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Two asynchronous distributed algorithms are presented for solving a linear equation of the form $Ax=b$ with at least one solution. The equation is simultaneously and asynchronously solved by $m$ agents assuming that each agent knows only a subset of the rows of the partitioned matrix $[A\ \ b]$, the estimates of the equation's solution generated by its neighbors, and nothing more. Neighbor relationships are characterized by a time-dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships. Each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by its neighbors which are transmitted with delays. The event time sequences of different agents are not assumed to be synchronized. It is shown that for any matrix-vector pair $(A, b)$ for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents’ event times, the algorithms cause all agents’ estimates to converge exponentially fast to the same solution to $Ax=b$ . The first algorithm requires a specific initialization step at each agent, and the second algorithm works for arbitrary initializations. Explicit expressions for convergence rates are provided, and a relation between local initializations and limiting consensus solutions is established, which is used to solve the least 2-norm solution. [ABSTRACT FROM PUBLISHER]

The distributed averaging problem is a consensus problem whose objective is to devise a protocol which will enable all the members of a group of autonomous agents to compute the average of the initial values of their individual consensus variables in a distributed manner. Periodic gossiping is a deterministic method for solving the distributed averaging problem by stipulating that each pair of agents which are allowed to gossip, do so repeatedly in accordance with a prespecified periodic schedule. Agent pairs which are allowed to gossip correspond to edges on a given connected, undirected graph. In general, the rate at which the agents' consensus variables converge to the desired average value depends on the order in which the gossips occur over a period. The main contributions of this paper are first to characterize the classes of periodic gossip sequences which have the same convergence rate and second to prove that if the graph of allowable gossips is a tree with each edge restricted to gossiping once per period, the convergence rate is quite surprisingly, fixed and invariant over all possible periodic gossip sequences the graph allows. To arrive at these results, a new and unusual graph theoretic concept, namely the transfer function of a node of an undirected graph, is used. Among all the trees with the same number of edges, optimal tree structures, which yield the fastest convergence rate, can then be sought. [ABSTRACT FROM AUTHOR]