Plurality Consensus in Arbitrary Graphs: Lessons Learned from Load Balancing

We consider plurality consensus in networks of n nodes. Initially, each node has one of k opinions. The nodes execute a (randomized) distributed protocol to agree on the plurality opinion (the opinion initially supported by the most nodes). In certain types of networks the nodes can be quite cheap and simple, and hence one seeks protocols that are not only time efficient but also simple and space efficient. Typically, protocols depend heavily on the employed communication mechanism, which ranges from sequential (only one pair of nodes communicates at any time) to fully parallel (all nodes communicate with all their neighbors at once) and everything in-between. We propose a framework to design protocols for a multitude of communication mechanisms. We introduce protocols that solve the plurality consensus problem and are, with probability 1-o(1), both time and space efficient. Our protocols are based on an interesting relationship between plurality consensus and distributed load balancing. This relationship allows us to design protocols that generalize the state of the art for a large range of problem parameters.

[1]  Eli Upfal,et al.  Balanced Allocations , 1999, SIAM J. Comput..

[2]  Thomas Sauerwald,et al.  Tight Bounds for Randomized Load Balancing on Arbitrary Network Topologies , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[3]  Colin Cooper,et al.  Fast Consensus for Voting on General Expander Graphs , 2015, DISC.

[4]  Luca Trevisan,et al.  Simple Dynamics for Majority Consensus , 2013, SPAA 2014.

[5]  Paul G. Spirakis,et al.  Determining majority in networks with local interactions and very small local memory , 2014, Distributed Computing.

[6]  Stefan Schmid,et al.  Distributed computation of the mode , 2008, PODC '08.

[7]  Milan Vojnovic,et al.  Using Three States for Binary Consensus on Complete Graphs , 2009, IEEE INFOCOM 2009.

[8]  P. Berenbrink,et al.  Randomized diffusion for indivisible loads , 2011, SODA 2011.

[9]  David Peleg,et al.  Local majorities, coalitions and monopolies in graphs: a review , 2002, Theor. Comput. Sci..

[10]  Andrea E. F. Clementi,et al.  Plurality Consensus in the Gossip Model , 2014, SODA.

[11]  Colin Cooper,et al.  The Power of Two Choices in Distributed Voting , 2014, ICALP.

[12]  Luca Cardelli,et al.  Programmable chemical controllers made from DNA. , 2013, Nature nanotechnology.

[13]  Moez Draief,et al.  Convergence Speed of Binary Interval Consensus , 2010, 2010 Proceedings IEEE INFOCOM.

[14]  Luca Cardelli,et al.  The Cell Cycle Switch Computes Approximate Majority , 2012, Scientific Reports.

[15]  Chen Avin,et al.  How to Explore a Fast-Changing World (Cover Time of a Simple Random Walk on Evolving Graphs) , 2008, ICALP.

[16]  Johannes Gehrke,et al.  Gossip-based computation of aggregate information , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[17]  David J. Aldous,et al.  Lower bounds for covering times for reversible Markov chains and random walks on graphs , 1989 .

[18]  David Eisenstat,et al.  The computational power of population protocols , 2006, Distributed Computing.

[19]  David Eisenstat,et al.  A Simple Population Protocol for Fast Robust Approximate Majority , 2007, DISC.

[20]  Andrea E. F. Clementi,et al.  Distributed community detection in dynamic graphs , 2013, Theor. Comput. Sci..

[21]  Stephen P. Boyd,et al.  Randomized gossip algorithms , 2006, IEEE Transactions on Information Theory.

[22]  Desh Ranjan,et al.  Balls and bins: A study in negative dependence , 1996, Random Struct. Algorithms.

[23]  S. Muthukrishnan,et al.  Dynamic Load Balancing by Random Matchings , 1996, J. Comput. Syst. Sci..

[24]  Martin Vetterli,et al.  Interval consensus: From quantized gossip to voting , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[25]  Elchanan Mossel,et al.  Majority dynamics and aggregation of information in social networks , 2012, Autonomous Agents and Multi-Agent Systems.

[26]  Dan Alistarh,et al.  Fast and Exact Majority in Population Protocols , 2015, PODC.

[27]  George Giakkoupis,et al.  Efficient Plurality Consensus, Or: the Benefits of Cleaning up from Time to Time , 2016, ICALP.

[28]  Martin Vetterli,et al.  The Distributed Multiple Voting Problem , 2011, IEEE Journal of Selected Topics in Signal Processing.

[29]  James Aspnes,et al.  An Introduction to Population Protocols , 2007, Bull. EATCS.