Advertisement

Hypergraph Coloring Games and Voter Models

  • Fan Chung
  • Alexander Tsiatas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7323)

Abstract

We analyze a network coloring game on hypergraphs which can also describe a voter model. Each node represents a voter and is colored according to its preferred candidate (or undecided). Each hyperedge is a subset of voters that can interact and influence one another. In each round of the game, one hyperedge is chosen randomly, and the voters in the hyperedge can change their colors according to some prescribed probability distribution. We analyze this interaction model based on random walks on the associated weighted, directed state graph. We consider three scenarios — a memoryless game, a partially memoryless game and the general game using the memoryless game for comparison and analysis. Under certain ‘memoryless’ restrictions, we can use semigroup spectral methods to explicitly determine the spectrum of the state graph, and the random walk on the state graph converges to its stationary distribution in O(m logn) steps, where n is the number of voters and m is the number of hyperedges. This can then be used to determine an appropriate cut-off time for voting: we can estimate probabilities that events occur within an error bound of ε by simulating the voting game for O(log(1/ε) m logn) rounds. Next, we consider a partially memoryless game whose associated random walk can be written as a linear combination of a memoryless random walk and another given random walk. In such a setting, we provide bounds on the convergence time to the stationary distribution, highlighting a tradeoff between the proportion of memorylessness and the time required. To analyze the general interaction model, we will first construct a companion memoryless process and then choose an appropriate damping constant β to build a partially memoryless process. The partially memoryless process can serve as an approximation of the actual interaction dynamics for determining the cut-off time if the damping constant is appropriately chosen either by using simulation or depending on the rules of interaction.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aldous, D., Fill, J.A.: Reversible Markov chains and random walks on graphs (preprint), http://www.stat.berkeley.edu/~aldous/RWG/book.html
  2. 2.
    Bidigare, T.P., Hanlon, P., Rockmore, D.N.: A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements. Duke Mathematical Journal 99(1), 135–174 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Brown, K.S.: Semigroups, rings, and Markov chains. Journal of Theoretical Probability 13(3), 837–938 (2000)CrossRefGoogle Scholar
  4. 4.
    Brown, K.S., Diaconis, P.: Random walks and hyperplane arrangements. Annals of Probability 26(4), 1813–1854 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chung, F.: Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics 9(1), 1–19 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chung, F., Graham, R.: Edge flipping in graphs. Advances in Applied Mathematics 48(1), 37–63 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Clifford, P., Sudbury, A.: A model for spatial conflict. Biometrika 60(3), 581–588 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ellison, G.: Learning, local interaction, and coordination. Econometrica 61(3), 1047–1071 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Fill, J.A.: An exact formula for the move-to-front rule for self-organizing lists. Journal of Theoretical Probability 9(1), 113–160 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Holley, R.A., Liggett, T.M.: Ergodic theorems for weakly interacting infinite systems and the voter model. Annals of Probability 3(4), 643–663 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Judd, S., Kearns, M.: Behavioral experiments in networked trade. In: Proceedings of the 9th ACM Conference on Electronic Commerce, pp. 150–159 (2008)Google Scholar
  12. 12.
    Judd, S., Kearns, M., Vorobeychik, Y.: Behavioral dynamics and influence in networked coloring and consensus. Proceedings of the National Academy of Sciences 107(34), 14978–14982 (2010)CrossRefGoogle Scholar
  13. 13.
    Kandori, M., Mailath, G., Rob, R.: Learning, mutation, and long run equilibria in games. Econometrica 61(1), 29–56 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kearns, M., Judd, S., Tan, T., Wortman, J.: Behavioral experiments on biased voting in networks. Proceedings of the National Academy of Sciences 106(5), 1347–1352 (2009)CrossRefGoogle Scholar
  15. 15.
    Kearns, M., Suri, S., Montfort, N.: An experimental study of the coloring problem on human subject networks. Science 313(5788), 824–827 (2006)CrossRefGoogle Scholar
  16. 16.
    Kearns, M., Tan, J.: Biased Voting and the Democratic Primary Problem. In: Papadimitriou, C., Zhang, S. (eds.) WINE 2008. LNCS, vol. 5385, pp. 639–652. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Klein-Barmen, F.: On a broader analysis of lattice theory. Mathematische Zeitschrift 46(1), 472–480 (1940) (in German)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Montanari, A., Saberi, A.: Convergence to equilibrium in local interaction games. In: Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 303–312 (2009)Google Scholar
  19. 19.
    Mossel, E., Schoenebeck, G.: Reaching consensus on social networks. In: Proceedings of the First Symposium on Innovations in Computer Science (ICS), pp. 214–229 (2010)Google Scholar
  20. 20.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press (1995)Google Scholar
  21. 21.
    Schützenberger, M.-P.: On left-regular bands. Comptes Rendus de l’Académie des Sciences 224, 777–778 (1947) (in French)Google Scholar
  22. 22.
    Suchecki, K., Eguíluz, V., San Miguel, M.: Voter model dynamics in complex networks: Role of dimensionality, disorder, and degree distribution. Physical Review E 72, 1–8 (2005)CrossRefGoogle Scholar
  23. 23.
    Tahbaz-Salehi, A., Jadbabaie, J.: Consensus over ergodic stationary graph processes. IEEE Transactions on Automatic Control 55(1), 225–230 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yildiz, M.E., Pagliari, A., Ozdaglar, A., Scaglione, A.: Voting models in random networks. In: Proceedings of the Information Theory and Applications Workshop (ITA), pp. 1–7 (2010)Google Scholar
  25. 25.
    Young, H.P.: The evolution of conventions. Econometrica 61(1), 57–84 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Zachary, W.W.: An information flow model for conflict and fission in small groups. Journal of Anthropological Research 33(4), 452–473 (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Fan Chung
    • 1
  • Alexander Tsiatas
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of CaliforniaSan DiegoUSA

Personalised recommendations