Dynamic sparse graphs with overlapping communities

Dynamic community detection concerns inferring how community memberships evolve over time, including the emergence, persistence, merging, and dissolution of groups in temporal networks. We propose a Bayesian nonparametric model for time-evolving sparse networks, which captures power-law degree distributions and dynamically overlapping communities. The model is constructed from vectors of completely random measures coupled through a latent Markov process governing the evolution of node affiliations. This construction provides a flexible and interpretable approach to model dynamic communities, naturally generalizing existing overlapping block models to the sparse and scale-free regimes. We establish asymptotic results characterizing sparsity and degree heterogeneity over time, and develop an approximate inference procedure for recovering time-varying community trajectories. Applications to synthetic and real-world dynamic networks show that the model accurately uncovers evolving community structure and yields interpretable temporal patterns.