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This paper considers the problem of solving a symmetric positive definitesystem of linear equations over a network of agents with arbitrary asynchronousinteractions and membership dynamics. The latter implies that each agent isallowed 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), adistributed asynchronous algorithm for solving such a problem. To design andanalyze SE, we introduce a novel time-varying Lyapunov-like function, definedon a state space with changing dimension, and a generalized concept of networkconnectivity, capable of handling such interactions and membership dynamics.Based on them, we establish the boundedness, asymptotic convergence, andexponential convergence of SE, along with a bound on its convergence rate.Finally, through extensive simulation, we show that SE is effective in avolatile agent network and that a special case of SE, termed GroupwiseEqualizing, is significantly more bandwidth/energy efficient than two existingalgorithms in multi-hop wireless networks.

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