A quality measure for multi-level community structure

Mining relational data often boils down to computing clusters, that is finding sub-communities of data elements forming cohesive sub-units, while being well separated from one another The clusters themselves are sometimes terms "communities" and the way clusters relate to one another is often referred to as a "community structure". We study a modularity criterion MQ introduced by Mancoridis et al. in order to infer community structure on relational data. We prove a fundamental and useful property of the modularity measure MQ, showing that it can be approximated by a gaussian distribution, making it a prevalent choice over less focused optimization criterion for graph clustering. This makes it possible to compare two different clusterings of a same graph as well as asserting the overall quality of a given clustering relying on the fact that MQ is gaussian. Moreover we introduce a generalization extending MQ to hierarchical clusterings of graphs which reduces to the original MQ when the hierarchy becomes flat.