A Probability Flux Approach for Binary Dynamics on Networks - Mohamad Riyadi; Agus Yodi Gunawan; Dewi Handayani ((IJCSAM) International Journal of Computing Science and Applied Mathematics, 2025)
Binary-state dynamics on networks provide a powerful framework for modeling epidemics and related spreading processes. Two main approaches are commonly used, namely exact continuous-time Markov chain (CTMC) formulations and meanfield approximations. The CTMC approach ensures stochastic accuracy but suffers from exponential state-space growth, whereas mean-field approximations lose reliability in heterogeneous or small networks. In this study, we formulate the master equation for binary dynamics using a probability flux approach, yielding an exact formulation for arbitrary networks. By integrating local transition rules, network topology, and state-space partitioning, the framework captures microscopic dynamics while enabling macroscopic analysis. Numerical simulations reveal that both state probabilities and expected infection levels are influenced not only by mean degree but also by structural heterogeneity. For instance, star and line topologies exhibit distinct behaviors despite having identical connectivity. Spectral analysis confirms the asymptotic stability of the disease-free equilibrium, while invariance under node relabeling emphasizes the role of graph symmetries in reducing state-space complexity. This work extends flux-based theory to network epidemics and provides a foundation for future studies on adaptive or time-varying networks. Index Terms - Binary dynamics, probability flux, master equation, CTMC, ep