# Acceleration of Gossip Algorithms through the Euler-Poisson-Darboux Equation

arXiv: https://arxiv.org/abs/2202.10742 · HTML: https://mlsanity.com/paper/2202.10742/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2022.

## Built on by (most cited first, 1 of 1)

- [On the Effectiveness of the z-Transform Method in Quadratic Optimization](https://mlsanity.com/paper/2507.03404/) (2025)

## Abstract (arXiv)

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 √t. Additionally, we provide simulations demonstrating that the gossip algorithms are accurately approximated by their limiting PDEs.