The diameter of sparse random graphs

We derive an expression of the form c ln n + o(ln n) for the diameter of a sparse random graph with a specified degree sequence. The result holds asymptotically almost surely, assuming that certain convergence and supercriticality conditions are met, and is applicable to the classical random graph G_n,p with np = Θ(1) + 1, as well as certain random power law graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

Scale Free Properties of Random <Emphasis Type="Italic">k</Emphasis>-Trees

Scale free graphs have attracted attention as their non-uniform structure that can be used as a model for many social networks including the WWW and the Internet. In this paper, we propose a simple random model for generating scale free k-trees. For any fixed integer k, a k-tree consists of a generalized tree parameterized by k, and is one of the basic notions in the area of graph minors. Our model is quite simple and natural; it first picks a maximal clique of size k + 1 uniformly at random, it then picks k vertices in the clique uniformly at random, and adds a new vertex incident to the k vertices. That is, the model only makes uniform random choices twice per vertex. Then (asymptotically) the distribution of vertex degree in the resultant k-tree follows a power law with exponent 2 + 1/k, the k-tree has a large clustering coefficient, and the diameter is small. Moreover, our experimental results indicate that the resultant k-trees have extremely small diameter, proportional to o(log n), where n is the number of vertices in the k-tree, and the o(1) term is a function of k.     

The degree sequence of a scale‐free random graph process

Recently, Barabási and Albert [2] suggested modeling complex real‐world networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabási and Albert suggested that after many steps the proportion P(d) of vertices with degree d should obey a power law P(d)αd^?γ. They obtained γ=2.9±0.1 by experiment and gave a simple heuristic argument suggesting that γ=3. Here we obtain P(d) asymptotically for all d≤n^1/15, where n is the number of vertices, proving as a consequence that γ=3. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 279–290, 2001     

A note on stable exponents

Given an element γ in a group γ, the stable exponent p+(γ) of γ is defined as p+(γ) =lim sup_n→∞P(γⁿ) denotes the exponent of P(γⁿ) = sup{k/ ?γ_o ∈ γ such that γⁿ = γ^k _o We prove that if γ acts properly discontinuously by isometrics on a proper geodesic Gromov-hyperbolic metric space and γ ∈ γ is of hyperbolic type, then P+(γ) is an integer. This implies that the stable exponent of every element of infinite order in a word hyperbolic group is an integer. We also show that, in a translation discrete group, the stable exponent of every element of infinite order is finite.     

On Threshold Probability for the Stability of Independent Sets in Distance Graphs

This paper considers the so-called distance graph G(n, r, s);its vertices can be identified with the r-element subsets of the set {1, 2,…,n}, and two vertices are joined by an edge if the size of the intersection of the corresponding subsets equals s. Note that, in the case s = 0, such graphs are known as Kneser graphs. These graphs are closely related to the Erd?s-Ko-Rado problem; they also play an important role in combinatorial geometry and coding theory.

We study properties of random subgraphs of the graph G(n, r, s) in the Erd?s-Rényi model, in which each edge is included in the subgraph with a certain fixed probability p independently of the other edges. It is known that if r > 2s + 1, then, for p = 1/2, the size of an independent set is asymptotically stable in the sense that the independence number of a random subgraph is asymptotically equal to that of the initial graph G(n, r, s). This gives rise to the question of how small p must be for asymptotic stability to cease. The main result of this paper is the answer to this question.

     

Visibility graphs of fractional Wu–Baleanu time series

ABSTRACT

We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos.     

Words and Almost Nilpotent Varieties of Groups #

For a word of a free group of rank n , the author obtains an invariant called its standard exponent, and shows that if any residually finite group satisfying the law defined by such a word is almost nilpotent, then the standard exponent of the word equals 1 .

Conversely, if the standard exponent of a word ω is 1 , then any residually finite or soluble group and any locally finite or soluble group satisfying the group law ω≡ 1 is nilpotent-of-bounded-class-by-bounded-exponent.     

The Arcsine Law

Let N _n denote the number of positive sums in the first n trials in a random walk (S _i) and let L _n denote the first time we obtain the maximum in S ₀,..., S _n. Then the classical equivalence principle states that N _n and L _n have the same distribution and the classical arcsine law gives necessary and sufficient condition for (1/n) L _n or (1/n) N _n to converge in law to the arcsine distribution. The objective of this note is to provide a simple and elementary proof of the arcsine law for a general class of integer valued random variables (T _n) and to provide a simple an elementary proof of the equivalence principle for a general class of integer valued random vectors (N _n, L _n).     

Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface Sg of genus g grows asymptotically like

c(g)n5(g−1)/2−1γnn!     

