Analyzing Information Cascading in Large Scale Networks: A Fixed Point Approach

Information cascading, referred as the phenomenon of an individual following the behavior of the preceding individual after observing its actions, is prevalent in real social networks and triggers intense research interests for the purpose of monitoring and controlling network epidemics. One of the typical lines of information cascading study belongs to the Influence Maximization Problem, which aims to algorithmically find the optimal seeds that can spread the information to the maximum number of nodes. Regardless of the tremendous efforts made in various algorithm design of finding such optimal seeds, it has not yet been well understood how the absolute influence power of the "optimal" source set affects the ultimate cascading, i.e., under which conditions the seeds are able or unable to influence an substantial fraction of the entire network. Most existing works have investigated the conditions of network scale influence under linear threshold model, where the activation of a node requires a large number of infected neighbors. Instead, in this paper we focus on the case of single source cascading, which is only possible to occur under the independent cascading model. We launch information cascading analysis from two aspects, i.e., the influence scale and network-scale cascading probability. Firstly, percolation analysis of the cascading outcome shows that estimating influence scale is equivalent to solving fixed point equations. Then, we investigate the speed and stability of information cascading based on fixed point analysis, which shows that the information cascading process almost surely terminates within logarithmic time complexity. Furthermore, the results are generalized to the stochastic block model, where we find that network-scale cascading is determined by the spectral radius of the community matrix. The analysis presented in this paper could help us better understand the conditions for different information cascading outcomes.