In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975.
One can imagine that there is a "voter" at each point on a connected graph, where the connections indicate that there is some form of interaction between a pair of voters (nodes). The opinions of any given voter on some issue changes at random times under the influence of opinions of his neighbours. A voter's opinion at any given time can take one of two values, labelled 0 and 1. At random times, a random individual is selected and that voter's opinion is changed according to a stochastic rule. Specifically, one of the chosen voter's neighbors is chosen according to a given set of probabilities and that neighbor’s opinion is transferred to the chosen voter.
An alternative interpretation is in terms of spatial conflict. Suppose two nations control the areas (sets of nodes) labelled 0 or 1. A flip from 0 to 1 at a given location indicates an invasion of that site by the other nation.
Note that only one flip happens each time. Problems involving the voter model will often be recast in terms of the dual system of coalescing Markov chains. Frequently, these problems will then be reduced to others involving independent Markov chains.
Definition
A voter model is a (continuous time) Markov process with state space and transition rates function , where is a d-dimensional integer lattice, and •,• is assumed to be nonnegative, uniformly bounded and continuous as a function of in the product topology on . Each component is called a configuration. To make it clear that stands for the value of a site x in configuration ; while means the value of a site x in configuration at time .
The dynamic of the process are specified by the collection of transition rates. For voter models, the rate at which there is a flip at from 0 to 1 or vice versa is given by a function of site . It has the following properties:
for every if or if
for every if for all
if and
is invariant under shifts in
Property (1) says that and are fixed points for the evolution. (2) indicates that the evolution is unchanged by interchanging the roles of 0's and 1's. In property (3), means , and implies if , and implies if .
Clustering and coexistence
The interest in is the limiting behavior of the models. Since the flip rates of a site depends on its neighbours, it is obvious that when all sites take the same value, the whole system stops changing forever. Therefore, a voter model has two trivial extremal stationary distributions, the point-masses and on or respectively, which represent consensus. The main question to be discussed is whether or not there are others, which would then represent coexistence of different opinions in equilibrium. It is said coexistence occurs if there is a stationary distribution that concentrates on configurations with infinitely many 0's and 1's. On the other hand, if for all and all initial configurations, then
It is said that clustering occurs.
It is important to distinguish clustering with the concept of cluster. Clusters are defined to be the connected components of or .
The linear voter model
Model description
This section will be dedicated to one of the basic voter models, the Linear Voter Model.
If •,• be the transition probabilities for an irreducible random walk on , then:
Then in Linear voter model, the transition rates are linear functions of :
Or if indicates that a flip happens at , then transition rates are simply:
A process of coalescing random walks is defined as follows. Here denotes the set of sites occupied by these random walks at time . To define , consider several (continuous time) random walks on with unit exponential holding times and transition probabilities •,•, and take them to be independent until two of them meet. At that time, the two that meet coalesce into one particle, which continues to move like a random walk with transition probabilities •,• .
The concept of Duality is essential for analysing the behavior of the voter models. The linear voter models satisfy a very useful form of duality, known as coalescing duality, which is:
where is the initial configuration of and is the initial state of the coalescing random walks .
Limiting behaviors of linear voter models
Let be the transition probabilities for an irreducible random walk on and , then the duality relation for such linear voter models says that