Using Three States for Binary Consensus on Complete Graphs

Abstract

We consider the binary consensus problem where each node in the network initially observes one of two states and the goal for each node is to eventually decide which one of the two states was initially held by the majority of the nodes. Each node contacts other nodes and updates its current state based on the state communicated by the last contacted node. We assume that both signaling (the information exchanged at node contacts) and memory (computation state at each node) are limited and restrict our attention to systems where each node can contact any other node (i.e., complete graphs). It is well known that for systems with binary signaling and memory, the probability of reaching incorrect consensus is equal to the fraction of nodes that initially held the minority state. We show that extending both the signaling and memory by just one state dramatically improves the reliability and speed of reaching the correct consensus. Specifically, we show that the probability of error decays exponentially with the number of nodes N and the convergence time is logarithmic in N for large N . We also examine the case when the state is ternary and signaling is binary. The convergence of this system to consensus is again shown to be logarithmic in N for large N , and is therefore faster than purely binary systems. The type of distributed consensus problems that we study arises in the context of decentralized peerto-peer networks, e.g. sensor networks and opinion formation in social networks -our results suggest that robust and efficient protocols can be built with rather limited signaling and memory.

Counting with Population Protocols

“…In [6], [7], the authors propose a four-state protocol that solves the majority problem with an expected convergence parallel time logarithmic in n. However, the expected convergence time is infinite when κ approaches 0. The authors in [3], [8] propose a three-state protocol that converges with some probability δ, and whose parallel time is logarithmic in n if κ is large enough, i.e κ = O( √ n log n). Finally, the closest work to ours is the one of Alistarh et al [1].…”

“…A lot of work has been devoted to determine the tasks that can be solved in the population protocol model, as well as to study their complexities in terms of memory and convergence time [1], [3], [6], [7], [8]. One of the most studied tasks is the majority task.…”

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

“…The closest problems to the one we address are the computation of the majority (see [4], [6], [2], [9], [1]). In this problem, all the agents start in one of two distinct states and they eventually converge to 1 if κ > 0 (i.e.…”

“…Moreover, the expected convergence time is infinite when n A and n B are close to each other (that is κ approaches 0). Angluin et al [2] and Perron et al [9] propose a three-state protocol that converges with high probability after a convergence parallel time logarithmic in n but only if κ is large enough, i.e when |n A − n B | ≥ √ n log n. Alistarh et al [1] present a nice protocol based on an average-andconquer method to solve the majority problem. The first type of interaction is close to the one used in this paper while the second one is used to diffuse the result of the computation to the agents that have not decided yet.…”

“…As motivated by Aspnes [3], the objective of this model is to analyze the conditions under which nodes can converge to a state from which some global property of the system can be locally computed. A considerable amount of work has been done so far to determine which properties can emerge from pairwise interactions between finite-state nodes, together with the derivation of lower bounds on the time and space needed to reach such properties (e.g., [1], [2], [4], [6], [9]). In this work, we are primarily interested in counting problems.…”