Spectral Characterization of Stochastic Epidemic Variability on Regular Networks via the Kirchhoff Index

This paper develops a stochastic framework for epidemic spreading on regular networks based on the Fokker–Planck equation and spectral graph theory. Starting from an Itô stochastic differential equation, we derive the corresponding Fokker–Planck equation and characterize the stationary distribution of epidemic fluctuations. A fundamental connection between the stationary covariance structure and the Kirchhoff index is established, showing that network topology plays a central role in determining the magnitude of stochastic variability. Using Lyapunov methods, we prove exponential stability of the disease-free equilibrium and show that the convergence rate is governed by the algebraic connectivity of the underlying network. In addition, an exact description of the mean-square dynamics is obtained, revealing how the spectral properties of regular graphs control both transient behavior and long-term fluctuations. Numerical experiments on complete, cycle, circulant, and random regular graphs confirm the theoretical predictions and illustrate the influence of network connectivity on epidemic variability and relaxation dynamics.