Degree heterogeneity shapes escape mechanisms in networks of diffusively coupled bistable elements

Abstract

For fully connected populations of diffusively coupled bistable elements, we identified three qualitatively distinct mechanisms of noise-induced escape as coupling strength varies [H. Ishii and H. Kori, arXiv:2512.01388 (2025)]. Here we generalize these results to a class of networked systems and demonstrate that degree heterogeneity (i.e., variability in node degree) shapes escape mechanisms alongside coupling strength. In applied contexts, networks of noisy bistable elements provide a minimal conceptual framework for understanding abrupt state transitions in complex systems. Theoretically, a quantitative approach to escape is challenging because nonlinearity, network interactions, and dynamical noise jointly shape the collective dynamics. We extend the analytical framework developed for the fully connected model to a class of networked systems based on the annealed network approximation. We derive three effective one-dimensional descriptions of collective escape dynamics. We validate our theoretical predictions for mean escape times by direct numerical simulations. Our analysis reveals that the validity and quantitative behavior of the reduced descriptions depend on degree heterogeneity in addition to coupling strength. This work extends the classification of escape mechanisms to networked bistable elements. Furthermore, our analytical framework provides tools for understanding synergistic phenomena arising from the interplay of nonlinearity, diffusive coupling, and dynamical noise.

Figures (4)

• Figure 1: The bifurcation point of DMFD, $K_2$, depends on the normalized degree heterogeneity $\kappa / N$. (a) Bifurcation diagram for $\kappa / N = 0.01$. (b) Bifurcation diagram for $\kappa / N = 0.2$. (c) The bifurcation point $K_2$ as a function of $\kappa / N$.
• Figure 2: Normalized degree heterogeneity $\kappa / N$ and mean degree of networks used in the numerical analysis.
• Figure 3: Numerically measured mean escape times (markers) and theoretical predictions (lines) as functions of normalized degree heterogeneity $\kappa / N$ at different values of coupling strength $K$. The mean escape time of the uncoupled system (i.e., $K = 0$) is denoted by $T_0$ [Eq. \ref{['eq:T0']}]. Marker shape and size indicate the network model and the number of nodes, respectively. Error bars represent standard errors. (a) The influence of $\kappa / N$ was not visible at $K = 0.01$. (b) Systems on networks with large $\kappa / N$ deviated from the prediction of NlinMFFPE at $K = 0.1$. (c) Network characteristics, including $\kappa / N$, strongly affected mean escape times at $K = 1$. (d) Deviations from the prediction of SMFD shrank as $K$ increased. (e) At $K = 100$, the numerical results and the prediction of SMFD agreed well and coincided with $T_\infty(\kappa / N)$ [Eq. \ref{['eq:T.infty']}].
• Figure 4: Dependence of the numerically measured mean escape times on the coupling strength $K$. Larger values of $\kappa / N$ (color-coded) suppressed the influence of $K$.