Katarzyna Rybarczyk

Diameter, connectivity, and phase transition of the uniform random intersection graph

We study properties of the uniform random intersection graph model G(n,m,d). We find asymptotic estimates on the diameter of the largest connected component of the graph near the phase transition and connectivity thresholds. Moreover we manage to prove an asymptotically tight bound for the connectivity and phase transition thresholds for all possible ranges of d, which has not been obtained before. The main motivation of our research is the usage of the random intersection graph model in the studies of wireless sensor networks.

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.

Geometric graphs with randomly deleted edges - connectivity and routing protocols

In the article we study important properties of random geometric graphs with randomly deleted edges which are natural models of wireless ad hoc networks with communication constraints. We concentrate on two problems which are most important in the context of theoretical studies on wireless ad hoc networks. The first is how to set parameters of the network (graph) to have it connected. The second is the problem of an effective message transmition i.e. the problem of construction of routing protocols in wireless networks. We provide a thorough mathematical analysis of connectivity property and a greedy routing protocol. The models we use are: an intersection of a random geometric graph with an Erdos-Renyi random graph and an intersection of a random geometric graph with a uniform random intersection graph. The obtained results are asymptotically tight up to a constant factor.

Equivalence of a random intersection graph and G ( n , p )

We solve the conjecture of Fill, Scheinerman and Singer-Cohen (Random Struct Algorithms 16 (2000), 156–176) and show equivalence of sharp threshold functions of a random intersection graph

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

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Random key predistribution for wireless sensor networks using deployment knowledge

**image** (n,m,p) with m ≥ n3 and a graph G(n,p) with independent edges. Moreover we prove sharper equivalence results under some additional assumptions.

Forward-secure key evolution in wireless sensor networks

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.

Clustering Coefficients of Random Intersection Graphs

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.

The Degree Distribution in Random Intersection Graphs

We consider a key distribution scheme for securing node-to-node communication in sensor networks. While most schemes in use are based on random predistribution, we consider a system of dynamic pairwise keys based on design due to Ren, Tanmoy and Zhou. We design and analyze a variation of this scheme, in which capturing a node does not lead to security threats for the past communication. Instead of bit-flipping, we use a cryptographic one-way function. While this immediately guarantees forward-security, it is not clear whether the pseudorandom transformation of the keys does not lead to subtle security risks due to a specific distribution of reachable keys, such as existence of small attractor subspaces. (This problem does not occur for the design of Ren, Tanmoy and Zhou.)We show, in a rigorous, mathematical way, that this is not the case: after a small number of steps probability distribution of keys leaves no room for potential attacks.

Isolated Vertices in 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.