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.