Random Dot Product Graphs a Model for Social Networks

We develop a new set of random graph models. The motivation for these models comes from social networks and is based on the idea of common interest. We represent a social network as a graph, in which vertices correspond to individuals. In the general model, an interest vector xv, drawn from a specific distribution, is associated with corresponding vertex v. The edge between vertices u and v exists with some probability P (xi, xj) = f(xu · xv); that is, it is equal to a function of the dot product of the vectors. The probability of a graph H is given by PX[H] = [∏ uv∈E(H) u 1) of the uniform distribution on [0, 1], and f is the identity function. In this case, with probability approaching one as n approaches infinity, all fixed graphs appear as subgraphs. In the sparse model, the vectors are again drawn from Ua[0, 1], however the probability function is f(s) = s nb (b ∈ (0,∞)). With this change, subgraphs appear at certain ii thresholds and we examine the sequence of their appearance. In both cases, we show that the models obey a power law degree distribution, exhibit clustering, and have a low diameter; these are all characteristics found in social networks. The third version is a discrete model. Here the vectors are drawn from {0, 1}t (t ∈ Z+)) and f(s) = st . Each coordinate of xv is independently assigned the value 1 with probability p and 0 otherwise (p ∈ [0, 1]). We define the probability order polynomial, or POP, of a graph H as a function that is asymptotic to P≥[H], the probability of H appearing as a (not necessarily induced) subgraph, and present geometric techniques for studying POP asymptotics. We give a general method for calculating the POP of H. We present formuals for the POPs and first moment results for trees, cycles, and complete graphs. We also prove a threshold result for K3 and describe a general method for proving threshold results when all the required POPs are known. Advisor: Edward R. Scheinerman Second Reader: Carey E. Priebe