A Probability Flux Approach for Binary Dynamics on Networks
DOI:
https://doi.org/10.12962/j24775401.ijcsam.v11i2.8848Kata Kunci:
Binary dynamics, probability flux,, master equation, CTMC,, epidemic processes, network topology, structural heterogeneityAbstrak
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
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