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## k-clique algorithm

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k-clique community detection algorithm

Updated

k-clique algorithm as defined in the paper "Uncovering the overlapping community structure of complex networks in nature and society" - G. Palla, I. Derényi, I. Farkas, and T. Vicsek - Nature 435, 814–818 (2005)

[X,Y,Z] = k_clique(k,A)

Inputs:
k - clique size

Outputs:
X - detected communities
Y - all cliques (i.e. complete subgraphs that are not parts of larger complete subgraphs)
Z - k-clique matrix

Fan Zhang

### Fan Zhang (view profile)

Hi,

In line 61, why only the 'i = size(C,1)' clique is recorded? Is it supposed to be 'for i = 1:size(C,1)'?

thanks!!!

Shukai Ni

Shukai Ni

SAMUEL HEROY

### SAMUEL HEROY (view profile)

I am confused. I created an adjacency matrix that generates a 2x4 scaffold with 8 pairwise adjacent 3-cliques. (My adj matrix is
0 1 1 1 0 0 0 0
1 0 0 1 0 0 0 0
1 0 0 1 1 0 0 0
1 1 1 0 1 1 0 0
0 0 1 1 0 1 1 1
0 0 0 1 1 0 0 1
0 0 0 0 1 0 0 1
0 0 0 0 1 1 1 0
).
However, the cliques output only returns four entries when I query for 3-cliques and associated communities:
[1,3,4]
[3,4,5]
[4,5,6]
[5,7,8]
By my understanding of this algorithm's intention, shouldn't there be four more entries here--and one large community that spans the network?

learn learn

### learn learn (view profile)

i found after execute the code source this result. but i can't well understand what's mean.
M

M =

1 1 0 0 0 0 0 0 0 1
1 1 1 1 1 1 1 0 0 1
0 1 1 1 0 0 1 0 0 0
0 1 1 1 1 1 1 0 0 0
0 1 0 1 1 1 1 1 0 0
0 1 0 1 1 1 1 1 0 0
0 1 1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1 1 1
0 0 0 0 0 0 1 1 1 1
1 1 0 0 0 0 1 1 1 1

>> m=4;
>> [X,Y,Z] = k_clique(m,M)

X =

[1x7 double]
[1x4 double]

Y =

[1x5 double]
[1x4 double]
[1x4 double]
[1x4 double]
[1x3 double]
[1x3 double]

Z =

1 1 1 0 0 0
1 1 0 0 0 0
1 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 0

>>

Anh-Dung Nguyen

### Anh-Dung Nguyen (view profile)

The result is correct. In fact, the algorithm first extracts all complete subgraphs that are not parts of larger complete subgraphs. These maximal complete subgraphs are called "cliques". I'll modify the description. Thank you.

YANG

### YANG (view profile)

I have a question
if A is 4*4 complete graph
[X,Y,Z] = k_clique(3,A);
Y is [1 2 3 4] and not find all cliques of size 3