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Random walks on dense graphs and graphons

Graph-limit theory focuses on the convergence of sequences of graphs when the number of nodes becomes arbitrarily large. This framework defines a continuous version of graphs allowing for the study of dynamical systems on very large graphs, where classical methods would become computationally intractable. Through an approximation procedure, the standard system of coupled ordinary differential equations is replaced by a nonlocal evolution equation on the unit interval. In this work, we adopt this methodology to explore the continuum limit of random walks, a popular model for diffusion on graphs. We focus on two classes of processes on dense weighted graph, in discrete and in continuous time, whose dynamics are encoded in the transition matrix and the random-walk Laplacian. We also show that previous works on the discrete heat equation, associated to the combinatorial Laplacian, fall within the scope of our approach. Finally, we apply the spectral theory of operators to characterize the relaxation time of the process in the continuum limit.