Hypergraph Coloring Games and Voter Models

Abstract

We analyze a network coloring game on hypergraphs that can also describe a voter model. Each node represents a voter and is colored according to its preferred candidate (or undecided). Each hyperedge is a subset of voters that can interact and influence one another. In each round of the game, one hyperedge is chosen randomly, and the voters in the hyperedge can change their colors according to some prescribed probability distribution. We analyze this _interaction model_ based on random walks on the associated weighted directed state graph. Under certain “memoryless” restrictions, we can use semigroup spectral methods to explicitly determine the spectrum of the state graph, and the random walk on the state graph converges to its stationary distribution in _O_(_m_log _n_) steps, where _n_ is the number of voters and _m_ the number of hyperedges. We can then estimate probabilities that events occur within an error bound of ε by simulating the voting game for _O_(log (1/ε)_m_log _n_) rounds. We also consider a partially memoryless game using the memoryless game for comparison and analysis, which serves as an approximation of the actual interaction dynamics.