Learning Global Temporal Dynamics in Sparse Networks via Cycle Counts

Statistical modeling and inference for temporal networks are increasingly important across modern applications, yet remain challenging in the sparse regime with evolving network sizes and temporal dependence. We propose a tractable model that captures these key features in a unified framework. Under this model, we establish geometric ergodicity and characterize the asymptotic behavior of cycle counts. These counts provide informative low-order summaries of temporal network dynamics. Moreover, we develop method-of-moments estimators based on cycle counts. We prove identifiability and uniqueness of the parameter estimates, and establish the strong consistency and asymptotic normality. To support uncertainty quantification, we further propose a simulation-based plug-in estimator of the asymptotic variance. Extensive simulations and a real data example demonstrate that our methods are accurate and effective for sparse, size-varying temporal networks.