Capturing Microdynamics of Temporal Network Growth: An HMM-Based Model of Triadic Closure

Complex networks with their nontrivial topological features and rich patterns of interactions are commonly used to model real-world systems, including social networks, biological systems, and communication networks. Triads, three-node subgraphs that encapsulate local connectivity, are the elementary structural motifs of such networks. Triadic closure governs the growth of social and complex networks by capturing the microlevel dynamics that drive their evolution and cohesion. In this work, we present a theoretical, generative framework that models triadic closure as a hidden stochastic process, governed by a six-state Hidden Markov Model (HMM). By aggregating the 64 possible directed 3-node microstates into interpretable macrolevel states, we construct an HMM that describes the probabilistic evolution of triadic motifs over time and the transitions that lead to closure events. We prove through theoretical results, including the existence of stationary distributions, bounds on closure probabilities, robustness under transition perturbations, and statistical anomaly detection via divergence measures. The primary contribution of this work is theoretical, but the work is further validated using symbolic examples and experiments. We used both simulated data and the Higgs Twitter interaction dataset to analyze model behavior and closure stability. The proposed HMM is bench-marked against three existing temporal network baselines, the Motif Transition Model MTM, Temporal Exponential Random Graph Model TERGM, and Dynamic Stochastic DSBM is compared by examining the predictive performance in terms of AUC and mean log-likelihood. The proposed HMM framework demonstrates strong predictive performance with an AUC of 0.88 and a stable generalization over various horizons of prediction. Theoretical and empirical work in tandem yield an interpretable and mathematically well-grounded analysis of the time-evolving networks’ local structure dynamics.