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In the mathematical theory of probability, the voter model is a stochastic process that is a specific type of interacting particle system (see Probabilistic Cellular Automata too). A voter model is a sequential dynamical system and it is similar to the contact process.
Contents
- Definition
- Clustering and coexistence
- Model description
- Limiting behaviors of linear voter models
- A special linear voter model
- Clusters in one dimension d 1
- Occupation time
- Limiting behaviors of threshold voter model
- Model with threshold T 1
- References
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 are changed according to a stochastic rule. Specifically, for one of the chosen voter's neighbors is chosen according to a given set of probabilities and that individual'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
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
-
c ( x , η ) = 0 for everyx ∈ Z d η ≡ 0 or ifη ≡ 1 -
c ( x , η ) = c ( x , ζ ) for everyx ∈ Z d η ( y ) + ζ ( y ) = 1 for ally ∈ Z d -
c ( x , η ) ≤ c ( x , ζ ) ifη ≤ ζ andη ( x ) = ζ ( x ) = 0 -
c ( x , η ) is invariant under shifts inZ d
Property (1) says that
Clustering and coexistence
What we are interested in is the limiting behavior of the models. Since the flip rates of a site depends 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
we will say that clustering occurs.
It is important to distinguish clustering with the concept of cluster. Clusters are defined to be the connected components of
Model description
This section will be dedicated to one of the basic voter models, the Linear Voter Model.
Let
Then in Linear voter model, the transition rates are linear functions of
Or if we use
We define a process of coalescing random walks
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
Limiting behaviors of linear voter models
Let
where
Therefore, the process clusters.
On the other hand, when
for some constant
Now let
Theorem 2.1
The linear voter model
- the process clusters if
d = 1 and∑ x | x | p ( 0 , x ) ≤ ∞ , or ifd = 2 and∑ x | x | 2 p ( 0 , x ) ≤ ∞ ; - the process coexists if
d ≥ 3 .
Remarks: To contrast this with the behavior of the threshold voter models that will be discussed in next section, note that whether the linear voter model clusters or coexists depends almost exclusively on the dimension of the set of sites, rather than on the size of the range of interaction.
Theorem 2.2 Suppose
- If
X ~ t μ S ( t ) ⇒ ρ δ 1 + ( 1 − ρ ) δ 0 as t → ∞ ; - If
X ~ t μ S ( t ) ⇒ μ ρ
where
A special linear voter model
One of the interesting special cases of the linear voter model, known as the basic linear voter model, is that for state space
So that
In this case,the process clusters if
Clusters in one dimension d = 1
For the special case with
As mentioned before, clusters of an
provided the limit exists.
Proposition 2.3
Suppose the voter model is with initial distribution
Occupation time
Define the occupation time functionals of the basic linear voter model as:
Theorem 2.4
Assume that for all site x and time t,
proof
By Chebyshev's inequality and the Borel–Cantelli lemma, we can get the equation below:
The theorem follows when letting
Model description
In this section, we will concentrate on a kind of non-linear voter models, known as the threshold voter model.
To define it, let
Simply put, the transition rate of site
For example, if
Limiting behaviors of threshold voter model
If a threshold voter model does not fixate, we should expect that the process will coexist for small threshold and cluster for large threshold, where large and small are interpreted as being relative to the size of the neighbourhood,
- If
T > | N | − 1 2 - If
d = 1 andT = | N | − 1 2 - If
T = θ | N | withθ sufficiently small(θ < 1 4 | N | sufficiently large, then the process coexists.
Here are two theorems corresponding to properties (1) and (2).
Theorem 3.1
If
Theorem 3.2
The threshold voter model in one dimension (
proof
The idea of the proof is to construct two sequences of random times
-
0 = V 0 < U 1 < V 1 < U 2 < V 2 < … , -
{ U k + 1 − V k , k ≥ 0 } are i.i.d.withE ( U k + 1 − V k ) < ∞ , -
{ V k − U k , k ≥ 1 } are i.i.d.withE ( V k − U k ) = ∞ , - the random variables in (b) and (c) are independent of each other,
- event A=
{ η t ( . ) is constant on{ − T , … , T } } , and evant A holds for everyt ∈ ∪ k = 1 ∞ [ U k , V k ] .
Once this construction is made, it will follow from renewal theory that
Hence,
Remarks: (a) Threshold models in higher dimensions do not necessarily cluster if
then no transition ever occur, and the process fixates.
(b) Under the assumption of Theorem 3.2, the process does not fixate. To see this, consider the initial configuration
Property 3 indicates that the threshold voter model is quite different from the linear voter model, in that coexistence occurs even in one dimension, provided that the neighbourhood is not too small. The threshold model has a drift toward the "local minority", which is not present in the linear case.
Most proofs of coexistence for threshold voter models are based on comparisons with hybrid model known as the threshold contact process with parameter
Proposition 3.3
For any
Model with threshold T = 1
The case that
In particular, we are interested in a kind of Threshold T=1 model with
By Theorem 3.2, the model with
Theorem 3.4
Suppose that
The proof of this theorem is given in a paper named "Coexistence in threshold voter models" by Thomas M. Liggett.