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.

Usage

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

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

Arguments

graph: The input graph, directed graphs will be considered as undirected ones, multiple edges and loops are ignored.
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.

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.

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.

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

Author

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

Examples


# this usually contains cliques of size six
g <- sample_gnp(100, 0.3)
clique_num(g)
#> [1] 6
cliques(g, min = 6)
#> [[1]]
#> + 6/100 vertices, from 75afb0d:
#> [1]   8  17  34  87  94 100
#> 
#> [[2]]
#> + 6/100 vertices, from 75afb0d:
#> [1]  8 17 22 34 87 94
#> 
#> [[3]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 26 31 58 61 63 94
#> 
#> [[4]]
#> + 6/100 vertices, from 75afb0d:
#> [1]  8 17 22 34 52 64
#> 
#> [[5]]
#> + 6/100 vertices, from 75afb0d:
#> [1]  8 17 22 34 64 94
#> 
#> [[6]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 41 54 56 66 78 81
#> 
#> [[7]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 10 11 12 25 95 99
#> 
#> [[8]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 16 32 46 60 74 77
#> 
largest_cliques(g)
#> [[1]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 12 99 10 25 95 11
#> 
#> [[2]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 16 60 46 77 32 74
#> 
#> [[3]]
#> + 6/100 vertices, from 75afb0d:
#> [1]  17 100  87  34  94   8
#> 
#> [[4]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 17 64 52  8 22 34
#> 
#> [[5]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 17 64 94 34  8 22
#> 
#> [[6]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 17 34 87 94  8 22
#> 
#> [[7]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 26 58 94 63 31 61
#> 
#> [[8]]
#> + 6/100 vertices, from 75afb0d:
#> [1] 41 81 66 54 78 56
#> 

