Solving Local Linear Systems with Boundary Conditions Using Heat Kernel Pagerank

TL;DR

This paper introduces an efficient algorithm leveraging the Dirichlet heat kernel pagerank to solve local linear systems with boundary conditions on graphs, providing approximate solutions with controlled error.

Contribution

It presents a novel method combining heat kernel pagerank with boundary conditions to efficiently approximate solutions to local linear systems in graphs.

Findings

Algorithm computes approximate solutions with multiplicative and additive error.

The method performs $O( ext{error}^{-5}s^3 ext{log}(s^3 ext{error}^{-1}) ext{log} n)$ random walk steps.

The approach is scalable for large graphs with local systems.

Abstract

We present an efficient algorithm for solving local linear systems with a boundary condition using the Green's function of a connected induced subgraph related to the system. We introduce the method of using the Dirichlet heat kernel pagerank vector to approximate local solutions to linear systems in the graph Laplacian satisfying given boundary conditions over a particular subset of vertices. With an efficient algorithm for approximating Dirichlet heat kernel pagerank, our local linear solver algorithm computes an approximate local solution with multiplicative and additive error $\epsilon$ by performing $O(\epsilon^{-5}s^3\log(s^3\epsilon^{-1})\log n)$ random walk steps, where $n$ is the number of vertices in the full graph and $s$ is the size of the local system on the induced subgraph.