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. Research supported by Lithuanian State Science and Studies Foundation Grant T-07149.     

Limit laws in the generalized random graphs with random vertex weights

In this note we investigate the number of edges and the vertex degree in the generalized random graphs with vertex weights, which are independent and identically distributed random variables.     

Connectivity of the uniform random intersection graph

A uniform random intersection graphG(n,m,k) is a random graph constructed as follows. Label each of n nodes by a randomly chosen set of k distinct colours taken from some finite set of possible colours of size m. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks, in particular when modelling the network graph of the well-known key predistribution technique due to Eschenauer and Gligor.The paper determines the threshold for connectivity of the graph G(n,m,k) when n→∞ in many situations. For example, when k is a function of n such that k≥2 and m=⌊n^α⌋ for some fixed positive real number α then G(n,m,k) is almost surely connected when

lim infk²n/mlogn>1,     

The neighborhood complex of a random graph

For a graph G, the neighborhood complex N[G] is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well-known result of Lovász that if ‖N[G]‖ is k-connected, then the chromatic number of G is at least k+3.We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between 1/2 and 2/3 of the expected clique number. We also show that the number of dimensions of nontrivial homology is almost always small, O(logd), compared to the expected dimension d of the complex itself.     

Optimal stopping in a search for a vertex with full degree in a random graph

We consider the following on-line decision problem. The vertices of a realization of the random graph G(n,p) are being observed one by one by a selector. At time m, the selector examines the mth vertex and knows the graph induced by the m vertices that have already been examined. The selector’s aim is to choose the currently examined vertex maximizing the probability that this vertex has full degree, i.e. it is connected to all other vertices in the graph. An optimal algorithm for such a choice (in other words, optimal stopping time) is given. We show that it is of a threshold type and we find the threshold and its asymptotic estimation.     

A random intersection digraph: Indegree and outdegree distributions

Let S(1),…,S(n),T(1),…,T(n) be random subsets of the set [m]={1,…,m}. We consider the random digraph D on the vertex set [n] defined as follows: the arc i→j is present in D whenever S(i)∩T(j)≠0?. Assuming that the pairs of sets (S(i),T(i)), 1≤i≤n, are independent and identically distributed, we study the in- and outdegree distributions of a typical vertex of D as n,m→∞.     

A note on the chromatic number of a dense random graph

Let G_n,p denote the random graph on n labeled vertices, where each edge is included with probability p independent of the others. We show that for all constant p

     

The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on n labeled vertices with prescribed degrees D = (d₁,…,d_n). For a wide class of tame degree sequences D we obtain a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0‐1 matrices with prescribed row and column sums. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

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 $\mathsf{NP}$-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 $k<c\log n$ 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.     

Improved random graph isomorphism

Canonical labeling of a graph consists of assigning a unique label to each vertex such that the labels are invariant under isomorphism. Such a labeling can be used to solve the graph isomorphism problem. We give a simple, linear time, high probability algorithm for the canonical labeling of a G(n,p) random graph for p[ω(ln⁴n/nlnlnn),1−ω(ln⁴n/nlnlnn)]. Our result covers a gap in the range of p in which no algorithm was known to work with high probability. Together with a previous result by Bollobás, the random graph isomorphism problem can be solved efficiently for p[Θ(lnn/n),1−Θ(lnn/n)].     

On the total variation distance between the binomial random graph and the random intersection graph

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman, and Singer‐Cohen introduced a random intersection graph by taking randomly assigned sets. The random intersection graph has n vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set M of size m where each element of M belongs to each random subset with probability p, independently of all other elements in M. In 2000, Fill, Scheinerman, and Singer‐Cohen showed that the total variation distance between the random graph and the Erdös‐Rényi graph tends to 0 for any if , where is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation distance still tends to 0 for any whenever .     

Continuous spin models on annealed generalized random graphs

We study Gibbs distributions of spins taking values in a general compact Polish space, interacting via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution.First we prove a variational formula for the corresponding annealed pressure and provide criteria for absence of phase transitions in the general case.We furthermore study classes of models with second order phase transitions which include rotation-invariant models on spheres and models on intervals, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model treated in Dommers et al. (2016). On the other hand, we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.     