Contact processes on scale-free networks

We investigate the contact process on random graphs generated from the configuration model for scale-free complex networks with the power law exponent β E （2, 3]. Using the neighborhood expansion method, we show that, with positive probability, any disease with an infection rate λ 〉 0 can survive for exponential time in the number of vertices of the graph. This strongly supports the view that stochastic scale-free networks are remarkably different from traditional regular graphs, such as, Z^d and classical Erdos-Renyi random graphs.     

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.     

Random walks on highly symmetric graphs

We consider uniform random walks on finite graphs withn nodes. When the hitting times are symmetric, the expected covering time is at least 1/2n logn-O(n log logn) uniformly over all such graphs. We also obtain bounds for the covering times in terms of the eigenvalues of the transition matrix of the Markov chain. For distance-regular graphs, a general lower bound of (n-1) logn is obtained. For hypercubes and binomial coefficient graphs, the limit law of the covering time is obtained as well.     

Small Drawings of Outerplanar Graphs,Series-Parallel Graphs,and Other Planar Graphs

In this paper, we study small planar drawings of planar graphs. For arbitrary planar graphs, Θ(n ²) is the established upper and lower bound on the worst-case area. A long-standing open problem is to determine for what graphs a smaller area can be achieved. We show here that series-parallel graphs can be drawn in O(n ^3/2) area, and outerplanar graphs can be drawn in O(nlog n) area, but 2-outerplanar graphs and planar graphs of proper pathwidth 3 require Ω(n ²) area. Our drawings are visibility representations, which can be converted to polyline drawings of asymptotically the same area.     

Generating Random Regular Graphs

Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [10] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d = O(n^1/3−ε), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd²). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author. The algorithm works for relatively large d in practical (quadratic) time and can be used to derive many properties of uniform random regular graphs. * Research supported in part by grant RB091G-VU from UCSD, by NSF grant DMS-0200357 and by an A. Sloan fellowship.     

Enumeration and limit laws for series–parallel graphs

We show that the number g_n of labelled series–parallel graphs on n vertices is asymptotically g_ngn^−5/2γⁿn!, where γ and g are explicit computable constants. We show that the number of edges in random series–parallel graphs is asymptotically normal with linear mean and variance, and that it is sharply concentrated around its expected value. Similar results are proved for labelled outerplanar graphs and for graphs not containing K_2,3 as a minor.     

Enumeration of Full Graphs: Onset of the Asymptotic Region

A full graph on n vertices, as defined by Fulkerson, is a representation of the intersection and containment relations among a system of n sets. It has an undirected edge between vertices representing intersecting sets and a directed edge from a to b if the corresponding set A contains B;. Kleitman, Lasaga, and Cowen gave a unified argument to show that asymptotically, almost all full graphs can be obtained by taking an arbitrary undirected graph on the n vertices, distinguishing a clique in this graph, which need not be maximal, and then adding directed edges going out from each vertex in the clique to all vertices to which there is not already an existing undirected edge. Call graphs of this type members of the dominant class. This article obtains the first upper and lower bounds on how large n has to be, so that the asymptotic behavior is indeed observed. It is shown that when n > 170, the dominant class predominates, while when n < 17, the full graphs in the dominant class compose less than half of the total number of realizable full graphs on n vertices.     

Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs in the so‐called rank‐1 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w_i, where w_i denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W. In this case, the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power‐law case, i.e., the case where \begin{align*}{\mathbb{P}}(W\geq k)\end{align*} is proportional to k^‐(τ‐1) for some power‐law exponent τ > 3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when \begin{align*}{\mathbb{P}}(W > k) \leq ck^{-(\tau-1)}\end{align*} for all k ≥ 1 and some τ > 4 and c > 0, the largest critical connected component in a graph of size n is of order n^2/3, as it is for the critical Erd?s‐Rényi random graph. When, instead, \begin{align*}{\mathbb{P}}(W > k)=ck^{-(\tau-1)}(1+o(1))\end{align*} for k large and some τ∈(3,4) and c > 0, the largest critical connected component is of the much smaller order n^{(τ‐2)/(τ‐1)}. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 480–508, 2013     

On the number of paths and cycles for almost all graphs and digraphs

In this paper it is deduced the number ofs-paths (s-cycles) havingk edges in common with a fixeds-path (s-cycle) of the complete graphK _n (orK* _n for directed graphs). It is also proved that the number of the common edges of twos-path (s-cycles) randomly chosen from the set ofs-paths (s-cycles) ofK _n (respectivelyK* _n ), are random variables, distributed asymptotically in accordance with the Poisson law whenever s/n exists, thus extending a result by Baróti. Some estimations of the numbers of paths and cycles for almost all graphs and digraphs are made by applying Chebyshev’s inequality.