Statistical properties of the voter model with fixed nodes

• Grant number: 07/07027-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: December 01, 2007
• End date: July 31, 2009
• Field of knowledge: Physical Sciences and Mathematics - Physics - General Physics
• Principal Investigator: Marcus Aloizio Martinez de Aguiar
• Grantee: Matheus Veronez
• Host Institution: Instituto de Física Gleb Wataghin (IFGW). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 03/12097-7 - Classical and quantum dynamical systems, AP.TEM

Abstract

The voter model consists in a simplified description of a social network where individuals (the elements of the network) try to decide on a given matter, as for example, whom to vote for in an election with two candidates. We assume the existence of two types of individuals: those with fixed opinion and those whose opinion can change according to their neighbors (free individuals). The microscopic state of the system is characterized by the individual states of its elements. The macroscopic state, on the other hand, takes into account only the total number of individuals with a given opinion. The system is updated at each time step by picking a randomly selected free individual and applying the following rule: with probability p its state remains the same and with probability (1-p) it adopts the opinion of one of its neighbors. The fixed individuals can be interpreted as external perturbations over the subsystem of free individuals. These perturbations can be small (when the fixed elements have small influence over the free ones), critic (when there is exactly one fixed individual of each opinion) and large (large number of fixed individuals). For each of these situations we can find stable or unstable equilibrium distributions, which can be characterized by the entropy of the system. Such property can also be analyzed in the thermodynamical limit, where the number of free individuals tend to infinity. We intend to study the dynamical and equilibrium properties of networks with different topologies, like random, scale-free and small-world. If we associate an energy to each state of the system we can also calculate other interesting thermodynamical properties, like free energy, magnetization and specific heat. (AU)