Random Intersection Graphs Research Articles

Hamilton cycles in the line graph of a random graph

We determine a threshold function for the property that the line graph of a random graph has a Hamilton cycle. Our work is motivated by the problem of determining a threshold function for existence of a Hamilton cycle in a uniform random intersection graph.

Poisson approximation of counts of induced subgraphs in random intersection graphs

Random intersection graphs are characterised by three parameters: n, m and p, where n is the number of vertices, m is the number of objects, and p is the probability that a given object is associated with a given vertex. Two vertices in a random intersection graph are adjacent if and only if they have an associated object in common. When m=⌊nα⌋ for constant α, we provide a condition, called strictlyα-balanced, for the Poisson convergence of the number of induced copies of a fixed subgraph.

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.

The Coupling Method for Inhomogeneous Random Intersection Graphs.

We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we improve and generalize the coupling method introduced for random intersection graphs so that it may be used for a wider range of parameters. Using the new approach we are able to tighten the best known results concerning random intersection graphs and establish threshold functions for some monotone properties of inhomogeneous random intersection graphs. Considered properties are: $k$-connectivity, matching containment and hamiltonicity.

Large Cliques in Sparse Random Intersection Graphs

An intersection graph defines an adjacency relation between subsets $S_1,\dots, S_n$ of a finite set $W=\{w_1,\dots, w_m\}$: the subsets $S_i$ and $S_j$ are adjacent if they intersect. Assuming that the subsets are drawn independently at random according to the probability distribution $\mathbb{P}(S_i=A)=P(|A|){\binom{m}{|A|}}^{-1}$, $A\subseteq W$, where $P$ is a probability on $\{0, 1, \dots, m\}$, we obtain the random intersection graph $G=G(n,m,P)$. We establish the asymptotic order of the clique number $\omega(G)$ of a sparse random intersection graph as $n,m\to+\infty$. For $m = \Theta(n)$ we show that the maximum clique is of size $(1-\alpha/2)^{-\alpha/2}n^{1-\alpha/2}(\ln n)^{-\alpha/2}(1+o_P(1))$ in the case where the asymptotic degree distribution of $G$ is a power-law with exponent $\alpha \in (1,2)$. It is of size $\frac {\ln n} {\ln \ln n}(1+o_P(1))$ if the degree distribution has bounded variance, i.e., $\alpha>2$. We construct a simple polynomial-time algorithm which finds a clique of the optimal order $\omega(G) (1-o_P(1))$.

Formation of a giant component in the intersection graph of a random chord diagram

We study the number of chords and the number of crossings in the largest component of a random chord diagram when the chords are sparsely crossing. This is equivalent to studying the number of vertices and the number of edges in the largest component of the random intersection graph. Denoting the number of chords by n and the number of crossings by m, when m/(nlog⁡n) tends to a limit in (0,2/π2), we show that the chord diagram chosen uniformly at random from all the diagrams with given parameters has a component containing almost all the crossings and a positive fraction of chords. On the other hand, when m≤n/14, the size of the largest component is of order O(log⁡n). One of the key analytical ingredients is an asymptotic expression for the number of chord diagrams with parameters n and m for m<(2/π2)nlog⁡n, based on the Touchard–Riordan formula and the Jacobi identity for the generating function of Euler partition function.

The chromatic number of random intersection graphs

In the random intersection graph model G(n,m,P(m)) to each vertex from a vertex set V (|V| = n) we assign, independently from all other vertices, a random set of its features W (v) from an auxiliary set W (|W| = m = m(n)). The cardinality of W (v) is chosen according to a given probability distribution P(m) and W (v) is chosen uniformly at random from all subsets of W of this cardinality. We connect vertices v and u by an edge if the sets W (v) and W (u) intersect. We present some recent results concerning colouring random intersection graphs. It was shown in [1] that for some range of parameters with high probability 1 the chromatic number χ(G(n,m,P(m))) is asymptotically equal to the clique number ω(G(n,m,P(m))), for P(m) binomial Bin(m, p). We reprove this result in a very simple way and we partially resolve the problem for a wider range of parameters. Our results are related to the problem of determining the chromatic index of the random hypergraph H = {W, E}, where E = {W (v), v ∈ V}, i.e. they are linked to the problem of generalising Vizing’s theorem for hypergraphs.

Hyperbolicity, degeneracy, and expansion of random intersection graphs

We establish the conditions under which several algorithmically exploitable structural features hold for random intersection graphs, a natural model for many real-world networks where edges correspond to shared attributes. Specifically, we fully characterize the degeneracy of random intersection graphs, and prove that the model asymptotically almost surely produces graphs with hyperbolicity at least log(n). Further, we prove that in the parametric regime where random intersection graphs are degenerate an even stronger notion of sparseness, so called bounded expansion, holds with high probability. We supplement our theoretical findings with experimental evaluations of the relevant statistics.

Clustering coefficient of random intersection graphs with infinite degree variance

For a random intersection graph with a power law degree sequence having a finite mean and an infinite variance we show that the global clustering coefficient admits a tunable asymptotic distribution.

Degree-degree distribution in a power law random intersection graph with clustering

The bivariate distribution of degrees of adjacent vertices (degree-degree distribution) is an important network characteristic defining the statistical dependencies between degrees of adjacent vertices. We show the asymptotic degree-degree distribution of a sparse inhomogeneous random intersection graph and discuss its relation to the clustering and power law properties of the graph.

