  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.

Data Structures and Algorithms