Phase transition of multi-state diffusion process in networks

Abstract

The spread of viruses, ideas, technologies or behaviors in networks has been widely studied using mathematical models of contagion [7, 5, 8]. Understanding these dynamical processes is important to control and prevent the spread of diseases, and to maximize the influence of a product in online social networks. One of the most studied contagion models so far is the Susceptible-Infected-Susceptible (SIS) model. In this model, there are k = 2 states and each node in the graph is in one of these two states: Susceptible or Infected. This model describes the spread of contagions like flu (without immunity) or idea. In practice, the SIS model can be quite restrictive since the degrees of interest in a contagion among individuals are different. For example, consider the case in which the diffusion process is designed to describe a product adoption [1, 6]. At some point of time after the product release, some people may have purchased the product while some may have not. For example, the consumer purchase decision process theory [2] suggests that there are five stages until a consumer purchases a product and influences others. The states include produce recognition, information search, alternative evaluation, purchase decision, and post-purchase behavior. This implies that one needs to further divide the susceptible state into several states according to the degree of interest. This is also intuitive because the adoption of a new product may need exposure from more than one customers. In this work, we propose a generalization of the SIS model by allowing the number of states of adoption (or infection) to be more than two (k ≥ 2). In particular, the states can range from 0 to k−1, where the state k−1 is the active state, that the node is infected and can influence other neighboring nodes. Nodes whose state is in 0 to k−2 can be promoted to a higher state if they are exposed to their infected neighbors (whose state is in k−1). We analyze the epidemic threshold dynamics, according to which initial condition leads to or prevents a disease outbreak. However, the traditional branching process approaches (that deal with a single initial spreader) [7] cannot be applied directly to this setup since we allow any fraction of initial infective nodes. Specifically, we use the multidimensional mean-field method to analyze our model and determine the condition of phase transition. We believe that our work is a step towards elucidating the complex interactions between nodes in the epidemic spreading. The key result of our research is that our method pre-