General Random Intersection Graph Research Articles

On Connectivity and Robustness in Random Intersection Graphs

Random intersection graphs have received much attention recently and been used in a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. For these graphs, each node is equipped with a set of objects in a random manner, and two nodes have an undirected edge in between if they have at least one object in common. In this paper, we investigate connectivity and robustness in a general random intersection graph model. Specifically, we establish sharp asymptotic zero-one laws for k-connectivity and k-robustness, as well as the asymptotically exact probability of k-connectivity, for any positive integer k. The k-connectivity property quantifies how resilient is the connectivity of a graph against node or edge failures, while k-robustness measures the effectiveness of local-information-based consensus algorithms (that do not use global graph topology information) in the presence of adversarial nodes. In addition to presenting the results under the general random intersection graph model, we consider two special cases of the general model, a binomial random intersection graph and a uniform random intersection graph, which both have numerous applications as well. For these two specialized graphs, our results on asymptotically exact probabilities of k-connectivity and asymptotic zero-one laws for k-robustness are also novel in the literature.

Zero-One Laws for Connectivity in Inhomogeneous Random Key Graphs

We introduce a new random key predistribution scheme for securing heterogeneous wireless sensor networks. Each of the n sensors in the network is classified into r classes according to some probability distribution {\mu} = {{\mu}_1 , . . . , {\mu}_r }. Before deployment, a class-i sensor is assigned K_i cryptographic keys that are selected uniformly at random from a common pool of P keys. Once deployed, a pair of sensors can communicate securely if and only if they have a key in common. We model the communication topology of this network by a newly defined inhomogeneous random key graph. We establish scaling conditions on the parameters P and {K_1 , . . . , K_r } so that this graph i) has no isolated nodes; and ii) is connected, both with high probability. The results are given in the form of zero-one laws with the number of sensors n growing unboundedly large; critical scalings are identified and shown to coincide for both graph properties. Our results are shown to complement and improve those given by Godehardt et al. and Zhao et al. for the same model, therein referred to as the general random intersection graph.

Degree Distributions in General Random Intersection Graphs

We study $G(n,m,F,H)$, a variant of the standard random intersection graph model in which random weights are assigned to both vertex types in the bipartite structure. Under certain assumptions on the distributions of these weights, the degree of a vertex is shown to depend on the weight of that particular vertex and on the distribution of the weights of the other vertex type.

Component Evolution in General Random Intersection Graphs

Given integers $n$, $m$ and a probability distribution $P_*$ on $[m]=\{1,\dots,m\}$, consider the random intersection graph on the vertex set $[n]$, where $i,j\in[n]$ are declared to be adjacent whenever $S(i)\cap S(j)\neq\emptyset$. Here $S(1),\dots,S(n)$ denote independent and identically distributed random subsets of $[m]$ with the distribution $\mathbf{P}(S(i)=A)={m\choose|A|}^{-1}P_*(|A|)$ for $A\subset[m]$. Assuming that $m$ is much larger than $n$, we show that the order of the largest connected component $N_1=n\rho+o_P(n)$ as $n,m\to\infty$. Here $\rho$ denotes the nonextinction probability of a related multitype Poisson branching process.

Large independent sets in general random intersection graphs

We investigate the existence and efficient algorithmic construction of close to optimal independent sets in random models of intersection graphs. In particular, (a) we propose a new model for random intersection graphs ( G n , m , p → ) which includes the model of [M. Karoński, E.R. Scheinerman, K.B. Singer-Cohen, On random intersection graphs: The subgraph problem, Combinatorics, Probability and Computing journal 8 (1999), 131–159] (the “uniform” random intersection graph models) as an important special case. We also define an interesting variation of the model of random intersection graphs, similar in spirit to random regular graphs. (b) For this model we derive exact formulae for the mean and variance of the number of independent sets of size k (for any k ) in the graph. (c) We then propose and analyse three algorithms for the efficient construction of large independent sets in this model. The first two are variations of the greedy technique while the third is a totally new algorithm. Our algorithms are analysed for the special case of uniform random intersection graphs. Our analyses show that these algorithms succeed in finding close to optimal independent sets for an interesting range of graph parameters.

Degree distribution of a typical vertex in a general random intersection graph

For a general random intersection graph, we show an approximation of the vertex degree distribution by a Poisson mixture.