On the Distance Between the Rumor Source and Its Optimal Estimate in a Regular Tree

Abstract: This paper addresses the rumor source identification problem, where the goal is to find the origin node of a rumor in a network among a given set of nodes with the rumor. In this paper, we focus on a network represented by a regular tree which does not have any cycle and in which all nodes have the same number of edges connected to a node. For this network, we clarify that, with quite high probability, the origin node is within the distance 3 from the node selected by the optimal estimator, where the distance is the number of edges of the unique path connecting two nodes. This is clarified by the probability distribution of the distance between the origin and the selected node.