Abstract

Since the introduction and widespread utility of the internet and World Wide Web began in the latter part of the twentieth century, the mathematical modeling of web based networks has been of interest to mathematicians, physicists, and computer scientists hoping to model such systems in a methodical way. Social networks, made popular by websites like Facebook and Twitter, present a particular challenge to modeling as the result of their specialized growth patterns that reflect human interaction. These patterns include non-trivial clustering and assortative mixing. Current modeling attempts of social networks have involved the utilization of random graphs as the primary methodology. Here, the network is simulated by generating a growing graph using a stochastic process. Despite a number of results in the current literature using binary random graphs, weighted network models, or models that take into account the strength of connection between members, have not been thoroughly studied. In this project, random weight graph models are extended to the fuzzy case, where fuzzy probability theory drives the stochastic process. To illustrate, suppose that an edge in a weighted graph is known to exist between two particular vertices but the strength of that edge is unclear. To determine the strength of this edge, we find the conditional expectation of a fuzzy random variable conditioned on the strength of mutual friends shared by the two vertices. This conditional expectation is based on an underlying joint probability distribution that implicitly characterizes the expected growth pattern of the individual network. The calculation of expected weight in this manner drives the stochastic process as a new vertex is connected randomly to the graph with each iteration. We find through simulation that our model is a reasonable predictor of number of connections in a given network. Also, we find that as κ and δ (representing a cutoff criteria and adding strength of each new vertex, respectively) vary, network growth exhibits a three dimensional sigmoidal growth pattern. Furthermore, at certain and values, our process exhibits non-trivial clustering similar to that seen in social networks.