Abstract: Continuous-time networks have attracted significant attention due to their widespread applications in various disciplines. A rich literature considers the community structure of the nodes, while few have accounted for the node heterogeneity of interaction propensities. To simultaneously account for both the self-exciting feature and the node heterogeneity, we propose a model based on the Hawkes process, which allows the interaction intensity to vary flexibly with incurred nodes and their affiliated communities. We derive the likelihood function using the immigration-birth representation of the Hawkes process and develop an innovative expectation-maximization algorithm with membership refinement to tackle the computational challenge. Further, we establish the consistency of parameter estimation under mild assumptions. The effectiveness of our model is validated by extensive simulation studies on synthetic data as well as two real-world applications.

Key words and phrases: Community structure, dynamic network, EM algorithm, Hawkes process, node heterogeneity.