Probabilistic analysis of counting protocols in large-scale asynchronous and anonymous systems

Abstract

Abstract-We consider a large system populated by n anonymous nodes that communicate through asynchronous and pairwise interactions. The aim of these interactions is for each node to converge toward a global property of the system, that depends on the initial state of each node. In this paper we focus on both the counting and proportion problems. We show that for any δ ∈ (0, 1), the number of interactions needed per node to converge is O(ln(n/δ)) with probability at least 1 − δ. We also prove that each node can determine, with any high probability, the proportion of nodes that initially started in a given state without knowing the number of nodes in the system. This work provides a precise analysis of the convergence bounds, and shows that using the 4-norm is very effective to derive useful bounds.