Handouts # 1 – Random Graphs Models

COMS W 4995 - 1 : Intro to Social Networks Fall 2012 Handouts # 1 – Random Graphs Models

Research Projects

Here is a collection of research projects, ranging from number theory to probability to statistics to random graphs.... Much of the background material is summarized from [MT-B], though most standard number theory textbooks would have these facts. Each chapter begins with a brief synopsis of the types of problems and background material needed. For more information, see the handouts on-line at ...

Vertex Models on Feynman Diagrams

The statistical mechanics of spin models, such as the Ising or Potts models, on generic random graphs can be formulated economically by considering the N → 1 limit of N ×N Hermitian matrix models. In this paper we consider the N → 1 limit in complex matrix models, which describes vertex models of different sorts living on random graphs. From the graph theoretic perspective one is using matrix m...

Random Threshold Graphs

We introduce a pair of natural, equivalent models for random threshold graphs and use these models to deduce a variety of properties of random threshold graphs. Specifically, a random threshold graph G is generated by choosing n IID values x1, . . . , xn uniformly in [0, 1]; distinct vertices i, j of G are adjacent exactly when xi + x j > 1. We examine various properties of random threshold gra...

ar X iv : h ep - l at / 9 70 40 20 v 1 3 0 A pr 1 99 7 Potts Models on Feynman Diagrams

We investigate numerically and analytically Potts models on " thin " random graphs – generic Feynman diagrams, using the idea that such models may be expressed as the N → 1 limit of a matrix model. The thin random graphs in this limit are locally tree-like, in distinction to the " fat " random graphs that appear in the planar Feynman diagram limit, N → ∞, more familiar from discretized models o...