A gossip-based algorithm for discrete consensus over heterogeneous networks

Quantized consensus assumes that the state of each node may only take nonnegative integer values. Reaching consensus under quantization is equivalent to determining a balanced assignment of identical tasks to nodes. In this paper we generalize this problem in two ways and denote the resulting framework discrete consensus. First, we consider tasks that are not identical: each one is characterized by its own weight. Secondly, we assume that nodes are not identical as well. As an example, in the case of task assignment, that we consider as a reference problem in this framework, nodes may have different speeds and should be assigned a total weight proportional to their speed. We provide a gossip-based distributed algorithm that aims to minimize the maximum execution time over nodes, whose convergence to a bounded set is guaranteed. We show that the convergence time of the proposed algorithm relies ultimately on the average meeting time between two agents performing a random walk on a graph.