Jerzy Jaworski

Random Intersection Graphs and Classification

We study properties of random intersection graphs generated by a random bipartite graph. We focus on the connectedness of these random intersection graphs and give threshold functions for this property and results for the size of the largest components in such graphs. The application of intersection graphs to find clusters and to test their randomness in sets of non-metric data is shortly discussed.

On a random mapping (T, Pj )

A random mapping (T, P,) of a finite set V into itself is studied. We give a new proof of the fundamental lemma of [6]. Our method leads to the derivation of several results which cannot be deduced from [6]. In particular we determine the distribution of the number of components, cyclical points and ancestors of a given point. RANDOM GRAPH

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

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.

Random key predistribution for wireless sensor networks using deployment knowledge

We consider a key distribution scheme for wireless sensor networks which uses deployment knowledge. Deployment is modeled as a grid of hexagonal clusters, into centers of which the sensor nodes are dropped according to a given probability distribution (e.g. a Gaussian one). We consider sensor connectivity in a random intersection graph model, instead of the more commonly used in literature G(n, p) graph model. While the latter is easier to analyze, the former is much more suitable to modeling sensor network key distribution. We provide analytical, asymptotic results showing how to pick parameters (key pool size |S|, the number of chosen keys d) depending on the number of deployed nodes in order to assure global connectivity of the network, and estimate the diameter of the network for the given parameters.

On a Random Digraph

We consider a random digraph on a vertex set V={1, 2, …, n} such that each vertex chooses independently its set of images as follows. First each vertex i ɛ V chooses its out-degree d + (i) according to a prescribed probability distribution {P j } n-1 j=o and then chooses at random its set of images from the family of all d + (i)-element subsets of V—{i}. If the prescribed distribution is degenerate in d then we obtain the case of random out-regular digraph D(n, d). One result of our study is concerned with relations between the general model and D(n, d i ), i=0, 1 when Another result treats the asymptotic behavior of the strength of connectedness of our general model if the above condition holds where d o and d i are constants. We prove that under some additional conditions its vertex connectivity, edge connectivity and minimum degree have asymptotically the same distribution.

The vertex degree distribution of passive random intersection graph models

In a random passive intersection graph model the edges of the graph are decided by taking the union of a fixed number of cliques of random size. We give conditions for a random passive intersection graph model to have a limiting vertex degree distribution, in particular to have a Poisson limiting vertex degree distribution. We give related conditions which, in addition to implying a limiting vertex degree distribution, imply convergence of expectation.

Clustering Coefficients of Random Intersection Graphs

Two general random intersection graph models (active and passive) were introduced by Godehardt and Jaworski (Exploratory Data Analysis in Empirical Research, Springer, Berlin, Heidelberg, New York, pp.68–81, 2002). Recently the models have been shown to have wide real life applications. The two most important ones are: non-metric data analysis and real life network analysis. Within both contexts, the clustering coefficient of the theoretical graph models is studied. Intuitively, the clustering coefficient measures how much the neighborhood of the vertex differs from a clique. The experimental results show that in large complex networks (real life networks such as social networks, internet networks or biological networks) there exists a tendency to connect elements, which have a common neighbor. Therefore it is assumed that in a good theoretical network model the clustering coefficient should be asymptotically constant. In the context of random intersection graphs, the clustering coefficient was first studied by Deijfen and Kets (Eng Inform Sci, 23:661–674, 2009). Here we study a wider class of random intersection graphs than the one considered by them and give the asymptotic value of their clustering coefficient. In particular, we will show how to set parameters – the sizes of the vertex set, of the feature set and of the vertices’ feature sets – in such a way that the clustering coefficient is asymptotically constant in the active (respectively, passive) random intersection graph.

Cycles in a Uniform Graph Process

We study the asymptotic properties of a “uniform” random graph process in which the minimum degree of U ( n, M ) grows at least as fast as ⌊M/n⌋. We show that if M — n → → ∞, almost surely U ( n, M ) consists of one giant component and some number of small unicyclic components. We go on to study the distribution of cycles in unicyclic components as they emerge at the beginning of the process and disappear when captured by the giant one.

A Random Bipartite Mapping

A random bipartite mapping ( T ; P j , Q j ) of a finite set V = V 1 ∪ V 2 into itself is considered. Wc determine the exact distributions of several numerical characteristics (for example the number of connected components, cyclical points, predecessors and successors of a given point) of such a random mapping. An asymptotical behaviour of the above random variables is studied in the special case ( P j ≡1/| V 1 |, Q 1 ≡1/| V 2 |).

Isolated Vertices in Random Intersection Graphs

For the structure analysis of non-metric data, it is natural to classify objects according to the properties they possess. An effective model to analyze the structure of similarities between objects is the random intersection graph generated by the random bipartite graph with bipartition \((\mathcal{V},\mathcal{W})\), where \(\mathcal{V}\) is a set of objects, \(\mathcal{W}\) is a set of properties, and according to some random procedure, edges join objects with their properties. In the related random intersection graph two vertices are joined by an edge if and only if they represent objects sharing at least s properties. In this paper we study the number of isolated vertices and its convergence to Poisson distribution. We generalize previous results obtained for special cases of the random model and for s = 1, only. Our approach leads us also to some interesting results on dependencies between the appearances of edges in the random intersection graph.