Graph dynamical geometrization: A Diffusion-Based Formulation

Abstract. We develop graph dynamical geometrization (GDG), a framework showing that any diffusion operator Q induces a time-dependent Euclidean geometry on a graph through the propagator K_{t} based on Mittag-Leffler matrix functions. The associated flow-resistance metric F_{t} is always Euclidean and yields a spherical embedding whose radius contracts monotonically with time. This construction unifies classical, fractional, long-range, and nonconservative diffusions under a common geometric viewpoint. Beyond distances, we introduce angular, wedge, and dihedral observables that capture higher-order correlations in the evolving diffusion states. These quantities reveal multiscale organization of transport, distinguishing normal, subdiffusive, and superdiffusive regimes, and clarifying how different operators imprint distinct geometric signatures on a graph. Altogether, GDG provides a principled correspondence between dynamics and geometry, offering new tools for analyzing operator-driven processes on networks and interpreting their structural and dynamical behavior.

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