Evolution equations on co-evolving graphs: long-time behaviour and the graph continuity equation
José Antonio Carrillo, Antonio Esposito, László Mikolás
TL;DR
The paper develops a rigorous framework for evolution equations on co-evolving graphs, linking vertex-mass dynamics to a nonlinear graph-continuity equation via flow maps solving characteristic equations. It proves well-posedness for the coupled nonlinear Euler system governing the vertex mass and edge weights, using contraction arguments and explicit representations of the edge weights. By embedding Euler dynamics into a Vlasov-type graph-continuity equation and leveraging disintegration, the authors obtain well-posedness and stability results, including contraction in $L^2$-based metrics under upwind flux. For compact graphs with pointwise, monotone velocities, they establish long-time convergence to a uniform mass distribution, with $\,\sigma_t$ converging to $\delta_{M/\mu(K)}\otimes\mu$ in the weak-* sense, highlighting consensus formation on graphs with co-evolving topology.
Abstract
We focus on evolution equations on co-evolving, infinite, graphs and establish a rigorous link with a class of nonlinear continuity equations, whose vector fields depend on the graphs considered. More precisely, weak solutions of the so-called graph-continuity equation are shown to be the push-forward of their initial datum through the flow map solving the associated characteristics' equation, which depends on the co-evolving graph considered. This connection can be used to prove contractions in a suitable distance, although the flow on the graphs requires a too limiting assumption on the overall flux. Therefore, we consider upwinding dynamics on graphs with pointwise and monotonic velocity and prove long-time convergence of the solutions towards the uniform mass distribution.