Red Light Green Light Method for Solving Large Markov Chains

TL;DR

This paper introduces a new distributed algorithm for computing stationary distributions of large Markov chains, unifying and improving upon existing methods with proven exponential convergence and enhanced efficiency.

Contribution

The authors propose a novel controlled, distributed algorithm that generalizes multiple existing methods and achieves faster convergence for large Markov chains.

Findings

Proves exponential convergence of the proposed method.

Demonstrates high efficiency and faster convergence than existing algorithms.

Includes a unified framework encompassing various known algorithms.

Abstract

Discrete-time discrete-state finite Markov chains are versatile mathematical models for a wide range of real-life stochastic processes. One of most common tasks in studies of Markov chains is computation of the stationary distribution. Without loss of generality, and drawing our motivation from applications to large networks, we interpret this problem as one of computing the stationary distribution of a random walk on a graph. We propose a new controlled, easily distributed algorithm for this task, briefly summarized as follows: at the beginning, each node receives a fixed amount of cash (positive or negative), and at each iteration, some nodes receive `green light' to distribute their wealth or debt proportionally to the transition probabilities of the Markov chain; the stationary probability of a node is computed as a ratio of the cash distributed by this node to the total cash distributed by all nodes together. Our method includes as special cases a wide range of known, very different, and previously disconnected methods including power iterations, Gauss-Southwell, and online distributed algorithms. We prove exponential convergence of our method, demonstrate its high efficiency, and derive scheduling strategies for the green-light, that achieve convergence rate faster than state-of-the-art algorithms.