Stochastic Dynamics on Hypergraphs and the Spatial Majority Rule Model

Nicolas Lanchier, J. Neufer

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)21-45
Number of pages25
JournalJournal of Statistical Physics
Volume151
Issue number1-2
DOIs
StatePublished - 2013

Fingerprint

Majority Rule
Stochastic Dynamics
Hypergraph
Voter Model
Spatial Model
apexes
Interacting Particle Systems
Spatial Process
Tie
Flip
Inertia
One Dimension
Higher Dimensions
Stochastic Processes
Two Dimensions
Odd
Model
Heuristics
inertia
Vertex of a graph

Keywords

  • Hypergraph
  • Interacting particle systems
  • Majority rule
  • Social group
  • Voter model

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Stochastic Dynamics on Hypergraphs and the Spatial Majority Rule Model. / Lanchier, Nicolas; Neufer, J.

In: Journal of Statistical Physics, Vol. 151, No. 1-2, 2013, p. 21-45.

Research output: Contribution to journalArticle

@article{26cdd6c24c99403f8043c6726d738af4,
title = "Stochastic Dynamics on Hypergraphs and the Spatial Majority Rule Model",
abstract = "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.",
keywords = "Hypergraph, Interacting particle systems, Majority rule, Social group, Voter model",
author = "Nicolas Lanchier and J. Neufer",
year = "2013",
doi = "10.1007/s10955-012-0543-5",
language = "English (US)",
volume = "151",
pages = "21--45",
journal = "Journal of Statistical Physics",
issn = "0022-4715",
publisher = "Springer New York",
number = "1-2",

}

TY - JOUR

T1 - Stochastic Dynamics on Hypergraphs and the Spatial Majority Rule Model

AU - Lanchier, Nicolas

AU - Neufer, J.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Hypergraph

KW - Interacting particle systems

KW - Majority rule

KW - Social group

KW - Voter model

UR - http://www.scopus.com/inward/record.url?scp=84875633221&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875633221&partnerID=8YFLogxK

U2 - 10.1007/s10955-012-0543-5

DO - 10.1007/s10955-012-0543-5

M3 - Article

AN - SCOPUS:84875633221

VL - 151

SP - 21

EP - 45

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -