Statistically optimal modular partitioning of directed graphs

Network models can be used to represent interacting subsystems in the brain or other biological systems. These subsystems can be identified by partitioning a graph representation of the network into highly connected modules. In this paper we describe a modularity-based partitioning method based on a Gaussian model of a directed graph. Using the degrees of each node, we first compute the conditional expected value of the connection weights. The resulting adjacency matrix forms a null model for the network which does not favor any particular partition. By comparing this null model to the true adjacency graph, we can perform a statistically optimal partitioning that maximizes modularity. Similarly to other modularity-based partitioning methods, the solution is found using spectral matrix decomposition. The process can be iterated to find multiple subgraphs. We demonstrate this approach through simulations and application to standard biological and other network data.