideration and also by allowing some labels to be fixed. In addition, it implements the fast label propagation alternative introduced by Traag and Šubelj (2023).
The algorithm works by iterating over nodes and updating the label of a node based on the labels of its neighbors. The labels that are most frequent among the neighbors are said to be dominant labels. The label of a node is always updated to a dominant label. The algorithm guarantees that the label for each is dominant when it terminates.
There are several variants implemented, which work slightly differently with
the dominance of labels. Nodes with a dominant label might no longer have a
dominant label if one of their neighbors change label. In IGRAPH_LPA_DOMINANCE an additional iteration over all nodes is made after
updating all labels to double check whether all nodes indeed have a dominant
label. When updating the label of a node, labels are always sampled from
among all dominant labels. The algorithm stops when all nodes have dominant
labels. In IGRAPH_LPA_RETENTION instead labels are only updated when they
are not dominant. That is, they retain their current label whenever the
current label is already dominant. The algorithm does not make an additional
iteration to check for dominance. Instead, it simply keeps track whether a
label has been updated, and terminates if no updates have been made. In IGRAPH_LPA_FAST labels are sampled from among all dominant labels, similar to
IGRAPH_LPA_DOMINANCE. Instead of iterating over all nodes, it keeps track
of a queue of nodes that should be considered. Nodes are popped from the
queue when they are considered for update. When the label of a node is
updated, the node's neighbors are added to the queue again (if they weren't
already in the queue). The algorithm terminates when the queue is empty. All
variants guarantee that the labels for all nodes are dominant.
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 large-scale 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
Traag, V. A., and Šubelj, L.: Large network community detection by fast label propagation. Scientific Reports, 13:1, (2023). https://doi.org/10.1038/s41598-023-29610-z https://arxiv.org/abs/2209.13338
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 ignored for these with a warning. Also note that label numbers by themselves have no meaning, and igraph may renumber labels. However, co-membership constraints will be respected: two vertices can be fixed to be in the same or in different communities. |
|
Which variant of label propagation algorithm to run.
One of
|
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, IGRAPH_LPA_DOMINANCE); /* Also calculate the modularity of the partition */ igraph_modularity( &graph, &membership, /* weights= */ NULL, /* resolution = */ 1, /* directed= */ false, &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. This algorithm takes edge directions into account.
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) (https://dx.doi.org/10.1073/pnas.0706851105, https://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). (https://dx.doi.org/10.1140/epjst/e2010-01179-1, https://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. Edge directions are taken into account. |
|
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 |
|
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 |
|
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.
igraph_error_t igraph_community_voronoi(
const igraph_t *graph,
igraph_vector_int_t *membership, igraph_vector_int_t *generators,
igraph_real_t *modularity,
const igraph_vector_t *lengths, const igraph_vector_t *weights,
igraph_neimode_t mode, igraph_real_t r);
This function is experimental and its signature is not considered final yet. We reserve the right to change the function signature without changing the major version of igraph. Use it at your own risk.
This function finds communities using a Voronoi partitioning of vertices based
on the given edge lengths divided by the edge clustering coefficient
(igraph_ecc()). The generator vertices are chosen to be those with the
largest local relative density within a radius r, with the local relative
density of a vertex defined as
s m / (m + k), where s is the strength of the vertex,
m is the number of edges within the vertex's first order neighborhood,
while k is the number of edges with only one endpoint within this
neighborhood.
References:
Deritei et al., Community detection by graph Voronoi diagrams, New Journal of Physics 16, 063007 (2014) https://doi.org/10.1088/1367-2630/16/6/063007
Molnár et al., Community Detection in Directed Weighted Networks using Voronoi Partitioning, Scientific Reports 14, 8124 (2024) https://doi.org/10.1038/s41598-024-58624-4
Arguments:
|
The input graph. It must be simple. |
|
If not |
|
If not |
|
If not |
|
Edge lengths, or |
|
Edge weights, or |
|
If |
|
The radius/resolution to use when selecting generator points. The larger this value, the fewer partitions there will be. Pass in a negative value to automatically select the radius that maximizes modularity. |
Returns:
Error code. |
See also:
Time complexity: TODO.
| ← Chapter 23. Vertex separators | Chapter 25. Graphlets → |