A Distributed Algorithm for Solving Positive Definite Linear Equations over Networks with Membership Dynamics

TL;DR

This paper introduces a novel distributed asynchronous algorithm called Subset Equalizing for solving positive definite linear systems over dynamic networks with agents that can join and leave arbitrarily, ensuring convergence and efficiency.

Contribution

The paper presents a new algorithm and analytical framework for solving linear equations over networks with dynamic membership, including a novel Lyapunov function and connectivity concept.

Findings

SE guarantees boundedness, convergence, and exponential convergence.

SE performs well in volatile agent networks.

Groupwise Equalizing reduces bandwidth and energy compared to existing algorithms.

Abstract

This paper considers the problem of solving a symmetric positive definite system of linear equations over a network of agents with arbitrary asynchronous interactions and membership dynamics. The latter implies that each agent is allowed to join and leave the network at any time, for infinitely many times, and lose all its memory upon leaving. We develop Subset Equalizing (SE), a distributed asynchronous algorithm for solving such a problem. To design and analyze SE, we introduce a novel time-varying Lyapunov-like function, defined on a state space with changing dimension, and a generalized concept of network connectivity, capable of handling such interactions and membership dynamics. Based on them, we establish the boundedness, asymptotic convergence, and exponential convergence of SE, along with a bound on its convergence rate. Finally, through extensive simulation, we show that SE is effective in a volatile agent network and that a special case of SE, termed Groupwise Equalizing, is significantly more bandwidth/energy efficient than two existing algorithms in multi-hop wireless networks.