Convergence time evaluation of a gossip algorithm over signed graphs

Convergence time is investigated for a gossip algorithm over a connected signed graph, where each edge of the graph has positive or negative sign. The algorithm is an iterative procedure. At each time, (i) two nodes directly connected with an edge are chosen randomly, (ii) they exchange their values according to the sign of the edge, and (iii) they update their values as the average of each node´s value and its received value. It is shown that the values of the algorithm always converge in mean square, where a bipartite consensus or a trivial consensus is achieved. A convergence time is defined as the smallest time such that it takes for the values of the algorithm to get within a given neighborhood of the consensus value with high probability, regardless of initial state. An upper bound of the convergence time is given in terms of a characteristic value of the given graph.