From Local Updates to Global Balance: A Framework for Distributed Matrix Scaling

TL;DR

This paper extends matrix scaling algorithms to decentralized settings, demonstrating convergence of local updates to a doubly stochastic matrix, with implications for distributed networks and agent-based models.

Contribution

It generalizes the Sinkhorn algorithm to arbitrary local updates and characterizes convergence using an extended Birkhoff's theorem, applied to decentralized graph weight adjustments.

Findings

Local normalization algorithms converge to doubly stochastic matrices.

Decentralized random walks on directed graphs achieve uniform stationary distributions.

Theoretical extension of Birkhoff's theorem for arbitrary normalization sequences.

Abstract

This paper investigates matrix scaling processes in the context of local normalization algorithms and their convergence behavior. Starting from the classical Sinkhorn algorithm, the authors introduce a generalization where only a single row or column is normalized at each step, without restrictions on the update order. They extend Birkhoff's theorem to characterize the convergence properties of these algorithms, especially when the normalization sequence is arbitrary. A novel application is explored in the form of a Decentralized Random Walk (DRW) on directed graphs, where agents modify edge weights locally without global knowledge or memory. The paper shows that such local updates lead to convergence towards a doubly stochastic matrix, ensuring a uniform stationary distribution across graph vertices. These results not only deepen the theoretical understanding of matrix scaling but also open avenues for distributed and agent-based models in networks.