# 2011 S3: David Schlueter

## Modeling Social Networks with Random and Fuzzy Graphs

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, or positive correlation between degrees of adjacent vertices.

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 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 discuss the efficacy of our approach as a modeling tool, interesting growth characteristics of the model, and possible modifications to the process.

Faculty Mentor: Jiyeon Suh, Mathematics