Triangulated Laman Graphs, Local Stochastic Matrices, and Limits of Their Products

TL;DR

This paper introduces restricted triangulated Laman graphs and provides conditions under which products of associated stochastic matrices converge to a unique invariant distribution, with an explicit formula for the limit.

Contribution

It defines a new class of graphs and derives explicit conditions and formulas for the convergence of products of stochastic matrices based on these graphs.

Findings

Conditions for convergence of stochastic matrix products

Explicit formula for the limit invariant distribution

Introduction of restricted triangulated Laman graphs

Abstract

We derive conditions on the products of stochastic matrices guaranteeing the existence of a unique limit invariant distribution. Belying our approach is the hereby defined notion of restricted triangulated Laman graphs. The main idea is the following: to each triangle in the graph, we assign a stochastic matrix. Two matrices can be adjacent in a product only if their corresponding triangles share an edge in the graph. We provide an explicit formula for the limit invariant distribution of the product in terms of the individual stochastic matrices.