On the clustering property of the random intersection graphs

A random intersection graph \mtl{\mcal{G}_{V,W,p}} is induced from a random bipartite graph \mtl{\mcal{G}^{*}_{V,W,p}} with vertices classes \mtl{V}, \mtl{W} and the edges incident between \mtl{v \in V} and \mtl{w \in W} with probability \mtl{p}. Two vertices in \mtl{V} are considered to be connected with each other if both of them connect with some common vertices in \mtl{W}. The clustering properties of the random intersection graph are investigated completely in this article. Suppose that the vertices number be \mtl{N = \mabs{V}} and \mtl{M=\mabs{W}} and \mtl{M = N^{\alpha},\ p=N^{-\beta}}, where \mtl{\alpha > 0,\, \beta > 0}, we derive the exact expressions of the clustering coefficient \mtl{C_{v}} of vertex \mtl{v} in \mtl{\mcal{G}_{V,W,p}}. The results show that if \mtl{\alpha < 2\beta} and \mtl{\alpha \neq \beta}, \mtl{C_{v}} decreases with the increasing of the graph size; if \mtl{\alpha = \beta} or \mtl{\alpha \geq 2\beta}, the graph has the constant clustering coefficients, in addition, if \mtl{\alpha > 2\beta}, the graph connecChangshui Zhangts almost completely. Therefore, we illustrate the phase transition for the clustering property in the random intersection graphs and give the condition that \mtl{\riG} being high clustering graph

Moment-based parameter estimation in binomial random intersection graph models

Binomial random intersection graphs can be used as parsimonious statistical models of large and sparse networks, with one parameter for the average degree and another for transitivity, the tendency of neighbours of a node to be connected. This paper discusses the estimation of these parameters from a single observed instance of the graph, using moment estimators based on observed degrees and frequencies of 2-stars and triangles. The observed data set is assumed to be a subgraph induced by a set of $n_0$ nodes sampled from the full set of $n$ nodes. We prove the consistency of the proposed estimators by showing that the relative estimation error is small with high probability for $n_0 \gg n^{2/3} \gg 1$. As a byproduct, our analysis confirms that the empirical transitivity coefficient of the graph is with high probability close to the theoretical clustering coefficient of the model.Comment: 15 pages, 6 figure

Correlation between clustering and degree in affiliation networks

We are interested in the probability that two randomly selected neighbors of a random vertex of degree (at least) $k$ are adjacent. We evaluate this probability for a power law random intersection graph, where each vertex is prescribed a collection of attributes and two vertices are adjacent whenever they share a common attribute. We show that the probability obeys the scaling $k^{-\delta}$ as $k\to+\infty$. Our results are mathematically rigorous. The parameter $0\le \delta\le 1$ is determined by the tail indices of power law random weights defining the links between vertices and attributes