Distributed Matrix Scaling and Application to Average Consensus in Directed Graphs

We propose a class of distributed iterative algorithms that enable the asymptotic scaling of a primitive column stochastic matrix, with a given sparsity structure, to a doubly stochastic form. We also demonstrate the application of these algorithms to the average consensus problem in networked multi-component systems. More specifically, we consider a setting where each node is in charge of assigning weights on its outgoing edges based on the weights on its incoming edges. We establish that, as long as the (generally directed) graph that describes the communication links between components is strongly connected, each of the proposed matrix scaling algorithms allows the system components to asymptotically assign, in a distributed fashion, weights that comprise a primitive doubly stochastic matrix. We also show that the nodes can asymptotically reach average consensus by executing a linear iteration that uses the time-varying weights (as they result at the end of each iteration of the chosen matrix scaling algorithm).