Long paths and cycles in random subgraphs of graphs with large minimum degree

For a given finite graph G of minimum degree at least k, let G_p be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if for a function that tends to infinity as k does, then G_p asymptotically almost surely contains a cycle (and thus a path) of length at least , and (ii) if , then G_p asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on k + 1 vertices. © Wiley Periodicals, Inc. Random Struct. Alg., 46, 320–345, 2015     

The degree sequence of a random graph. I. The models

We show that the joint distribution of the degrees of a random graph can be accurately approximated by several simpler models derived from a set of independent binomial distributions. On the one hand, we consider the distribution of degree sequences of random graphs with n vertices and ½m edges. For a wide range of values of m, this distribution is almost everywhere in close correspondence with the conditional distribution {(X₁,…,X_n) | ∑ X_i=m}, where X₁,…,X_n are independent random variables, each having the same binomial distribution as the degree of one vertex. We also consider random graphs with n vertices and edge probability p. For a wide range of functions p=p(n), the distribution of the degree sequence can be approximated by {(X₁,…,X>_n) | ∑ X_i is even}, where X₁,…,X_n are independent random variables each having the distribution Binom (n−1, p′), where p′ is itself a random variable with a particular truncated normal distribution. To facilitate computations, we demonstrate techniques by which statistics in this model can be inferred from those in a simple model of independent binomial random variables. Where they apply, the accuracy of our method is sufficient to determine asymptotically all probabilities greater than n^−k for any fixed k. In this first paper, we use the geometric mean of degrees as a tutorial example. In the second paper, we will determine the asymptotic distribution of the tth largest degree for all functions t=t(n) as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 97–117 (1997)     

Simultaneous intersection representation of pairs of graphs

We characterize the pairs (G₁, G₂) of graphs on a shared vertex set that are intersection polysemic: those for which the vertices may be assigned subsets of a universal set such that G₁ is the intersection graph of the subsets and G₂ is the intersection graph of their complements. We also consider several special cases and explore bounds on the size of the universal set. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 171–190, 1999     

Large cycles in random generalized Johnson graphs

This paper studies thresholds in random generalized Johnson graphs for containing large cycles, i.e. cycles of variable length growing with the size of the graph. Thresholds are obtained for different growth rates.     

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.     

On the cycle space of a random graph

Write for the cycle space of a graph G, for the subspace of spanned by the copies of the κ‐cycle in G, for the class of graphs satisfying , and for the class of graphs each of whose edges lies in a . We prove that for every odd and , so the 's of a random graph span its cycle space as soon as they cover its edges. For κ = 3 this was shown in [6].     

The order of the largest complete minor in a random graph

Let ccl(G) denote the order of the largest complete minor in a graph G (also called the contraction clique number) and let G_n,p denote a random graph on n vertices with edge probability p. Bollobás, Catlin, and Erd?s (Eur J Combin 1 (1980), 195–199) asymptotically determined ccl(G_n,p) when p is a constant. ?uczak, Pittel and Wierman (Trans Am Math Soc 341 (1994) 721–748) gave bounds on ccl(G_n,p) when p is very close to 1/n, i.e. inside the phase transition. We show that for every ε > 0 there exists a constant C such that whenever C/n < p < 1 ‐ ε then asymptotically almost surely ccl(G_n,p) = (1 ± ε)n/ , where b := 1/(1 ‐ p). If p = C/n for a constant C > 1, then ccl(G_n,p) = Θ( ). This extends the results in (Bollobás, Catlin, and P. Erd?s, Eur J Combin 1 (1980), 195–199) and answers a question of Krivelevich and Sudakov (preprint, 2006). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Joint probability generating function for degrees of active/passive random intersection graphs

Correlations of active and passive random intersection graphs are studied in this paper. We present the joint probability generating function for degrees of G ^active(n, m, p) and G ^passive(n, m, p), which are generated by a random bipartite graph G*(n, m, p) on n + m vertices.