Journal of Statistical Physics

, Volume 151, Issue 1–2, pp 21–45 | Cite as

Stochastic Dynamics on Hypergraphs and the Spatial Majority Rule Model

  • N. LanchierEmail author
  • J. Neufer


This article starts by introducing a new theoretical framework to model spatial systems which is obtained from the framework of interacting particle systems by replacing the traditional graphical structure that defines the network of interactions with a structure of hypergraph. This new perspective is more appropriate to define stochastic spatial processes in which large blocks of vertices may flip simultaneously, which is then applied to define a spatial version of the Galam’s majority rule model. In our spatial model, each vertex of the lattice has one of two possible competing opinions, say opinion 0 and opinion 1, as in the popular voter model. Hyperedges are updated at rate one, which results in all the vertices in the hyperedge changing simultaneously their opinion to the majority opinion of the hyperedge. In the case of a tie in hyperedges with even size, a bias is introduced in favor of type 1, which is motivated by the principle of social inertia. Our analytical results along with simulations and heuristic arguments suggest that, in any spatial dimensions and when the set of hyperedges consists of the collection of all n×⋯×n blocks of the lattice, opinion 1 wins when n is even while the system clusters when n is odd, which contrasts with results about the voter model in high dimensions for which opinions coexist. This is fully proved in one dimension while the rest of our analysis focuses on the cases when n=2 and n=3 in two dimensions.


Interacting particle systems Hypergraph Social group Majority rule Voter model 



The authors would like to thank two anonymous referees for many comments that helped to improve the clarity of this article.


  1. 1.
    Bramson, M., Durrett, R.: A simple proof of the stability criterion of Gray and Griffeath. Probab. Theory Relat. Fields 80, 293–298 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bramson, M., Griffeath, D.: Renormalizing the 3-dimensional voter model. Ann. Probab. 7, 418–432 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bramson, M., Griffeath, D.: Clustering and dispersion rates for some interacting particle systems on ℤ. Ann. Probab. 8, 183–213 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009) ADSCrossRefGoogle Scholar
  5. 5.
    Clifford, P., Sudbury, A.: A model for spatial conflict. Biometrika 60, 581–588 (1973) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cox, J.T., Griffeath, D.: Occupation time limit theorems for the voter model. Ann. Probab. 11, 876–893 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cox, J.T., Griffeath, D.: Diffusive clustering in the two-dimensional voter model. Ann. Probab. 14, 347–370 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Durrett, R.: Multicolor particle systems with large threshold and range. J. Theor. Probab. 5, 127–152 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Durrett, R.: Ten lectures on particle systems. In: Lectures on Probability Theory, Saint-Flour, 1993. Lecture Notes in Math., vol. 1608, pp. 97–201. Springer, Berlin (1995) CrossRefGoogle Scholar
  10. 10.
    Durrett, R., Steif, J.E.: Fixation results for threshold voter systems. Ann. Probab. 21, 232–247 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Galam, S.: Minority opinion spreading in random geometry. Eur. Phys. J. B 25, 403–406 (2002) ADSGoogle Scholar
  12. 12.
    Harris, T.E.: Nearest neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9, 66–89 (1972) zbMATHCrossRefGoogle Scholar
  13. 13.
    Holley, R.A., Liggett, T.M.: Ergodic theorems for weakly interacting systems and the voter model. Ann. Probab. 3, 643–663 (1975) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lanchier, N.: 2012, Stochastic spatial model of producer-consumer systems on the lattice. Preprint Google Scholar
  15. 15.
    Zähle, I.: Renormalization of the voter model in equilibrium. Ann. Probab. 29, 1262–1302 (2001) MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

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