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We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Social Network Analysis

On robust estimation of the location parameter

We establish asymptotic vertex degree distribution and examine its relation to the clustering coecient in two popular random intersection graph models of Godehardt and Jaworski (2001). For sparse graphs with positive clustering coecient, we examine statistical dependence between the (local) clustering coecient and the degree. Our results are mathematically rigorous. They are consistent with the empirical observation of Foudalis et al. (2011) that \clustering correlates negatively with degree." Moreover, they explain empirical results on k 1 scaling of the local clustering coecient of a vertex of degree k reported in Ravasz and Barab asi (2003).

/pdf/degree-and-clustering-coefficient-in-sparse-random-1kx40sel8a.pdf

Degree and clustering coefficient in sparse random intersection graphs

We study a connectivity property of a secure wireless network that uses random pre-distribution of keys A network is composed of n sensors Each sensor is assigned a collection of d different keys drawn uniformly at random from a given set of m keys Two sensors are joined by a communication link if they share a common key We show that for large n with high probability the connected component of size Ω(n) emerges in the network when the probability of a link exceeds the threshold 1-n Similar component evolution is shown for networks where sensors communicate if they share at least s common keys © 2008 Wiley Periodicals, Inc NETWORKS, 2009

Component evolution in a secure wireless sensor network

We establish asymptotic vertex degree distribution and examine its relation to the clustering coefficient in two popular random intersection graph models of Godehardt and Jaworski [Electron. Notes Discrete Math. 10 (2001) 129-132]. For sparse graphs with a positive clustering coefficient, we examine statistical dependence between the (local) clustering coefficient and the degree. Our results are mathematically rigorous. They are consistent with the empirical observation of Foudalis et al. [In Algorithms and Models for Web Graph (2011) Springer] that, "clustering correlates negatively with degree." Moreover, they explain empirical results on $k^{-1}$ scaling of the local clustering coefficient of a vertex of degree k reported in Ravasz and Barabasi [Phys. Rev. E 67 (2003) 026112].

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: Properties of Random Intersection Graphs

We consider sparse random intersection graphs with the property that the clustering coefficient does not vanish as the number of nodes tends to infinity. We find explicit asymptotic expressions for  the correlation coefficient of degrees of adjacent nodes (called the assortativity coefficient), the expected number of common neighbours of adjacent nodes, and  the expected degree of a neighbour of a node of a given degree k. These expressions are written in terms of the asymptotic degree distribution and, alternatively, in terms of the parameters defining the underlying random graph model.

/pdf/assortativity-and-clustering-of-sparse-random-intersection-2e0kjois0u.pdf

Mindaugas Bloznelis

Papers

Degree and clustering coefficient in sparse random intersection graphs

Component evolution in a secure wireless sensor network

Degree and clustering coefficient in sparse random intersection graphs

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

Assortativity and clustering of sparse random intersection graphs