Recognizing random intersection graphs

Random graphs with latent geometric structure are popular models of social and biological networks, with applications ranging from network user profiling to circuit design. These graphs are also of purely theoretical interest within computer science, probability and statistics. A fundamental initial question regarding these models is: when are these random graphs affected by their latent geometry and when are they indistinguishable from simpler models without latent structure, such as the Erdős–Rényi graph $${\mathcal {G}}(n, p)$$ ? We address this question for two of the most well-studied models of random graphs with latent geometry—the random intersection and random geometric graph. Our results are as follows: (a) The random intersection graph is defined by sampling n random sets $$S_1, S_2, \ldots , S_n$$ by including each element of a set of size d in each $$S_i$$ independently with probability $$\delta $$ , and including the edge $$\{i, j\}$$ if $$S_i \cap S_j \ne \emptyset $$ . We prove that the random intersection graph converges in total variation to an Erdős–Rényi graph if $$d = {\tilde{\omega }}(n^3)$$ , and does not if $$d = o(n^3)$$ , for both dense and sparse edge densities p. This resolves an open problem in Fill et al. (Random Struct Algorithms 16(2):156–176, 2000), Rybarczyk (Random Struct Algorithms 38(1–2):205–234, 2011) and Kim et al. (Random Struct Algorithms 52(4):662–679, 2018). The same result was obtained simultaneously and independently by Bubeck et al. (When random intersection graphs lose geometry. Manuscript, 2019). (b) We strengthen the preceding argument to show that the matrix of random intersection sizes $$|S_i \cap S_j|$$ converges in total variation to a symmetric matrix with independent Poisson entries. This yields the first total variation convergence result for $$\tau $$ -random intersection graphs, where the edge $$\{i, j\}$$ is included if $$|S_i \cap S_j| \ge \tau $$ . More precisely, our results imply that, if p is bounded away from 1, then the $$\tau $$ -random intersection graph with edge density p converges to $${\mathcal {G}}(n, p)$$ if $$d = \omega (\tau ^3 n^3)$$ . (c) The random geometric graph on $${\mathbb {S}}^{d - 1}$$ is defined by sampling $$X_1, X_2, \dots , X_n$$ uniformly at random from $${\mathbb {S}}^{d - 1}$$ and including the edge $$\{i, j\}$$ if $$\Vert X_i - X_j \Vert _2 \le \tau $$ . A result of Bubeck et al. (Random Struct Algorithms 49:503–532, 2016) showed that this model converges to $${\mathcal {G}}(n, p)$$ in total variation, where p is chosen so that the models have matching edge densities, as long as $$d = \omega (n^3)$$ . It was conjectured in Bubeck et al. (2016) that this threshold decreases drastically for p small. We make the first progress towards this conjecture by showing convergence if $$d = {\tilde{\omega }}\left( \min \{ pn^3, p^2 n^{7/2} \} \right) $$ . Our proofs are a hybrid of combinatorial arguments, direct couplings and applications of information inequalities. Previous upper bounds on the total variation distance between random graphs with latent geometry and $${\mathcal {G}}(n, p)$$ have typically not been both combinatorial and information-theoretic, while this interplay is essential to the sharpness of our bounds.

Expander properties and the cover time of random intersection graphs

Coupling methods to establish threshold functions in random graphs

Let H be an edge-colored hypergraph. We say that H contains a rainbow copy of a hypergraph S if it contains an isomorphic copy of S with all edges of distinct colors. We consider the following setting. A randomly edge-colored random hypergraph \(H \sim \mathcal{H}_{c}^{k}(n,p)\) is obtained by adding each k-subset of [n] with probability p, and assigning it a color from [c] uniformly, independently at random. As a first result we show that a typical \(H \sim \mathcal{H}_{c}^{2}(n,p)\) (that is, a random edge-colored graph) contains a rainbow Hamilton cycle, provided that \(c = (1 + o(1))n\) and \(p = \frac{\log n+\log \log n+\omega (1)} {n}\). This is asymptotically best possible with respect to both parameters, and improves a result of Frieze and Loh. Secondly, based on an ingenious coupling idea of McDiarmid, we provide a general tool for tackling problems related to finding “nicely edge-colored” structures in random graphs/hypergraphs. We illustrate the generality of this statement by presenting two interesting applications. In one application we show that a typical \(H \sim \mathcal{H}_{c}^{k}(n,p)\) contains a rainbow copy of a hypergraph S, provided that \(c = (1 + o(1))\vert E(S)\vert\) and p is (up to a multiplicative constant) a threshold function for the property of containment of a copy of S. In the second application we show that a typical \(G \sim \mathcal{H}_{c}^{2}(n,p)\) contains \((1 - o(1))np/2\) edge-disjoint Hamilton cycles, each of which is rainbow, provided that c = ω(n) and \(p =\omega (\log n/n)\).

We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have different densities. In the graph case, we prove a stronger result, on packing a random graph with a fixed graph.

We study the evolution of the chromatic number of a random intersection graph and show that, in a certain range of parameters, these random graphs can be colored optimally with high probability using different greedy algorithms. Experiments on real network data confirm the positive theoretical predictions and suggest that heuristics for the clique and the chromatic number can work hand in hand proving mutual optimality.

In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.

