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)
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)
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.- 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 aweight
vertex attribute and this argument isNULL
, 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
).
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
See also
Other cliques:
ivs()
,
weighted_cliques()
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 181c4cf:
#> [1] 6 7 20 76 87 94
#>
#> [[2]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 7 20 28 53 76 94
#>
#> [[3]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 17 24 50 57 87 93
#>
#> [[4]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 8 15 31 40 56 63
#>
#> [[5]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 8 31 40 56 63 97
#>
#> [[6]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 8 13 68 69 75 97
#>
#> [[7]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 8 13 56 68 75 97
#>
#> [[8]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 56 63 68 81 82 97
#>
#> [[9]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 15 28 31 66 85 93
#>
#> [[10]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 3 38 50 68 96 99
#>
#> [[11]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 28 72 79 94
#>
#> [[12]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 54 72 79 94
#>
#> [[13]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 63 79 94 97
#>
#> [[14]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 63 77 94 97
#>
#> [[15]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 13 14 33 79 97
#>
#> [[16]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 17 63 77 97
#>
#> [[17]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 8 14 18 60 63 97
#>
#> [[18]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 17 24 46 57 84 91
#>
#> [[19]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 17 24 46 47 57 91
#>
largest_cliques(g)
#> [[1]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 3 99 50 96 68 38
#>
#> [[2]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 28 72 79 94
#>
#> [[3]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 54 79 72 94
#>
#> [[4]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 97 17 63 77
#>
#> [[5]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 97 79 13 33
#>
#> [[6]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 97 79 63 94
#>
#> [[7]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 5 14 97 77 94 63
#>
#> [[8]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 6 7 94 87 20 76
#>
#> [[9]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 7 53 94 28 76 20
#>
#> [[10]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 8 97 13 68 75 56
#>
#> [[11]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 8 97 13 68 75 69
#>
#> [[12]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 8 97 63 14 60 18
#>
#> [[13]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 8 97 63 40 56 31
#>
#> [[14]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 8 15 63 56 40 31
#>
#> [[15]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 15 93 31 85 28 66
#>
#> [[16]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 17 93 50 87 57 24
#>
#> [[17]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 17 46 91 57 24 84
#>
#> [[18]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 17 46 91 57 24 47
#>
#> [[19]]
#> + 6/100 vertices, from 181c4cf:
#> [1] 56 97 82 68 81 63
#>
# 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 924c4ec:
#> [1] 49
#>
#> [[2]]
#> + 1/100 vertex, from 924c4ec:
#> [1] 64
#>
#> [[3]]
#> + 1/100 vertex, from 924c4ec:
#> [1] 28
#>
#> [[4]]
#> + 1/100 vertex, from 924c4ec:
#> [1] 27
#>
#> [[5]]
#> + 1/100 vertex, from 924c4ec:
#> [1] 56
#>
#> [[6]]
#> + 1/100 vertex, from 924c4ec:
#> [1] 35
#>
#> [[7]]
#> + 1/100 vertex, from 924c4ec:
#> [1] 4
#>
#> [[8]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 89 80
#>
#> [[9]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 38 57
#>
#> [[10]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 70 1
#>
#> [[11]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 12 31
#>
#> [[12]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 93 22
#>
#> [[13]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 21 63
#>
#> [[14]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 73 50
#>
#> [[15]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 96 31
#>
#> [[16]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 25 23
#>
#> [[17]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 99 94
#>
#> [[18]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 32 75
#>
#> [[19]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 100 39
#>
#> [[20]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 44 16
#>
#> [[21]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 62 61
#>
#> [[22]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 36 2
#>
#> [[23]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 50 87
#>
#> [[24]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 51 3
#>
#> [[25]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 33 85
#>
#> [[26]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 33 65
#>
#> [[27]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 34 98
#>
#> [[28]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 34 58
#>
#> [[29]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 37 88
#>
#> [[30]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 37 85
#>
#> [[31]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 37 65
#>
#> [[32]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 39 72
#>
#> [[33]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 39 57
#>
#> [[34]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 39 46
#>
#> [[35]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 40 66
#>
#> [[36]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 40 61
#>
#> [[37]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 40 60
#>
#> [[38]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 40 31
#>
#> [[39]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 41 82
#>
#> [[40]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 41 18
#>
#> [[41]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 41 8
#>
#> [[42]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 42 79
#>
#> [[43]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 42 76
#>
#> [[44]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 43 97
#>
#> [[45]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 43 74
#>
#> [[46]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 43 58
#>
