Analysis of community structure in networks of correlated data

We present a reformulation of modularity that allows the analysis of the community structure in networks of correlated data. The modularity preserves the probabilistic semantics of the original definition even when the network is directed, weighted, signed, and has self-loops. This is the most general condition one can find in the study of any network, in particular those defined from correlated data. We apply our results to a real network of correlated data between stores in the city of Lyon (France).

Figure 1

Network with well-defined community structure and its correlation matrix.

Figure 2

Network with two well-defined communities. Solid lines correspond to positive links, and the dashed line to the only negative link, with weight $v<0$.