Spectral Curvature of Replicator Diffusions on Networks Authors

This work studies the geometric structure of stochastic replicator dynamics on interaction networks. We model evolutionary game dynamics as a diffusion process on the probability simplex endowed with the Shahshahani metric and analyze the curvature properties of the associated diffusion generator.

Our main result establishes an explicit lower bound for the Bakry–Émery curvature of the stochastic replicator diffusion. The bound links the intrinsic Ricci curvature of the Shahshahani simplex with the spectral properties of the interaction network through the largest eigenvalue of a weighted graph Laplacian induced by the simplex geometry.

As consequences of this curvature estimate, we derive logarithmic Sobolev inequalities, establish the existence of a strict spectral gap, and obtain exponential decay of relative entropy toward equilibrium under suitable conditions. These results provide a geometric interpretation of convergence rates in evolutionary dynamics and reveal a direct connection between information geometry, stochastic processes, and spectral graph theory.

Numerical simulations on several network topologies (ring, complete, star, and random graphs) illustrate how the Laplacian spectrum of the interaction network influences entropy dissipation and convergence behavior in stochastic replicator systems.