Faster Local Solvers for Graph Diffusion Equations

TL;DR

This paper introduces a new framework for efficiently solving graph diffusion equations locally, significantly speeding up computations and enabling scalable applications in large graphs and graph neural networks.

Contribution

It presents a novel local diffusion process framework that improves upon existing methods, providing provably sublinear, parallelizable algorithms for large-scale graph diffusion problems.

Findings

Achieves up to 100x speedup in computing diffusion vectors.

Effective for large-scale and dynamic graphs.

Facilitates more efficient local message passing in GNNs.

Abstract

Efficient computation of graph diffusion equations (GDEs), such as Personalized PageRank, Katz centrality, and the Heat kernel, is crucial for clustering, training neural networks, and many other graph-related problems. Standard iterative methods require accessing the whole graph per iteration, making them time-consuming for large-scale graphs. While existing local solvers approximate diffusion vectors through heuristic local updates, they often operate sequentially and are typically designed for specific diffusion types, limiting their applicability. Given that diffusion vectors are highly localizable, as measured by the participation ratio, this paper introduces a novel framework for approximately solving GDEs using a local diffusion process. This framework reveals the suboptimality of existing local solvers. Furthermore, our approach effectively localizes standard iterative solvers by designing simple and provably sublinear time algorithms. These new local solvers are highly parallelizable, making them well-suited for implementation on GPUs. We demonstrate the effectiveness of our framework in quickly obtaining approximate diffusion vectors, achieving up to a hundred-fold speed improvement, and its applicability to large-scale dynamic graphs. Our framework could also facilitate more efficient local message-passing mechanisms for GNNs.