A study of a time-graph friendship model

Modeling friendship is a challenging task in social networking given the opportunistic behavior of human relationships that is hard to model. In this paper a simple two-state markov chain model is introduced attempting to give further insight on friendship and particularly how relations evolve as time passes, given that the corresponding graph is a time evolving one with interesting properties. Based on this model four distinct behavioral categories are identified and studied. As it is analytically shown, and subsequently confirmed by simulations, any network of nodes having the same friendship characteristics (e.g., a network consisted exclusively of nodes of one of the behavioral categories characterized in this work) eventually results to a network with properties similar to that of random graphs. Since modern society is characterized by power-law distributions, it is shown by simulations that there exists a certain mix of the previously mentioned categories such that the resulting graph has similar to power-law distribution.