The planted clique conjecture states that no polynomial-time algorithm can find a hidden clique of size k ≪ √n in an n-vertex Erdős–Rényi random graph with a k-clique planted. In this paper, we prove the equivalence among many (in fact, most) variants of planted clique conjectures, such as search versions with a success probability exponentially close to 1 and with a non-negligible success probability, a worst-case version (the k-clique problem on incompressible graphs), decision versions with small and large success probabilities, and decision versions with adversarially chosen k and binomially distributed k. In particular, we establish the equivalence between the planted clique problem introduced by Jerrum and Kučera and its decision version suggested by Saks in the 1990s. Moreover, the equivalence among decision versions identifies the optimality of a simple edge counting algorithm: By counting the number of edges, one can efficiently distinguish an n-vertex random graph from a random graph with a k-clique planted with probability Θ(k2/n) for any k ≤ √n. We show that for any k, no polynomial-time algorithm can distinguish these two random graphs with probability ≫ k2 / n if and only if the planted clique conjecture holds. The equivalence among search versions identifies the first one-way function that admits a polynomial-time security-preserving self-reduction from exponentially weak to strong one-way functions. These results reveal a detection-recovery gap in success probabilities for the planted clique problem. We also present another equivalence between the existence of a refutation algorithm for the planted clique problem and an average-case polynomial-time algorithm for the k-clique problem with respect to the Erdős–Rényi random graph.

This chapter presents a random graph model for image representation. The first contribution the authors propose is a Geometric-Edge (G-E) Random Graph Model for image representation. The second contribution is that of casting image matching into G-E Random Graph matching by using the random dot product graph based matching algorithm. Experimental results show that the proposed G-E Random Graph model and matching algorithm are effective and robust to structural variations.

For a number of optimization problems on random graphs and hypergraphs, e.g., k-colorings, there is a very big gap between the largest average degree for which known polynomial-time algorithms can find solutions, and the largest average degree for which solutions provably exist. We study this phenomenon by examining how sets of solutions evolve as edges are added. We prove in a precise mathematical sense that, for each problem studied, the barrier faced by algorithms corresponds to a phase transition in the problems solution-space geometry. Roughly speaking, at some problem-specific critical density, the set of solutions shatters and goes from being a single giant ball to exponentially many, well-separated, tiny pieces. All known polynomial-time algorithms work in the ball regime, but stop as soon as the shattering occurs. Besides giving a geometric view of the solution space of random instances our results provide novel constructions of one-way functions.

In cluster analysis, the random graph model G n,p and G n,p-based multigraph models have been used for purposes of statistical modelling of data and testing the randomness of outlined clusters. While being appropriate for non-metric data, such models supposing independence of all edges do not take into account the triangle inequality which is valid for metric data. We will introduce graph models I n,dand I t,n,(d1,…,dt) for random intersection graphs in R 1 and multigraphs in R t under which the triangle inequality holds. We derive limit theorems for the distribution of random variables which describe important properties of these random intersection graphs. While being asymptotically equivalent for some properties like the limit distribution of the number of isolated points, the G n,p-model and the I n,d-model differ in numerous aspects.

Hitting times, commute times, and cover times for random walks on random hypergraphs

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

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.KeywordsRandom Intersection GraphsClustering CoefficientGodehardtLarge Complex NetworksBloznelisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we investigate the efficiency of local search algorithms for some relatively unstructured networks. By local search, we mean that each vertex has a knowledge of its very close neighborhood, typically the identities, degrees and possibly the contents. Assuming there is no preprocessing step, we are interested in seeking a vertex of a given identity starting from any vertex. The time complexity of search is expressed as the number of vertices to explore before reaching the target or a neighbor of the target. For an arbitrary graph and a worst-case labelling of vertices, time complexity can be of the same order as the size of the graph itself. However, there is some hope that for large classes of graphs, the size of the exploration sequence would drastically decrease. Recent literature exhibits non trivial properties shared by the topologies of numerous real-world distributed networks: the combination of a low diameter, a specific degree distribution of the vertices and a sparse network are typically representative of a small-world topology. On the other hand, in a seminal paper, Kleinberg [Kle00] focuses on the ability of a user or a mobile agent to route along shorts paths within a class of random small-world graphs, the navigable small-worlds† using a greedy distributed algorithm (distances within a spanning subgraph are known or can be computed locally). Under some conditions, Kleinberg also proposes polynomial lower bounds valid for any distributed algorithm using the same knowledge of the network even if the diameter is polylogarithmic. However, Kleinberg’s model lacks an important property observed in real networks which are often scalefree. The degree distribution of the vertices tends to follow a power-law distribution, that is the number of vertices of degree δ is proportional to n ( 1 δ )k for an n-vertex graph and k a constant strictly greater than one. Moreover, it has been observed that for real-life networks, k tends to belong to the range [2,3]. For graphs built using Kleinberg’s model, the degree distribution is very different (close to a Poisson distribution). All local‡ distributed search algorithms presented in the literature work as follows: given the set of visited vertices, one should choose the next vertex to explore. The process stops as soon as the target is reached. In this paper, the underlying network is a random scale-free labeled graph whose node identities belong to the range [1,n]. Each vertex knows its own identity and its degree and we assume two models of local knowledge: some information about the neighbors can be known (strong model) or not (weak model). The first model is the more realistic one but our starting point is the second model. Note that Kleinberg’s model assumes more information than our strong model. Up to now, it is still an open problem to know if scale-free graphs are navigable in the polylogarithmic sense. In this paper, we answer negatively to this question for some random scale-free graphs built using a