Inter-Cluster Transmission Control Using Graph Modal Barriers

Abstract

In this paper we consider the problem of transmission across a graph and develop a method for effectively controlling/restricting it with limited resources. Transmission can represent information transfer across a social network, spread of a malicious virus across a computer network, or spread of an infectious disease across communities. The key insight is to appropriately reduce the capacity for transmission on inter-cluster edges of the graph. To that end, the key contribution of the paper is to develop algorithms for computing a real, positive-valued distribution over the edges that gives a measure of the role each edge plays in being an inter-cluster edge. We refer to this distribution as the “ resistance ” and its computation is based on the eigenvectors of the graph Laplacian. Selectively reducing the weights (implying reduced transmission rate) on the critical edges based on this computed distribution helps establish barriers between the clusters ( modal barriers ) and thus limit the transmission from one cluster to another. Our proposed method for edge weight reduction for inter-cluster transmission control depends only on the graph topology and not on the exact model of the transmission dynamics. Unlike other work on graph partitioning and clustering, we completely circumvent the associated computational complexities by assigning real values to edges instead of performing discrete graph cuts. This allows us to provide strong theoretical results on our proposed methods. We also develop approximations that allow low complexity distributed computation of the barrier weights using only neighborhood communication.