Using Three States for Binary Consensus on Complete Graphs

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 a variety of applications including those of 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.

tr-2008-114.pdf
PDF file

In  IEEE Infocom 2009

Publisher  IEEE Communications Society
Copyright © 2007 IEEE. Reprinted from IEEE Communications Society. This material is posted here with permission of the IEEE. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.

Details

TypeProceedings
> Publications > Using Three States for Binary Consensus on Complete Graphs