Discovering block structure in networks

Abstract

A generalisation of modularity, called block modularity, is defined. This is a quality function which evaluates a label assignment against an arbitrary block pattern. Therefore, unlike standard modularity or its variants, arbitrary network structures can be compared and an optimal block matrix can be determined. Some simple algorithms for optimising block modularity are described and applied on networks with planted structure. In many cases the planted structure is recovered. Cases where it is not are analysed and it is found that strong degree-correlations explain the planted structure so that the discovered pattern is more ‘surprising’ than the planted one under the configuration model. Some well studied networks are analysed with this new method, which is found to automatically deconstruct the network in a very useful way for creating a summary of its key features.