Conservation laws for voter-like models on random directed networks

M. Angeles Serrano, Konstantin Klemm, Federico Vazquez, Víctor M. Eguíluz, Maxi San Miguel

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

We study the voter model, under node and link update, and the related invasion process on a single strongly connected component of a directed network. We implement an analytical treatment in the thermodynamic limit using the heterogeneous mean-field assumption. From the dynamical rules at the microscopic level, we find the equations for the evolution of the relative densities of nodes in a given state on heterogeneous networks with arbitrary degree distribution and degree-degree correlations. We prove that conserved quantities as weighted linear superpositions of spin states exist for all three processes and, for uncorrelated directed networks, we derive their specific expressions. We also discuss the time evolution of the relative densities that decay exponentially to a homogeneous stationary value given by the conserved quantity. The conservation laws obtained in the thermodynamic limit for a system that does not order in that limit determine the probabilities of reaching the absorbing state for a finite system. The contribution of each degree class to the conserved quantity is determined by a local property. Depending on the dynamics, the highest contribution is associated with influential nodes reaching a large number of outgoing neighbors, not too influenceable ones with a low number of incoming connections, or both at the same time.

Original languageEnglish
Article numberP10024
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2009
Issue number10
DOIs
Publication statusPublished - 2009
Externally publishedYes

Fingerprint

Directed Network
Conserved Quantity
Random Networks
Vote
conservation laws
Conservation Laws
Thermodynamic Limit
Vertex of a graph
Voter Model
thermodynamics
Local Properties
Heterogeneous Networks
Invasion
Degree Distribution
Absorbing
Connected Components
Mean Field
Superposition
Update
Model

Keywords

  • Disordered systems (theory)
  • Network dynamics
  • Tochastic processes

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistical and Nonlinear Physics
  • Statistics, Probability and Uncertainty

Cite this

Conservation laws for voter-like models on random directed networks. / Angeles Serrano, M.; Klemm, Konstantin; Vazquez, Federico; Eguíluz, Víctor M.; San Miguel, Maxi.

In: Journal of Statistical Mechanics: Theory and Experiment, Vol. 2009, No. 10, P10024, 2009.

Research output: Contribution to journalArticle

Angeles Serrano, M. ; Klemm, Konstantin ; Vazquez, Federico ; Eguíluz, Víctor M. ; San Miguel, Maxi. / Conservation laws for voter-like models on random directed networks. In: Journal of Statistical Mechanics: Theory and Experiment. 2009 ; Vol. 2009, No. 10.
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