Solving Local Linear Systems with Boundary Conditions Using Heat Kernel Pagerank

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.