Functions to find cliques, i.e. complete subgraphs in a graph

These functions find all, the largest or all the maximal cliques in an undirected graph. The size of the largest clique can also be calculated.

Tests if all pairs within a set of vertices are adjacent, i.e. whether they form a clique. An empty set and singleton set are considered to be a clique.

cliques(graph, min = 0, max = 0)

largest_cliques(graph)

max_cliques(graph, min = NULL, max = NULL, subset = NULL, file = NULL)

count_max_cliques(graph, min = NULL, max = NULL, subset = NULL)

clique_num(graph)

largest_weighted_cliques(graph, vertex.weights = NULL)

weighted_clique_num(graph, vertex.weights = NULL)

clique_size_counts(graph, min = 0, max = 0, maximal = FALSE)

is_clique(graph, candidate, directed = FALSE)

Arguments

graph

The input graph.

min

Numeric constant, lower limit on the size of the cliques to find. NULL means no limit, i.e. it is the same as 0.

max

Numeric constant, upper limit on the size of the cliques to find. NULL means no limit.

subset

If not NULL, then it must be a vector of vertex ids, numeric or symbolic if the graph is named. The algorithm is run from these vertices only, so only a subset of all maximal cliques is returned. See the Eppstein paper for details. This argument makes it possible to easily parallelize the finding of maximal cliques.

file

If not NULL, then it must be a file name, i.e. a character scalar. The output of the algorithm is written to this file. (If it exists, then it will be overwritten.) Each clique will be a separate line in the file, given with the numeric ids of its vertices, separated by whitespace.

vertex.weights

Vertex weight vector. If the graph has a weight vertex attribute, then this is used by default. If the graph does not have a weight vertex attribute and this argument is NULL, then every vertex is assumed to have a weight of 1. Note that the current implementation of the weighted clique finder supports positive integer weights only.

maximal

Specifies whether to look for all weighted cliques (FALSE) or only the maximal ones (TRUE).

candidate

The vertex set to test for being a clique.

directed

Whether to consider edge directions.

Value

cliques(), largest_cliques() and clique_num() return a list containing numeric vectors of vertex ids. Each list element is a clique, i.e. a vertex sequence of class igraph.vs.

max_cliques() returns NULL, invisibly, if its file argument is not NULL. The output is written to the specified file in this case.

clique_num() and count_max_cliques() return an integer scalar.

clique_size_counts() returns a numeric vector with the clique sizes such that the i-th item belongs to cliques of size i. Trailing zeros are currently truncated, but this might change in future versions.

is_clique() returns TRUE if the candidate vertex set forms a clique.

Details

cliques() find all complete subgraphs in the input graph, obeying the size limitations given in the min and max arguments.

largest_cliques() finds all largest cliques in the input graph. A clique is largest if there is no other clique including more vertices.

max_cliques() finds all maximal cliques in the input graph. A clique is maximal if it cannot be extended to a larger clique. The largest cliques are always maximal, but a maximal clique is not necessarily the largest.

count_max_cliques() counts the maximal cliques.

clique_num() calculates the size of the largest clique(s).

clique_size_counts() returns a numeric vector representing a histogram of clique sizes, between the given minimum and maximum clique size.

is_clique() tests whether all pairs within a vertex set are connected.

References

For maximal cliques the following algorithm is implemented: David Eppstein, Maarten Loffler, Darren Strash: Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time. https://arxiv.org/abs/1006.5440

See also

Other cliques: is_complete(), ivs(), weighted_cliques()

Author

Tamas Nepusz ntamas@gmail.com and Gabor Csardi csardi.gabor@gmail.com

Related documentation in the C library

cliques(), largest_cliques(), clique_number(), largest_weighted_cliques(), weighted_clique_number(), maximal_cliques_hist(), clique_size_hist(), is_clique().

