A Probability Flux Approach for Binary Dynamics on Networks - Mohamad Riyadi; Agus Yodi Gunawan; Dewi Handayani (LPPM Institut Teknologi Sepuluh Nopember, 2025)
Binary-state dynamics on networks provide a powerfulframework for modeling epidemics and related spreading processes.Two main approaches are commonly used, namely exactcontinuous-time Markov chain (CTMC) formulations and meanfieldapproximations. The CTMC approach ensures stochastic accuracybut suffers from exponential state-space growth, whereasmean-field approximations lose reliability in heterogeneous orsmall networks. In this study, we formulate the master equationfor binary dynamics using a probability flux approach, yieldingan exact formulation for arbitrary networks. By integrating localtransition rules, network topology, and state-space partitioning,the framework captures microscopic dynamics while enablingmacroscopic analysis. Numerical simulations reveal that bothstate probabilities and expected infection levels are influencednot only by mean degree but also by structural heterogeneity.For instance, star and line topologies exhibit distinct behaviorsdespite having identical connectivity. Spectral analysis confirmsthe asymptotic stability of the disease-free equilibrium, whileinvariance under node relabeling emphasizes the role of graphsymmetries in reducing state-space complexity. This work extendsflux-based theory to network epidemics and provides a foundationfor future studies on adaptive or time-varying networks.Index Terms - Binary dynamics, probability flux, master equation,CTMC, ep