Recent Progress in Complex Network Analysis: Properties of Random Intersection Graphs

Abstract

Experimental results show that in large complex networks (such as internet, social or biological networks) there exists a tendency to connect elements which have a common neighbor. In theoretical random graph models, this tendency is described by the clustering coefficient being bounded away from zero. Complex networks also have power-law degree distributions and short average distances (small world phenomena). These are desirable features of random graphs used for modeling real life networks. We survey recent results concerning various random intersection graph models showing that they have tunable clustering coefficient, a rich class of degree distributions including power-laws, and short average distances.

Recent Progress in Complex Network Analysis: Models of Random Intersection Graphs

“…Our analysis shows that random intersection graph models provide remarkably good approximations to some real networks (such as the actor network) as long as the degree and clustering properties are considered. These empirical observations are supported by theoretical findings (see Bloznelis et al 2015). The study of random intersection graphs has just started and many interesting properties are still unexplored.…”

“…C1. Detailed results concerning degree distribution, clustering and other properties are given in the accompanying paper (Bloznelis et al 2015).…”

“…An attractive property of these models is that they capture important features of real networks, the power-law degree distribution (also called the "scale-free" property), small typical distances between vertices (the "small-world" property, see Strogatz and Watts 1998), and a high statistical dependency of neighboring adjacency relations expressed in terms of clustering and assortativity coefficients (we give a detailed account of these properties in the accompanying paper Bloznelis et al 2015). In the literature a network possessing these properties is called a complex network.…”

Degree-Degree Distribution in a Power Law Random Intersection Graph with Clustering

“…Now an attribute w i ∈ W is included in the collection S v j at random and with probability proportional to the attractiveness x i and activity y j (cf., [14], [17]). In this way we obtain a random graph on the vertex set V sometimes called the inhomogeneous random intersection graph, see [5] and references therein. Before giving a detailed definition of this random graph model we mention a recent publication [16], which argues convincingly that in some social networks the 'heavy-tailed degree distribution is causally determined by similarly skewed distribution of human activity'.…”

On the total variation distance between the binomial random graph and the random intersection graph

“…Random intersection graphs have received a lot of attention due to a great diversity of applications in areas such as epidemics [9], circuit design [20], network user profiling [23], and analysis of complex networks [3,4,7,10]. For more information, we refer the reader to the survey papers [5,6,28]. For instance, G(n, m; p) is applicable for gate matrix circuit design, which is related to the optimization problem of finding a permutation of the order of gate lines that minimizes the number of horizontal tracks required to lay out the circuit.…”