Dynamics and asymptotic behaviour of directed modularity in heterogeneous networks

In various real-world networks, the set of nodes is partitioned in groups or types forming densely connected clusters commonly called 'communities'. Several methods have been developed to find community structure in networks; one of them is based on modularity, a measure that represents the fraction of edges between nodes of the same type minus the expected fraction of such edges if they are established by a random process. Our work introduces a dynamic generation model of directed heterogeneous networks in which the set of nodes is partitioned in two groups. Networks based on our model grow by the addition of new nodes and the creation of new edges at each instant of time according to the combination of two mechanisms: (i) affinity-weighted preferential attachment and (ii) reciprocity response. We characterise the dynamics and limit values for the sum of the in- and out-degree of the nodes of the network which help us to determine the tail of the degree distributions (in and out) for each type of nodes. We show that both distributions follow a power law with equal scaling exponent. Using the limit values associated with the degree properties of our networks, we derive the dynamics and the asymptotic behaviour of the directed network modularity and establish conditions to guarantee the occurrence of communities according to the type of nodes.