How to count all cliques (not just maximal ones) in graphs?

Question

How can I count the numbers of $r$-cliques in a graph? In other words, the number of triangles, the number of $K_4$, the number of $K_5$ etc. I have tried for example FindClique[G,{4},All], which finds all "isolated" $K_4$s but does not find $K_4$s in a larger clique. So for example if $G = K_5$ then FindClique[G,{4},All] does not find the 4-cliques within G but FindClique[G,{5},All] does find the 5-clique. Is there a simple way of doing this with a Mathematica command or is it necessary to compute the number of smaller cliques from the number of larger cliques?

Answer 1

Mathematica only finds maximal cliques, i.e. cliques (complete subgraphs) that are not part of a larger clique.

Computing the number of all cliques given the maximal ones is not trivial because some of the maximal cliques may be overlapping.

The simplest way to find all cliques is to use one of several packages that can do this.

g = ExampleData[{"NetworkGraph", "ZacharyKarateClub"}];

IGraph/M

<<IGraphM`

cliques = IGCliques[g, Infinity]

{{2}, {1}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}, {11}, {12}, {13}, \
{14}, {17}, {18}, {20}, {22}, {26}, {24}, {25}, {28}, {29}, {30}, \
{27}, {31}, {32}, {33}, {15}, {16}, {19}, {21}, {23}, {34}, {2, 
  1}, {2, 3}, {2, 4}, {2, 8}, {2, 14}, {2, 18}, {2, 20}, {2, 22}, {2, 
  31}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1, 8}, {1, 9}, {1, 
  11}, {1, 12}, {1, 13}, {1, 14}, {1, 18}, {1, 20}, {1, 22}, {1, 
  32}, {3, 4}, {3, 8}, {3, 9}, {3, 10}, {3, 14}, {3, 28}, {3, 29}, {3,
   33}, {4, 8}, {4, 13}, {4, 14}, {5, 7}, {5, 11}, {6, 7}, {6, 
  11}, {6, 17}, {7, 17}, {9, 31}, {9, 33}, {9, 34}, {10, 34}, {14, 
  34}, {20, 34}, {26, 24}, {26, 25}, {26, 32}, {24, 28}, {24, 
  30}, {24, 33}, {24, 34}, {25, 28}, {25, 32}, {28, 34}, {29, 
  32}, {29, 34}, {30, 27}, {30, 33}, {30, 34}, {27, 34}, {31, 
  33}, {31, 34}, {32, 33}, {32, 34}, {33, 15}, {33, 16}, {33, 
  19}, {33, 21}, {33, 23}, {33, 34}, {15, 34}, {16, 34}, {19, 
  34}, {21, 34}, {23, 34}, {2, 1, 3}, {2, 1, 4}, {2, 1, 8}, {2, 1, 
  14}, {2, 1, 18}, {2, 1, 20}, {2, 1, 22}, {2, 3, 4}, {2, 3, 8}, {2, 
  3, 14}, {2, 4, 8}, {2, 4, 14}, {1, 3, 4}, {1, 3, 8}, {1, 3, 9}, {1, 
  3, 14}, {1, 4, 8}, {1, 4, 13}, {1, 4, 14}, {1, 5, 7}, {1, 5, 
  11}, {1, 6, 7}, {1, 6, 11}, {3, 4, 8}, {3, 4, 14}, {3, 9, 33}, {6, 
  7, 17}, {9, 31, 33}, {9, 31, 34}, {9, 33, 34}, {26, 25, 32}, {24, 
  28, 34}, {24, 30, 33}, {24, 30, 34}, {24, 33, 34}, {29, 32, 
  34}, {30, 27, 34}, {30, 33, 34}, {31, 33, 34}, {32, 33, 34}, {33, 
  15, 34}, {33, 16, 34}, {33, 19, 34}, {33, 21, 34}, {33, 23, 34}, {2,
   1, 3, 4}, {2, 1, 3, 8}, {2, 1, 3, 14}, {2, 1, 4, 8}, {2, 1, 4, 
  14}, {2, 3, 4, 8}, {2, 3, 4, 14}, {1, 3, 4, 8}, {1, 3, 4, 14}, {9, 
  31, 33, 34}, {24, 30, 33, 34}, {2, 1, 3, 4, 8}, {2, 1, 3, 4, 14}}

 CountsBy[cliques, Length]
 (* <|1 -> 34, 2 -> 78, 3 -> 45, 4 -> 11, 5 -> 2|> *)

Update: In IGraph/M 0.1.4 or later, we can directly count cliques, without listing them:

IGCliqueSizeCounts[g]
(* {34, 78, 45, 11, 2} *)

This release also significantly speeds up clique finding.

Since there is significant overhead in storing and transferring all the cliques back to Mathematica after computing them, IGCliqueSizeCounts can be many times faster than IGCliques. If you only need counts, use IGCliqueSizeCounts.

IGraphR

This needs a bit more work as the output of R/igraph's cliques command doesn't easily transfer to Mathematica.

<<IGraphR`
REvaluate["cliqCounter <- function (g) { as.vector(table(sapply(cliques(g), length))) };"]

IGraph["cliqCounter"][g]
(* {34, 78, 45, 11, 2} *)

This gives you the number of cliques with sizes 1, 2, 3, 4 and 5.

MmaCliquer

This is a partial interface to Cliquer that I made for my own use. Currently it can only count cliques of different sizes, but not return the vertices they contain. This library is very fast for counting all cliques (but not so much for counting maximal ones).

Due to GPL issues you need to compile it yourself ...

<<Cliquer`

CliquerDistribution[g]
(* {34, 78, 45, 11, 2} *)

---

Finally, just for some fun with IGraph/M's isomorphism functions:

Table[IGVF2SubisomorphismCount[CompleteGraph[i], g]/IGBlissAutomorphismCount@CompleteGraph[i], {i, 5}]

(* {34, 78, 45, 11, 2} *)

Of course this is extremely inefficient :)

Answer 2

1. Mathematica can return the set of all maximal cliques of a graph.
2. Any subset of clique vertices is also a clique.

So, this gives all cliques:

findAllCliques[g_] := 
 DeleteDuplicates[Flatten[Map[Subsets, FindClique[g, Infinity, All]], 1]]

Try it:

n = RandomInteger[25];
m = RandomInteger[Binomial[n, 2]];
rgr = Graph[RandomGraph[{n, m}], VertexLabels -> "Name"];
allCliques = findAllCliques[rgr];
Print["n = ", n, "   m = ", m]
Print[Length@allCliques, " Cliques found"]
rgr
allCliques
Histogram@Map[Length, allCliques]

Answer 3

A naive approach is to enumerate all ${n \choose r}$ subsets, and for each check whether the subset induces an $r$-clique. Obviously, this does not scale nicely when $r$ gets larger, nor when $n$ starts to get larger.

For special cases, i.e., small values of $r$ you can do better. A neat algebraic graph-theoretic way of computing the number of triangles is given by $\text{tr}(A^3)/6$, where $A$ is the adjacency matrix of a graph $G$. For counting 3-cliques, we can simply do

TriangleCount[g_] := Tr[MatrixPower[AdjacencyMatrix[g], 3]]/6;