Graphons, Geometry, and Dynamics: Forward and Inverse Perspectives

Abstract: In this work, we explore the interplay between graph limit theory, the geometry of underlying probability spaces, spectral theory, and network dynamical systems. We investigate two primary questions concerning forward and inverse perspectives: first, whether a graphon retains information about the geometry of the space on which it is defined, and second, whether spectral properties can distinguish graphons that originate from different geometric spaces. To address these questions, we differentiate between combinatorial equivalence and geometric structure, highlighting how these concepts are captured simultaneously by the class of pure graphons. Furthermore, we construct explicit examples of isospectral graphons -- graphons whose integral operators share the same spectrum -- that differ in their underlying geometry. By utilizing the heat kernels of Neumann- and Dirichlet-isospectral drums, we demonstrate that these graphons are not combinatorially equivalent. Finally, we establish new connections between the geometric aspects of graph limit theory and dynamical systems by analyzing a continuum Kuramoto model with graphon-defined interactions. We demonstrate that while isospectrality implies identical stability properties in certain cases, this correspondence breaks down when the differing boundary conditions of our specific Neumann and Dirichlet constructions are considered.