Democratic Consensus and the Local Majority Rule

In this paper we study a rather generic communication/ coordination/ computation problem: in a finite network of agents, each initially having one of the two possible states, can the majority initial state be computed and agreed upon (i.e., can a democratic consensus be reached) by means of iterative application of the local majority rule. We show that this task is failure-free only in the networks that are nowhere truly local. In other words, the idea of solving a truly global task (reaching consensus on majority ) by means of truly local computation only (local majority rule) is doomed for failure. We also show that even well connected networks of agents that are nowhere truly local might fail to reach democratic consensus when the local majority rule is applied iteratively. Structural properties of democratic consensus computers, i.e., the networks in which iterative application of the local majority rule always yields consensus in the initial majority state, are presented.

[1]  Svatopluk Poljak,et al.  On pre-periods of discrete influence systems , 1986, Discret. Appl. Math..

[2]  Peter D. Mosses.  A Modular SOS for ML Concurrency Primitives , 1999 .

[3]  David Peleg,et al.  Size Bounds for Dynamic Monopolies , 1998, Discret. Appl. Math..

[4]  David Peleg,et al.  Extremal bounds for probabilistic polling in graphs , 2000, SIROCCO.

[5]  Gabriel Bracha,et al.  An O(log n) expected rounds randomized byzantine generals protocol , 1987, JACM.

[6]  Shay Kutten,et al.  Fault-Local Distributed Mending , 1999, J. Algorithms.

[7]  David Peleg,et al.  Distributed Probabilistic Polling and Applications to Proportionate Agreement , 1999, ICALP.

[8]  M. Degroot.  Reaching a Consensus , 1974 .

[9]  Christian N. S. Pedersen,et al.  Pseudoknots in RNA secondary structures , 2000, RECOMB '00.

[10]  Masafumi Yamashita,et al.  A probabilistic local majority polling game on weighted directed graphs with an application to the distributed agreement problem , 2000, Networks.

[11]  Eric Goles Ch.,et al.  Periodic behaviour of generalized threshold functions , 1980, Discret. Math..

[12]  Eli Upfal,et al.  Fault Tolerance in Networks of Bounded Degree , 1988, SIAM J. Comput..

[13]  Leslie Lamport,et al.  The Byzantine Generals Problem , 1982, TOPL.

[14]  G. Sullivan.  The complexity of system-level fault diagnosis and diagnosability , 1986 .

[15]  Fabrizio Luccio,et al.  Monotone Dynamos in Tori , 1999, SIROCCO.

[16]  J. French.  A formal theory of social power. , 1956, Psychology Review.

[17]  Svatopluk Poljak,et al.  On an application of convexity to discrete systems , 1986, Discret. Appl. Math..

[18]  Gadi Moran.  Parametrization for stationary patterns of the r-majority operators on 0-1 sequences , 1994, Discret. Math..

[19]  Fabrizio Luccio,et al.  Irreversible Dynamos in Tori , 1998, Euro-Par.

[20]  Eric Goles,et al.  Positive Automata Networks , 1986 .

[21]  Shay Kutten,et al.  Tight fault locality , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[22]  David K. Gifford,et al.  Weighted voting for replicated data , 1979, SOSP '79.

[23]  G. Moran.  ON THE PERIOD-TWO-PROPERTY OF THE MAJORITY OPERATOR IN INFINITE GRAPHS , 1995 .

[24]  Eric Goles,et al.  Neural and automata networks , 1990 .

[25]  Hector Garcia-Molina,et al.  How to assign votes in a distributed system , 1985, JACM.

[26]  Krzysztof Diks,et al.  System Diagnosis with Smallest Risk of Error , 1996, WG.

[27]  Eric Goles Ch.,et al.  Comportement periodique des fonctions a seuil binaires et applications , 1981, Discret. Appl. Math..

[28]  Rasmus Pagh.  A New Trade-Off for Deterministic Dictionaries , 2000, SWAT.

[29]  Noah E. Friedkin,et al.  A formal theory of social power , 1986 .

[30]  Piotr Berman,et al.  Voting as the Optimal Static Pessimistic Scheme for Managing Replicated Data , 1994, IEEE Trans. Parallel Distributed Syst..

[31]  Igor Walukiewicz.  Local Logics for Traces , 2002, J. Autom. Lang. Comb..

[32]  Marcin Paprzycki,et al.  Distributed Computing: Fundamentals, Simulations and Advanced Topics , 2001, Scalable Comput. Pract. Exp..

[33]  M. P. Herlihy.  REPLICATION METHODS FOR ABSTRACT DATA TYPES , 1984 .

[34]  D. Peleg.  Local Majority Voting, Small Coalitions and Controlling Monopolies in Graphs: A Review , 1996 .

[35]  Fabrizio Luccio,et al.  Irreversible Dynamos in Butterflies , 1999, Colloquium on Structural Information & Communication Complexity.

[36]  Eli Berger.  Dynamic Monopolies of Constant Size , 2001, J. Comb. Theory, Ser. B.

[37]  Peter D. Mosses.  A Modular SOS for Action Notation , 1999 .

[38]  Ivan Damgård,et al.  Improved Non-committing Encryption Schemes Based on a General Complexity Assumption , 2000, CRYPTO.

[39]  Hector Garcia-Molina,et al.  Consistency in a partitioned network: a survey , 1985, CSUR.

[40]  Masafumi Yamashita,et al.  Probabilistic Local Majority Voting for the Agreement Problem on Finite Graphs , 1999, COCOON.

[41]  Gadi Moran.  The r-majority vote action on 0-1 sequences , 1994, Discret. Math..

[42]  Andrzej Pelc.  Efficient fault location with small risk , 1996, SIROCCO.

[43]  Ivan Damgård,et al.  Efficient Protocols based on Probabilistic Encryption using Composite Degree Residue Classes , 2000, IACR Cryptol. ePrint Arch..

[44]  Anna Philippou,et al.  Tools and Algorithms for the Construction and Analysis of Systems , 2018, Lecture Notes in Computer Science.

[45]  Hagit Attiya,et al.  Distributed Computing: Fundamentals, Simulations and Advanced Topics , 1998 .