The Voter Model
Table of Contents
Programmatic Details
In this article, I will go into the mathematics behind the voter model. If you are interested in how the voter model is implemented, you can read the article I have written about it.
Idea
In the Voter Model, we consider a graph whose nodes have two possible states or opinions. After some time any node can adopt the opinion of its neighbor if its neighbor has a different opinion.
We call these node states opinions, since we can think of the nodes as a voter population and the edges as their interactions. We are interested in whether, after a sufficient amount of time has passed, there is only one option left.
Stochastic Properties
The times at which a node can adopt its neighbor’s opinion is the jump times of a Poisson Process of rate , at such a jump time a neighbor is selected uniformly and the opinion is copied. By the thinning and superposition property of the Poisson Process, we can model the same process by attaching a Poisson Process to each outgoing edge of any node of the graph if is the number of neighbors of , this Poisson Process needs to have rate . In the same way we can also just have one Poisson Process for the entire graph and uniformly select an edge along which an opinion is copied, this Poisson Process would have to have a rate of, the number of edges.
Findings
We find that in the case of a finite, undirected, connected graph the nodes all have the same opinion after a sufficient amount of time has passed, moreover, this is true for and , but not for any with .
Interpretation
Now since we already view these nodes as voters and their states as opinions, it is reasonable to wonder what that result means for people in general. Certainly, we can model all humans as nodes in a graph where the nodes are connected if the humans that are represented by these nodes know each other. Then that might be a finite, undirected, connected graph. Does that mean after a certain amount of time every human on earth will have the same opinion? Probably not.
The first obvious problem in the model is that we require an undirected graph, but we can certainly imagine a directed relationship between humans. Think of a celebrity, they can influence many people but are not influenced by most of them. We also need to account for things that can change opinions like books, reading a book would also be an undirected relationship. This also means that we would have to model all things that can change someone’s opinion in the graph as well. Lastly, we might also have to consider that humans might be able to change their opinions spontaneously, or based on some event that happens in their life.