Examples

# this usually contains cliques of size six
g <- sample_gnp(100, 0.3)
clique_num(g)
#> [1] 7
cliques(g, min = 6)
#> [[1]]
#> + 6/100 vertices, from 2470883:
#> [1] 27 34 39 62 63 97
#> 
#> [[2]]
#> + 6/100 vertices, from 2470883:
#> [1] 27 39 62 63 76 97
#> 
#> [[3]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 27 39 63 76 97
#> 
#> [[4]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 18 27 63 76 97
#> 
#> [[5]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 32 63 76 81 97
#> 
#> [[6]]
#> + 6/100 vertices, from 2470883:
#> [1]  3 40 63 69 78 97
#> 
#> [[7]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 32 63 78 81 97
#> 
#> [[8]]
#> + 6/100 vertices, from 2470883:
#> [1]   8  22  93  95  96 100
#> 
#> [[9]]
#> + 6/100 vertices, from 2470883:
#> [1]  1 56 58 93 97 98
#> 
#> [[10]]
#> + 6/100 vertices, from 2470883:
#> [1] 13 23 49 56 58 80
#> 
#> [[11]]
#> + 6/100 vertices, from 2470883:
#> [1] 10 13 23 49 56 80
#> 
#> [[12]]
#> + 6/100 vertices, from 2470883:
#> [1] 13 23 30 49 56 80
#> 
#> [[13]]
#> + 6/100 vertices, from 2470883:
#> [1]  2  4 21 32 78 81
#> 
#> [[14]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 21 32 78 81 97
#> 
#> [[15]]
#> + 6/100 vertices, from 2470883:
#> [1]  8 21 32 44 67 68
#> 
#> [[16]]
#> + 6/100 vertices, from 2470883:
#> [1] 24 25 52 59 88 97
#> 
#> [[17]]
#> + 6/100 vertices, from 2470883:
#> [1] 37 62 66 73 79 97
#> 
#> [[18]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 14 66 78 81 97
#> 
#> [[19]]
#> + 6/100 vertices, from 2470883:
#> [1] 14 40 63 78 81 97
#> 
#> [[20]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 14 32 63 78 81
#> 
#> [[21]]
#> + 7/100 vertices, from 2470883:
#> [1]  4 14 32 63 78 81 97
#> 
#> [[22]]
#> + 6/100 vertices, from 2470883:
#> [1] 14 32 63 78 81 97
#> 
#> [[23]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 14 63 78 81 97
#> 
#> [[24]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 14 32 63 78 97
#> 
#> [[25]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 14 32 78 81 97
#> 
#> [[26]]
#> + 6/100 vertices, from 2470883:
#> [1]  4 14 32 63 81 97
#> 
#> [[27]]
#> + 6/100 vertices, from 2470883:
#> [1]  6 27 34 39 63 97
#> 
#> [[28]]
#> + 6/100 vertices, from 2470883:
#> [1]  4  6 27 39 63 76
#> 
#> [[29]]
#> + 7/100 vertices, from 2470883:
#> [1]  4  6 27 39 63 76 97
#> 
#> [[30]]
#> + 6/100 vertices, from 2470883:
#> [1]  6 27 39 63 76 97
#> 
#> [[31]]
#> + 6/100 vertices, from 2470883:
#> [1]  4  6 18 27 63 76
#> 
#> [[32]]
#> + 7/100 vertices, from 2470883:
#> [1]  4  6 18 27 63 76 97
#> 
#> [[33]]
#> + 6/100 vertices, from 2470883:
#> [1]  6 18 27 63 76 97
#> 
#> [[34]]
#> + 6/100 vertices, from 2470883:
#> [1]  4  6 27 63 76 97
#> 
#> [[35]]
#> + 6/100 vertices, from 2470883:
#> [1]  4  6 39 63 76 97
#> 
#> [[36]]
#> + 6/100 vertices, from 2470883:
#> [1]  4  6 18 63 76 97
#> 
#> [[37]]
#> + 6/100 vertices, from 2470883:
#> [1]  4  6 27 39 63 97
#> 
#> [[38]]
#> + 6/100 vertices, from 2470883:
#> [1]  4  6 18 27 63 97
#> 
#> [[39]]
#> + 6/100 vertices, from 2470883:
#> [1]  4  6 27 39 76 97
#> 
#> [[40]]
#> + 6/100 vertices, from 2470883:
#> [1]  4  6 18 27 76 97
#> 
largest_cliques(g)
#> [[1]]
#> + 7/100 vertices, from 2470883:
#> [1]  4 97 14 81 78 63 32
#> 
#> [[2]]
#> + 7/100 vertices, from 2470883:
#> [1]  4 97 76 63  6 27 18
#> 
#> [[3]]
#> + 7/100 vertices, from 2470883:
#> [1]  4 97 76 63  6 27 39
#> 