#> [[47]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 43 46
#>
#> [[48]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 45 30
#>
#> [[49]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 45 10
#>
#> [[50]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 47 19
#>
#> [[51]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 47 9
#>
#> [[52]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 47 6
#>
#> [[53]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 48 29
#>
#> [[54]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 48 5
#>
#> [[55]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 52 95
#>
#> [[56]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 52 6
#>
#> [[57]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 53 87
#>
#> [[58]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 53 9
#>
#> [[59]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 54 69
#>
#> [[60]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 54 24
#>
#> [[61]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 54 8
#>
#> [[62]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 55 78
#>
#> [[63]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 55 67
#>
#> [[64]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 55 26
#>
#> [[65]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 57 80
#>
#> [[66]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 57 69
#>
#> [[67]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 57 3
#>
#> [[68]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 58 90
#>
#> [[69]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 58 79
#>
#> [[70]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 58 59
#>
#> [[71]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 59 85
#>
#> [[72]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 60 86
#>
#> [[73]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 61 80
#>
#> [[74]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 61 75
#>
#> [[75]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 61 23
#>
#> [[76]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 63 78
#>
#> [[77]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 63 77
#>
#> [[78]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 63 19
#>
#> [[79]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 65 20
#>
#> [[80]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 65 16
#>
#> [[81]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 66 15
#>
#> [[82]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 66 13
#>
#> [[83]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 67 30
#>
#> [[84]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 67 16
#>
#> [[85]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 68 92
#>
#> [[86]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 68 85
#>
#> [[87]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 68 24
#>
#> [[88]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 68 2
#>
#> [[89]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 69 7
#>
#> [[90]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 71 87
#>
#> [[91]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 71 13
#>
#> [[92]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 71 11
#>
#> [[93]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 72 88
#>
#> [[94]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 72 78
#>
#> [[95]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 72 9
#>
#> [[96]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 74 84
#>
#> [[97]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 74 26
#>
#> [[98]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 75 90
#>
#> [[99]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 75 26
#>
#> [[100]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 76 87
#>
#> [[101]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 77 90
#>
#> [[102]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 77 87
#>
#> [[103]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 77 29
#>
#> [[104]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 77 9
#>
#> [[105]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 78 92
#>
#> [[106]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 78 13
#>
#> [[107]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 78 7
#>
#> [[108]]
#> + 3/100 vertices, from 924c4ec:
#> [1] 81 17 84
#>
#> [[109]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 81 5
#>
#> [[110]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 82 86
#>
#> [[111]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 82 19
#>
#> [[112]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 83 22
#>
#> [[113]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 83 13
#>
#> [[114]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 84 97
#>
#> [[115]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 84 24
#>
#> [[116]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 84 22
#>
#> [[117]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 84 14
#>
#> [[118]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 85 26
#>
#> [[119]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 86 92
#>
#> [[120]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 86 90
#>
#> [[121]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 87 15
#>
#> [[122]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 90 94
#>
#> [[123]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 90 22
#>
#> [[124]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 91 14
#>
#> [[125]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 91 13
#>
#> [[126]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 94 98
#>
#> [[127]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 94 17
#>
#> [[128]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 95 26
#>
#> [[129]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 97 23
#>
#> [[130]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 98 23
#>
#> [[131]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 98 5
#>
#> [[132]]
#> + 3/100 vertices, from 924c4ec:
#> [1] 2 13 26
#>
#> [[133]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 3 26
#>
#> [[134]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 6 26
#>
#> [[135]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 6 14
#>
#> [[136]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 10 31
#>
#> [[137]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 10 11
#>
#> [[138]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 15 30
#>
#> [[139]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 16 26
#>
#> [[140]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 18 20
#>
#> [[141]]
#> + 2/100 vertices, from 924c4ec:
#> [1] 19 23
#>