Finding Relation between PageRank and Voter Model

Estimating influence of a node is an important problem in social network analyses. We address this problem in a particular class of model for opinion propagation in which a node adopts its opinion based on not only its direct neighbors but also the average opinion share over the whole network, which we call an extended Voter Model with uniform adoption (VM). We found a similarity of this model with the well known PageRank (PR) and explored the relationships between the two. Since the uniform adoption implies the random opinion adoption of all nodes in the network, it corresponds to the random surfer jump of PR. For an undirected network, both VM and PR give the same ranking score vector because the adjacency matrix is symmetric, but for a directed network, the score vector is different for both because the adjacency matrix is asymmetric. We investigated the effect of the uniform adoption probability on ranking and how the ranking correlation between VM and PR changes using four real world social networks. The results indicate that there is little correlation between VM and PR when the uniform adoption probability is small but the correlation becomes larger when both the uniform adoption and the random surfer jump probabilities become larger. We identified that the recommended value for the uniform adoption probability is to be around 0.25 to obtain a stable solution.