Thermodynamics of Community Structure

We introduce an approach to partitioning networks into communities that not only determines the best community structure, but also provides a range of characterization techniques to assess how significant that structure is. We study the thermodynamics of community structure by producing equilibrium ensembles of partitions, in which each partition is represented with a well-defined statistical weight. Thus we are able to study the temperature dependence of thermodynamic properties, namely the modularity $Q$ and heat capacity, with particular emphasis on the transition between high-temperature, essentially random partitions and low-temperature partitions with high modularity. We also look at frequency matrices that measure the likelihood that two nodes belong to the same community, and introduce an order parameter to measure the `blockiness' of the frequency matrix, and therefore the uniqueness of the community structure. These methods have been applied to a number of model networks in order to understand the effects of the degree distribution, spatial embedding and randomization. Finally, we apply these methods to a metabolic network known to have strong community structure and find hierarchical community structure, with some communities being more robust than others.