Convergence of Periodic Gossiping Algorithms

In deterministic gossiping, pairs of nodes in a network holding in general different values of a variable share information with each other and set the new value of the variable at each node to the average of the previous values. This occurs by cycling, sometimes periodically, through a designated sequence of nodes. There is an associated undirected graph, whose vertices are defined by the nodes and whose edges are defined by the node pairs which gossip over the cycle. Provided this graph is connected, deterministic gossiping asymptotically determines the average value of the initial values of the variables across all the nodes. The main result of the paper is to show that for the case when the graph is a tree, all periodic gossiping sequences including all edges of the tree just once actually have the same rate of convergence. The relation between convergence rate and topology of the tree is also considered.