The mixing time of the giant component of a random graph

We show that the total variation mixing time of the simple random walk on the giant component of supercritical and is . This statement was proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is interesting in its own right. We show that these graphs are “decorated expanders” — an expander glued to graphs whose size has constant expectation and exponential tail, and such that each vertex in the expander is glued to no more than a constant number of decorations. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 383–407, 2014     

The cover time of the giant component of a random graph

We study the cover time of a random walk on the largest component of the random graph G_n,p. We determine its value up to a factor 1 + o(1) whenever np = c > 1, c = O(lnn). In particular, we show that the cover time is not monotone for c = Θ(lnn). We also determine the cover time of the k‐cores, k ≥ 2. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

The order of the giant component of random hypergraphs

We establish central and local limit theorems for the number of vertices in the largest component of a random d‐uniform hypergraph H_d(n,p) with edge probability p = c/ , where c > (d ‐ 1)^‐1 is a constant. The proof relies on a new, purely probabilistic approach. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

On a random graph with immigrating vertices: Emergence of the giant component

A randomly evolving graph, with vertices immigrating at rate n and each possible edge appearing at rate 1/n, is studied. The detailed picture of emergence of giant components with O(n^2/3) vertices is shown to be the same as in the Erdős–Rényi graph process with the number of vertices fixed at n at the start. A major difference is that now the transition occurs about a time t=π/2, rather than t=1. The proof has three ingredients. The size of the largest component in the subcritical phase is bounded by comparison with a certain multitype branching process. With this bound at hand, the growth of the sum‐of‐squares and sum‐of‐cubes of component sizes is shown, via martingale methods, to follow closely a solution of the Smoluchowsky‐type equations. The approximation allows us to apply results of Aldous [Brownian excursions, critical random graphs and the multiplicative coalescent, Ann Probab 25 (1997), 812–854] on emergence of giant components in the multiplicative coalescent, i.e., a nonuniform random graph process. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 79–102, 2000     

On the normality of giant components

We consider a general family of random graph processes, which begin with an empty graph, and where at every step an edge is added at random according to some rule. We show that when certain general conditions are satisfied, the order of the giant component tends to a normal distribution. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 43, 452–485, 2013     

The size of the giant high‐order component in random hypergraphs

The phase transition in the size of the giant component in random graphs is one of the most well‐studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of the notion of a connected component. We consider the following: two j‐sets (sets of j vertices) are j‐connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. A hypergraph is j‐connected if all j‐sets are pairwise j‐connected. In this paper, we determine the asymptotic size of the unique giant j‐connected component in random k‐uniform hypergraphs for any and .     

Corrigendum: The cover time of the giant component of a random graph,Random Structures and Algorithms 32 (2008), 401–439

The above paper gives an asymptotically precise estimate of the cover time of the giant component of a sparse random graph. The proof as it stands is not tight enough, and in particular, Eq. (64) is not strong enough to prove (65). The o(1) term in (64) needs to be improved to o(1/(lnn)²) for (65) to follow. The following section, which replaces Section 3.6, amends this oversight. The notation and definitions are from the paper. A further correction is needed. Property P2 claims that the conductance of the giant is whp , Ω(1/lnn). The proof of P2 in the appendix of the paper is not quite complete. A complete proof of Property P2 can be found in 1 , 2 . © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

On tree census and the giant component in sparse random graphs

For each of the two models of a sparse random graph on n vertices, G(n, # of edges = cn/2) and G(n, Prob (edge) = c/n) define t_n(k) as the total number of tree components of size k (1 ≤ k ≤ n). the random sequence {[t_n(k) - nh(k)]n^?1/2} is shown to be Gaussian in the limit n →∞, with h(k) = k^k?2c^k?1e^?kc/k! and covariance function being dependent upon the model. This general result implies, in particular, that, for c> 1, the size of the giant component is asymptotically Gaussian, with mean nθ(c) and variance n(1 ? T)^?2(1 ? 2Tθ)θ(1 ? θ) for the first model and n(1 ? T)^?2θ(1 ? θ) for the second model. Here Te^?T = ce^?c, T<1, and θ = 1 ? T/c. A close technique allows us to prove that, for c < 1, the independence number of G(n, p = c/n) is asymptotically Gaussian with mean nc^?1(β + β²/2) and variance n[c^?1(β + β²/2) ?c^?2(c + 1)β²], where βe^β = c. It is also proven that almost surely the giant component consists of a giant two-connected core of size about n(1 ? T)β and a “mantle” of trees, and possibly few small unicyclic graphs, each sprouting from its own vertex of the core.     