# To have a bit less maximal cliques, about 100-200 usually
g <- sample_gnp(100, 0.03)
max_cliques(g)
#> [[1]]
#> + 1/100 vertex, from 80e0057:
#> [1] 27
#> 
#> [[2]]
#> + 1/100 vertex, from 80e0057:
#> [1] 51
#> 
#> [[3]]
#> + 1/100 vertex, from 80e0057:
#> [1] 87
#> 
#> [[4]]
#> + 1/100 vertex, from 80e0057:
#> [1] 21
#> 
#> [[5]]
#> + 1/100 vertex, from 80e0057:
#> [1] 84
#> 
#> [[6]]
#> + 1/100 vertex, from 80e0057:
#> [1] 11
#> 
#> [[7]]
#> + 2/100 vertices, from 80e0057:
#> [1] 41 30
#> 
#> [[8]]
#> + 2/100 vertices, from 80e0057:
#> [1] 16 36
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#> [[9]]
#> + 2/100 vertices, from 80e0057:
#> [1] 91 73
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#> [[10]]
#> + 2/100 vertices, from 80e0057:
#> [1] 44 48
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#> [[11]]
#> + 2/100 vertices, from 80e0057:
#> [1] 44 14
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#> [[12]]
#> + 2/100 vertices, from 80e0057:
#> [1] 94 54
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#> [[13]]
#> + 2/100 vertices, from 80e0057:
#> [1] 96  7
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#> [[14]]
#> + 2/100 vertices, from 80e0057:
#> [1] 35 58
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#> [[15]]
#> + 2/100 vertices, from 80e0057:
#> [1] 17 97
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#> [[16]]
#> + 2/100 vertices, from 80e0057:
#> [1] 62 37
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#> [[17]]
#> + 2/100 vertices, from 80e0057:
#> [1] 1 9
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#> [[18]]
#> + 2/100 vertices, from 80e0057:
#> [1] 29  2
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#> [[19]]
#> + 2/100 vertices, from 80e0057:
#> [1] 65 60
#> 
#> [[20]]
#> + 2/100 vertices, from 80e0057:
#> [1]  9 56
#> 
#> [[21]]
#> + 2/100 vertices, from 80e0057:
#> [1] 40 93
#> 
#> [[22]]
#> + 2/100 vertices, from 80e0057:
#> [1] 54 14
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#> [[23]]
#> + 2/100 vertices, from 80e0057:
#> [1] 30 86
#> 
#> [[24]]
#> + 2/100 vertices, from 80e0057:
#> [1] 30 72
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#> [[25]]
#> + 2/100 vertices, from 80e0057:
#> [1] 30 34
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#> [[26]]
#> + 2/100 vertices, from 80e0057:
#> [1] 31 86
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#> [[27]]
#> + 2/100 vertices, from 80e0057:
#> [1] 31 69
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#> [[28]]
#> + 2/100 vertices, from 80e0057:
#> [1] 31 36
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#> [[29]]
#> + 2/100 vertices, from 80e0057:
#> [1] 32 60
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#> [[30]]
#> + 2/100 vertices, from 80e0057:
#> [1] 32 49
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#> [[31]]
#> + 2/100 vertices, from 80e0057:
#> [1] 33 49
#> 
#> [[32]]
#> + 2/100 vertices, from 80e0057:
#> [1] 33  5
#> 
#> [[33]]
#> + 2/100 vertices, from 80e0057:
#> [1] 34 90
#> 
#> [[34]]
#> + 2/100 vertices, from 80e0057:
#> [1] 34 85
#> 
#> [[35]]
#> + 2/100 vertices, from 80e0057:
#> [1] 34 10
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#> [[36]]
#> + 2/100 vertices, from 80e0057:
#> [1] 36 72
#> 
#> [[37]]
#> + 2/100 vertices, from 80e0057:
#> [1] 36 50
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#> [[38]]
#> + 2/100 vertices, from 80e0057:
#> [1] 36 28
#> 
#> [[39]]
#> + 2/100 vertices, from 80e0057:
#> [1] 36 20
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#> [[40]]
#> + 2/100 vertices, from 80e0057:
#> [1] 37 69
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#> [[41]]
#> + 2/100 vertices, from 80e0057:
#> [1] 37  6
#> 
#> [[42]]
#> + 2/100 vertices, from 80e0057:
#> [1] 38 45
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#> [[43]]
#> + 3/100 vertices, from 80e0057:
#> [1] 38 26 63
#> 
#> [[44]]
#> + 2/100 vertices, from 80e0057:
#> [1] 38  2
#> 
#> [[45]]
#> + 2/100 vertices, from 80e0057:
#> [1] 39 83
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#> [[46]]
#> + 2/100 vertices, from 80e0057:
#> [1] 39 58
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#> [[47]]
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#> [1] 42 68
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#> + 2/100 vertices, from 80e0057:
#> [1] 42 55
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#> [[49]]
#> + 2/100 vertices, from 80e0057:
#> [1] 43 67
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#> [[50]]
#> + 2/100 vertices, from 80e0057:
#> [1] 43  2
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#> [[51]]
#> + 2/100 vertices, from 80e0057:
#> [1] 45 50
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#> [[52]]
#> + 2/100 vertices, from 80e0057:
#> [1] 45 22
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#> [1] 46 80
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#> [1] 46 75
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#> [1] 46 71
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#> [1] 46 60
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#> [1] 47 82
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#> [1] 47 61
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#> [[59]]
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#> [1] 47  4
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#> [1] 49 89
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#> [1] 49 88
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#> [1] 49 53
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#> [1] 52 18
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#> [1] 53 70
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#> [1] 53  7
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#> [1] 55 80
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#> [1] 56 79
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#> [1] 56  7
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#> [1] 57 98
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#> [1] 57 93
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#> [1] 58 22
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#> [1] 58 18
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#> [1] 59 22
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#> [1] 61 98
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#> [1] 61 97 12
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#> + 3/100 vertices, from 80e0057:
#> [1]  61  97 100
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#> [1] 61 25
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#> 
#> [[145]]
#> + 2/100 vertices, from 80e0057:
#> [1]  8 20
#> 
#> [[146]]
#> + 2/100 vertices, from 80e0057:
#> [1] 19 28
#> 

# Check that all returned vertex sets are indeed cliques
all(sapply(max_cliques(g), function (c) is_clique(g, c)))
#> [1] TRUE