Acceleration of gossip algorithms through the Euler–Poisson–Darboux Equation

Gossip algorithms and their accelerated versions have been studied exclusively in discrete time on graphs. In this work, we take a different approach and consider the scaling limit of gossip algorithms in both large graphs and large number of iterations. These limits lead to well-known partial differential equations (PDEs) with insightful properties. On lattices, we prove that the non-accelerated gossip algorithm of Boyd et al. (2006) converges to the heat equation, and the accelerated Jacobi polynomial iteration of Berthier et al. (2020) converges to the Euler–Poisson–Darboux (EPD) equation—a damped wave equation. Remarkably, with appropriate parameters, the fundamental solution of the EPD equation has the ideal gossip behaviour: a uniform density over an ellipsoid, whose radius increases at a rate proportional to $t$—the fastest possible rate for locally communicating gossip algorithms. This is in contrast with the heat equation where the density spreads on a typical scale of $\sqrt{t}$. Additionally, we provide simulations demonstrating that the gossip algorithms are accurately approximated by their limiting PDEs.