### 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 language | English |
---|---|

Article number | P10024 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2009 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

### Fingerprint

### 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

*Journal of Statistical Mechanics: Theory and Experiment*,

*2009*(10), [P10024]. https://doi.org/10.1088/1742-5468/2009/10/P10024

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

Research output: Contribution to journal › Article

*Journal of Statistical Mechanics: Theory and Experiment*, vol. 2009, no. 10, P10024. https://doi.org/10.1088/1742-5468/2009/10/P10024

}

TY - JOUR

T1 - Conservation laws for voter-like models on random directed networks

AU - Angeles Serrano, M.

AU - Klemm, Konstantin

AU - Vazquez, Federico

AU - Eguíluz, Víctor M.

AU - San Miguel, Maxi

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Disordered systems (theory)

KW - Network dynamics

KW - Tochastic processes

UR - http://www.scopus.com/inward/record.url?scp=71049125232&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=71049125232&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/2009/10/P10024

DO - 10.1088/1742-5468/2009/10/P10024

M3 - Article

VL - 2009

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 10

M1 - P10024

ER -