Long-memory and cyclical link prediction in temporal social networks

<title>Abstract</title> Temporal social networks contain persistent tie activation and periodic communication rhythms, but most link prediction models either emphasize short-term recurrence or rely on black-box embeddings that offer no interpretable summary of temporal structure. We propose a Bayesian discrete temporal link prediction framework that captures such dependence through Gegenbauer long-memory operators acting on a network-level latent dynamic process governing edge formation. Edges are modeled as conditionally independent Bernoulli variables given the latent process, embedded within the broader multivariate Bernoulli family for possible extensions to higher-order edge dependence. The Gegenbauer parameter-ization uses a memory parameter d to control hyperbolic decay and a frequency parameter u to locate cyclical components, yielding interpretable summaries of temporal dependence in binary network data. Two extensions accommodate non-stationarity: a B-spline time-varying specification for smooth structural drift, and a Markov-switching specification for abrupt regime changes; for the latter we establish strict stationarity and ergodicity of the finite-truncation implementation. Bayesian inference via Hamiltonian Monte Carlo provides posterior uncertainty for memory parameters and forward link predictions. Application to the SNAP email-Eu-core temporal communication dataset shows that long-memory MulBer specifications substantially improve held-out link prediction over static and traditional baselines, while the full Gegenbauer specification recovers a sharply concentrated weekly cyclical frequency (ˆ u = 0.622, implied period 1 ≈ 7.0 days), providing an interpretable network-level summary of recurring communication rhythms.