14. A General Model for Studying Time Evolution of Transition Networks

We consider a class of complex networks whose nodes assume one of several possible states at any time and may change their states from time to time. Such networks, referred to as transition networks in this chapter, represent practical networks of rumor spreading, disease spreading, language evolution, and so on. Here, we derive a general analytical model describing the dynamics of a transition network and derive a simulation algorithm for studying the network evolutionary behavior. By using this model, we can analytically compute the probability that (1) the next transition will happen at a given time; (2) a particular transition will occur; (3) a particular transition will occur with a specific link. This model, derived at a microscopic level, can reveal the transition dynamics of every node. A numerical simulation is taken as an “experiment” or “realization” of the model. We use this model to study the disease propagation dynamics in four different prototypical networks, namely, the regular nearest-neighbor (RN) network, the classical Erdös-Renyí (ER) random graph, the Watts-Strogátz small-world (SW) network, and the Barabási-Albert (BA) scalefree network. We find that the disease propagation dynamics in these four networks generally have different properties but they do share some common features. Furthermore, we utilize the transition network model to predict user growth in the Facebook network. Simulation shows that our model agrees with the historical data. The study can provide a useful tool for a more thorough understanding of the dynamics of transition networks.