# 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 5aec96a:
#> [1] 65
#> 
#> [[2]]
#> + 1/100 vertex, from 5aec96a:
#> [1] 7
#> 
#> [[3]]
#> + 1/100 vertex, from 5aec96a:
#> [1] 47
#> 
#> [[4]]
#> + 1/100 vertex, from 5aec96a:
#> [1] 29
#> 
#> [[5]]
#> + 1/100 vertex, from 5aec96a:
#> [1] 27
#> 
#> [[6]]
#> + 1/100 vertex, from 5aec96a:
#> [1] 14
#> 
#> [[7]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 21 73
#> 
#> [[8]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 21 42
#> 
#> [[9]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 51 85
#> 
#> [[10]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 52  6
#> 
#> [[11]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 34 61
#> 
#> [[12]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 15 28
#> 
#> [[13]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 82 72
#> 
#> [[14]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 36 58
#> 
#> [[15]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 56 57
#> 
#> [[16]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 56 25
#> 
#> [[17]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  9 66
#> 
#> [[18]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  9 32
#> 
#> [[19]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 24 55
#> 
#> [[20]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 12 64
#> 
#> [[21]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 63 53
#> 
#> [[22]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  92 100
#> 
#> [[23]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 94 70
#> 
#> [[24]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 43 88
#> 
#> [[25]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 99 50
#> 
#> [[26]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 19 46
#> 
#> [[27]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 100  49
#> 
#> [[28]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 46 39
#> 
#> [[29]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 72 62
#> 
#> [[30]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 74 31
#> 
#> [[31]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 50 95
#> 
#> [[32]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 50 91
#> 
#> [[33]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 50 87
#> 
#> [[34]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 53 60
#> 
#> [[35]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 53  8
#> 
#> [[36]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 54 70
#> 
#> [[37]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 54 10
#> 
#> [[38]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 55 68
#> 
#> [[39]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 55  8
#> 
#> [[40]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 58 98
#> 
#> [[41]]
#> + 3/100 vertices, from 5aec96a:
#> [1] 58 35 81
#> 
#> [[42]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 59 39
#> 
#> [[43]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 59 30
#> 
#> [[44]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 60 78
#> 
#> [[45]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 60 69
#> 
#> [[46]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 62 77
#> 
#> [[47]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 62 33
#> 
#> [[48]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 62 16
#> 
#> [[49]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 64 71
#> 
#> [[50]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 64 10
#> 
#> [[51]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 66 83
#> 
#> [[52]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 66 78
#> 
#> [[53]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 66 71
#> 
#> [[54]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 66 26
#> 
#> [[55]]
#> + 3/100 vertices, from 5aec96a:
#> [1] 66 22 91
#> 
#> [[56]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 66  5
#> 
#> [[57]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 67 91
#> 
#> [[58]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 67 84
#> 
#> [[59]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 67  3
#> 
#> [[60]]
#> + 3/100 vertices, from 5aec96a:
#> [1] 67  1 28
#> 
#> [[61]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 68 87
#> 
#> [[62]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 68 42
#> 
#> [[63]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 68 17
#> 
#> [[64]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 68  3
#> 
#> [[65]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 69 10
#> 
#> [[66]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 70 84
#> 
#> [[67]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 70 81
#> 
#> [[68]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 70 42
#> 
#> [[69]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 70 39
#> 
#> [[70]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 75 93
#> 
#> [[71]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 75 90
#> 
#> [[72]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 75 84
#> 
#> [[73]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 75 83
#> 
#> [[74]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 75 23
#> 
#> [[75]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 75  6
#> 
#> [[76]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 76 98
#> 
#> [[77]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 76 97
#> 
#> [[78]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 76 40
#> 
#> [[79]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 76 11
#> 
#> [[80]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 76  5
#> 
#> [[81]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 77 84
#> 
#> [[82]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 77 20
#> 
#> [[83]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 78 40
#> 
#> [[84]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 78  6
#> 
#> [[85]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 79 95
#> 
#> [[86]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 79 85
#> 
#> [[87]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 79 48
#> 
#> [[88]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 79 39
#> 
#> [[89]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 79 31
#> 
#> [[90]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 80 33
#> 
#> [[91]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 80 17
#> 
#> [[92]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 81 31
#> 
#> [[93]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 83 13
#> 
#> [[94]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 84 96
#> 
#> [[95]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 85 44
#> 
#> [[96]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 86 37
#> 
#> [[97]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 86 28
#> 
#> [[98]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 86  6
#> 
#> [[99]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 86  2
#> 
#> [[100]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 87 98
#> 
#> [[101]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 87 28
#> 
#> [[102]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  1 48
#> 
#> [[103]]
#> + 3/100 vertices, from 5aec96a:
#> [1]  1 18 28
#> 
#> [[104]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  1 17
#> 
#> [[105]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 89 38
#> 
#> [[106]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 89  3
#> 
#> [[107]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 90 45
#> 
#> [[108]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 91 41
#> 
#> [[109]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 91 39
#> 
#> [[110]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 93 48
#> 
#> [[111]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 93 40
#> 
#> [[112]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 93 30
#> 
#> [[113]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 96 97
#> 
#> [[114]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 96 11
#> 
#> [[115]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 98 41
#> 
#> [[116]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 88 44
#> 
#> [[117]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 88 38
#> 
#> [[118]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 88 20
#> 
#> [[119]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 88 17
#> 
#> [[120]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  2 33
#> 
#> [[121]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 2 8
#> 
#> [[122]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  3 26
#> 
#> [[123]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  3 16
#> 
#> [[124]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  4 33
#> 
#> [[125]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  4 28
#> 
#> [[126]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  5 48
#> 
#> [[127]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  5 42
#> 
#> [[128]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  5 28
#> 
#> [[129]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  6 10
#> 
#> [[130]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  8 28
#> 
#> [[131]]
#> + 2/100 vertices, from 5aec96a:
#> [1]  8 11
#> 
#> [[132]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 11 49
#> 
#> [[133]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 11 23
#> 
#> [[134]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 13 48
#> 
#> [[135]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 16 48
#> 
#> [[136]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 20 37
#> 
#> [[137]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 26 48
#> 
#> [[138]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 26 45
#> 
#> [[139]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 31 42
#> 
#> [[140]]
#> + 2/100 vertices, from 5aec96a:
#> [1] 35 49
#>