I'm trying to detect quasi cliques in an undirected graph. My problem is that I don't want any overlap between cluster.

I'm currently trying to detect community using Louvain algorithm, but it detects subgraphs with a low clustering coefficient (0.60) and when I detect maximal or quasi cliques they tend to overlap.

How can I compute a clique decomposition of an undirected graph with no overlap between clusters ? Is there an algorithm with an implementation of it which can deal with my problem ?


  • $\begingroup$ I'm not sure what you're looking for. The input is a graph but what's the output supposed to be? A set of non-overlapping cliques? If so, what does "maximal" mean in this context? No clique can be extended and no new cliques can be added? $\endgroup$ May 23, 2016 at 19:32
  • $\begingroup$ Sorry i misused the term clique. The output is supposed to be a set of non-overlapping densely connected subgraph. A maximal clique is a clique that cannot be extended by including one more adjacent vertex. $\endgroup$
    – adp7
    May 23, 2016 at 20:16
  • $\begingroup$ Please edit the question to clarify what problem exactly you wish to solve. $\endgroup$
    – Raphael
    May 23, 2016 at 22:14
  • $\begingroup$ Generally cliques allow overlaps; there are heaps of methods for detecting communities such as modularity optimization and the codes are available. You need to do some research and find the method that suits you. $\endgroup$
    – orezvani
    May 24, 2016 at 11:10

1 Answer 1


If you want to find cliques, or quasi-cliques, don't expect non-overlapping communities, as their definitions imply that there might be overlap between clusters. When it comes to community detection or clustering, you need to make a formal definition of a community or cluster (in this case it is a clique). There are two types of definitions:

Non-overlapping definitions - There are some definitions that do not allow any overlap between communities such as $k$-cores. Basically, the following properties prevent $k$-cores of being overlapping:

  1. $k$-cores are maximal
  2. Union of two $k$-cores is a $k$-core.

There are lots of different techniques such as modularity optimization, seed expansion, etc. If you are looking for stricter non-overlapping community definitions, you can dig further in this paper (and the sequence of other papers) and find the one ghat works well on coefficient clustering score:

Fortunato, Santo. "Community detection in graphs." Physics reports 486.3 (2010): 75-174.

Overlapping definitions - However, some definitions such as cliques or quasi-cliques, allow overlap between clusters, even though you consider maximality. Below, I show a simple example where few maximal cliques overlap with each other (link for image).

enter image description here


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