On the Chromatic Number of Non-Sparse Random Intersection Graphs

An intersection graph of n vertices assumes that each vertex is equipped with a subset of a global label set. Two vertices share an edge when their label sets intersect. Random intersection graphs (RIGs) (as defined in Karonski et al. Comb. Probab. Comput. J. 8, 131---159 (1999); Singer-Cohen (1995)) consider label sets formed by the following experiment: each vertex, independently and uniformly, examines all the labels (m in total) one by one. Each examination is independent and the vertex succeeds to put the label in her set with probability p. Such graphs can capture interactions in networks due to sharing of resources among nodes. In this paper, we discuss various structural and algorithmic results concerning random intersection graphs and we focus on the computational problem of properly coloring random instances of the binomial random intersection graphs model. For the latter, we consider a range of parameters m, p for which RIGs differ substantially from Erdoźs-Renyi random graphs and for which greedy approaches fail.

Random Subcube Intersection Graphs I: Cliques and Covering

We study random subcube intersection graphs, that is, graphs obtained by selecting a random collection of subcubes of a fixed hypercube $Q_d$ to serve as the vertices of the graph, and setting an edge between a pair of subcubes if their intersection is non-empty. Our motivation for considering such graphs is to model 'random compatibility' between vertices in a large network.For both of the models considered in this paper, we determine the thresholds for covering the underlying hypercube $Q_d$ and for the appearance of $s$-cliques. In addition we pose a number of open problems.

Clustering coefficient of random intersection graphs with infinite degree variance

AbstractFor a random intersection graph with a power law degree sequence having a finite mean and an infinite variance we show that the global clustering coefficient admits a tunable asymptotic distribution.

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.

&lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$k$ &lt;/tex-math&gt;&lt;/inline-formula&gt;-Connectivity in Random Key Graphs With Unreliable Links

Random key graphs form a class of random intersection graphs and are naturally induced by the random key predistribution scheme of Eschenauer and Gligor for securing wireless sensor network (WSN) communications. Random key graphs have received much interest recently, owing in part to their wide applicability in various domains including recommender systems, social networks, secure sensor networks, clustering and classification analysis, and cryptanalysis to name a few. In this paper, we study connectivity properties of random key graphs in the presence of unreliable links. Unreliability of the edges are captured by independent Bernoulli random variables, rendering edges of the graph to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdos-Renyi graph, and is expected to be useful in capturing various real-world networks; e.g., with secure WSN applications in mind, link unreliability can be attributed to harsh environmental conditions severely impairing transmissions. We present conditions on how to scale this model's parameters so that i) the minimum node degree in the graph is at least k, and ii) the graph is k-connected, both with high probability as the number of nodes becomes large. The results are given in the form of zeroone laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on the k-connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on the 1-connectivity of random key graphs with unreliable links.

On the kernel size of clique cover reductions for random intersection graphs

Covering all edges of a graph by a minimum number of cliques is a well known NP-hard problem. For the parameter k being the maximal number of cliques to be used, the problem becomes fixed parameter tractable. However, assuming the Exponential Time Hypothesis, there is no kernel of subexponential size in the worst-case.We study the average kernel size for random intersection graphs with n vertices, edge probability p, and clique covers of size k. We consider the well-known set of reduction rules of Gramm, Guo, Hüffner, and Niedermeier (2009) [17] and show that with high probability they reduce the graph completely if p is bounded away from 1 and k<clog⁡n for some constant c>0. This shows that for large probabilistic graph classes like random intersection graphs the expected kernel size can be substantially smaller than the known exponential worst-case bounds.

On the Chromatic Index of Random Uniform Hypergraphs

Let $\mathbb{H}^{(k)}(n, N)$, where $k \ge 2$, be a random hypergraph on the vertex set $[n] = \{1, 2, \dots, n\}$ with $N$ edges drawn independently with replacement from all subsets of $[n]$ of size $k$. For $\bar{d} = k N/n$ and any $\varepsilon > 0$ we show that if $k = o(\ln ({\bar d}/\ln n))$ and $k = o(\ln (n/\ln {\bar d}))$, then with probability $1-o(1)$ a random greedy algorithm produces a proper edge coloring of $\mathbb{H}^{(k)}(n, N)$ with at most $\bar{d} (1+\varepsilon)$ colors. This yields the asymptotic chromatic number of the corresponding uniform random intersection graph.

Random Intersection Graph Process

We introduce a random intersection graph process aimed at modeling sparse evolving affiliation networks. We establish the asymptotic degree distribution and provide explicit asymptotic formulas for assortativity and clustering coefficients showing how these edge dependence characteristics vary over time.

POISSON APPROXIMATION FOR THE NUMBER OF ISOLATED COMPLETE GRAPHS IN A RANDOM INTERSECTION GRAPH

Let Wn,r be the number of isolated complete graphs of order r in a random intersection graph G(n,m,p). In this paper, we demonstrate that Wn,r can be approximated by Poisson distribution and give the bound of this approximation by using the Stein-Chen method.

Formula omitted]-connectivity of uniform s-intersection graphs

Let W1,…,Wn be independent random subsets of [m]={1,…,m}. Assuming that each Wi is uniformly distributed in the class of d-subsets of [m] we study the uniform random intersection graph Gs(n,m,d) on the vertex set {W1,…Wn}, defined by the adjacency relation: Wi∼Wj whenever ∣Wi∩Wj∣≥s. We show that as n,m→∞ the edge density threshold for the property that each vertex of Gs(n,m,d) has at least k neighbours is asymptotically the same as that for Gs(n,m,d) being k-connected.