A Queueing Network-Based Distributed Laplacian Solver

A stochastic process on a network with connections to Laplacian systems of equations

“…We call such systems 'one-sink' Laplacian systems. In our subsequent work [8,9] we discuss this connection in detail.…”

“…In this work, we also show how the steady-state equation of the Data Collection Process maps to such Laplacian systems of equations. This provides a framework that we were able to use in a subsequent work to design and analyze a simple and efficient Laplacian solver [8].…”

“…0 . We know that for all u ∈ V \ {u s }, βJ(u) < J, β J(u), so by using induction and evolving queues using the one-step queue evolution equation (8), we can show that Q…”

“…The fact that the Data Collection Process mixes fast to its stationary distribution when started from the all-empty setting can be exploited to solve systems of equations such as (16) simply by allowing the process to get close enough to stationarity and then estimating the η by keeping track of the number of time slots for which each queue is occupied. This opens up the possibilities of distributed algorithms for effective resistance and other problems, some of which we have explored in [8][9][10]. Even if we consider graph problems on very large graphs, Laplacian systems of equations become tractable via this method, since random walks can be simulated very fast in modern computing systems for graphs with nodes in the millions (see, e.g., [28]).…”