A probabilistic approach to the leader problem in random graphs

We study the fixation time of the identity of the leader, that is, the most massive component, in the general setting of Aldous's multiplicative coalescent, which in an asymptotic sense describes the evolution of the component sizes of a wide array of near‐critical coalescent processes, including the classical Erd?s‐Rényi process. We show tightness of the fixation time in the “Brownian” regime, explicitly determining the median value of the fixation time to within an optimal O(1) window. This generalizes ?uczak's result for the Erd?s‐Rényi random graph using completely different techniques. In the heavy‐tailed case, in which the limit of the component sizes can be encoded using a thinned pure‐jump Lévy process, we prove that only one‐sided tightness holds. This shows a genuine difference in the possible behavior in the two regimes.     

Giant components in Kronecker graphs

Let \begin{align*}n\in\mathbb{N}\end{align*}, 0 <α,β,γ< 1. Define the random Kronecker graph K(n,α,γ,β) to be the graph with vertex set \begin{align*}\mathbb{Z}_2^n\end{align*}, where the probability that u is adjacent to v is given by p_u,v =α ^{u ? v} γ^{( 1‐u )?( 1‐v )}β^{n‐ u ? v ‐( 1‐u )?( 1‐v )}. This model has been shown to obey several useful properties of real‐world networks. We establish the asymptotic size of the giant component in the random Kronecker graph.© 2011 Wiley Periodicals, Inc. Random Struct. Alg.,2011     

Quasi-stationary distributions as centrality measures for the giant strongly connected component of a reducible graph

A random walk can be used as a centrality measure of a directed graph. However, if the graph is reducible the random walk will be absorbed in some subset of nodes and will never visit the rest of the graph. In Google PageRank the problem was solved by the introduction of uniform random jumps with some probability. Up to the present, there is no final answer to the question about the choice of this probability. We propose to use a parameter-free centrality measure which is based on the notion of a quasi-stationary distribution. Specifically, we suggest four quasi-stationary based centrality measures, analyze them and conclude that they produce approximately the same ranking.     

A new approach to chordal graphs

By a chordal graph is meant a graph with no induced cycle of length ⩾ 4. By a ternary system is meant an ordered pair (W, T), where W is a finite nonempty set, and T ⊆ W × W × W. Ternary systems satisfying certain axioms (A1)–(A5) are studied in this paper; note that these axioms can be formulated in a language of the first-order logic. For every finite nonempty set W, a bijective mapping from the set of all connected chordal graphs G with V(G) = W onto the set of all ternary systems (W, T) satisfying the axioms (A1)–(A5) is found in this paper.     

A Bayesian-martingale approach to the general disorder problem

We consider a Bayesian-martingale approach to the general change-point detection problem. In our setting the change-point represents a random time of bifurcation of two probability measures given on the space of right-continuous functions. We derive a reflecting backward stochastic differential equation (RBSDE) for the value process related to the disorder problem and show that in classical cases of the Wiener and Poisson disorder problems this RBSDE is equivalent to free-boundary problems for parabolic differential and differential–difference operators respectively.     

A new approach to Cagniard's problem

Cagniard problem refers to the class of linear reflection and transmission problem for pulsed line and point sources, which have solution methods leading to exact algebraic representations of the wave fields. All previous methods have relied heavily on integral or differential transforms. We present in this paper a new and direct approach to the problem which involves only the wave equation and its associated characteristic equation. We illustrate the new method by applying it to the problem of the reflection and transmission of acoustic waves radiating from a line source in the vicinity of a plane boundary separating two uniform acoustic media.     

A graphon approach to limiting spectral distributions of Wigner‐type matrices

We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner‐type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes transforms explicitly with weaker assumptions on the convergence of variance profiles than previous results. As applications, we give a new proof of the semicircle law for generalized Wigner matrices and determine the limiting spectral distributions for three sparse inhomogeneous random graph models with sparsity ω(1/n): inhomogeneous random graphs with roughly equal expected degrees, W‐random graphs and stochastic block models with a growing number of blocks. Furthermore, we show our theorems can be applied to random Gram matrices with a variance profile for which we can find the limiting spectral distributions under weaker assumptions than previous results.     

The Stationary Distribution of a Continuous-Time Random Graph Process

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A new Lagrangean approach to the pooling problem

We present a new Lagrangean approach for the pooling problem. The relaxation targets all nonlinear constraints, and results in a Lagrangean subproblem with a nonlinear objective function and linear constraints, that is reformulated as a linear mixed integer program. Besides being used to generate lower bounds, the subproblem solutions are exploited within Lagrangean heuristics to find feasible solutions. Valid cuts, derived from bilinear terms, are added to the subproblem to strengthen the Lagrangean bound and improve the quality of feasible solutions. The procedure is tested on a benchmark set of fifteen problems from the literature. The proposed bounds are found to outperform or equal earlier bounds from the literature on 14 out of 15 tested problems. Similarly, the Lagrangean heuristics outperform the VNS and MALT heuristics on 4 instances. Furthermore, the Lagrangean lower bound is equal to the global optimum for nine problems, and on average is 2.1% from the optimum. The Lagrangean heuristics, on the other hand, find the global solution for ten problems and on average are 0.043% from the optimum.