On the rate of convergence of a regular martingale related to a branching random walk

Let M _n , n = 1, 2, ..., be a supercritical branching random walk in which the number of direct descendants of an individual may be infinite with positive probability. Assume that the standard martingale W _n related to M _n is regular and W is a limit random variable. Let a(x) be a nonnegative function regularly varying at infinity with index greater than −1. We present sufficient conditions for the almost-sure convergence of the series . We also establish criteria for the finiteness of EW ln⁺ Wa(ln⁺ W) and E ln⁺|Z _∞|a(ln⁺|Z _∞|), where and (M _n , Q _n ) are independent identically distributed random vectors not necessarily related to M _n . __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 326–342, March, 2006.     

The Spectrum of a Random Geometric Graph is Concentrated

Consider n points, x ₁,... , x _n , distributed uniformly in [0, 1] ^d . Form a graph by connecting two points x _i and x _j if . This gives a random geometric graph, , which is connected for appropriate r(n). We show that the spectral measure of the transition matrix of the simple random walk on is concentrated, and in fact converges to that of the graph on the deterministic grid.      

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 phase transition in random graphs: A simple proof

The classical result of Erd?s and Rényi asserts that the random graph G(n,p) experiences sharp phase transition around \begin{align*}p=\frac{1}{n}\end{align*} – for any ε > 0 and \begin{align*}p=\frac{1-\epsilon}{n}\end{align*}, all connected components of G(n,p) are typically of size O_ε(log n), while for \begin{align*}p=\frac{1+\epsilon}{n}\end{align*}, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime \begin{align*}p=\frac{1+\epsilon}{n}\end{align*}, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Dirac’s Type Sufficient Conditions for Hamiltonicity and Pancyclicity

Let G = (V, E) be a any simple, undirected graph on n ≥ 3 vertices with the degree sequence . We consider the class of graphs satisfying the condition where , is a positive integer. It is known that is hamiltonian if θ ≤ δ. In this paper,

Asymptotic enumeration by degree sequence of graphs with degreeso(n 1/2)

We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree iso(|E(G)|^1/3). The previously best enumeration, by the first author, required maximum degreeo(|E(G)|^1/4). In particular, ifk=o(n ^1/2), the number of regular graphs of degreek and ordern is asymptotically $$\frac{{(nk)!}}{{(nk/2)!2^{nk/2} (k!)^n }}\exp \left( { - \frac{{k^2 - 1}}{4} - \frac{{k^3 }}{{12n}} + 0\left( {k^2 /n} \right)} \right).$$ Under slightly stronger conditions, we also determine the asymptotic number of unlabelled graphs with a given degree sequence. The method used is a switching argument recently used by us to uniformly generate random graphs with given degree sequences.     

On Interpolation Sequences of One Class of Functions Analytic in the Unit Disk

We establish a criterion for the existence of a solution of the interpolation problem f( _n ) = b _n in the class of functions f analytic in the unit disk and satisfying the relation

Dirac's theorem for random graphs

A classical theorem of Dirac from 1952 asserts that every graph on n vertices with minimum degree at least \begin{align*}\left\lceil n/2 \right\rceil\end{align*} is Hamiltonian. In this paper we extend this result to random graphs. Motivated by the study of resilience of random graph properties we prove that if p ? log n/n, then a.a.s. every subgraph of G(n,p) with minimum degree at least (1/2 + o (1) )np is Hamiltonian. Our result improves on previously known bounds, and answers an open problem of Sudakov and Vu. Both, the range of edge probability p and the value of the constant 1/2 are asymptotically best possible. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

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     

Mean-Field Conditions for Percolation on Finite Graphs

Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if

lim sup  n n1/3 ?t = 1n1/3 tpt(v,v) < ￥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}      12. The order of the giant component of random hypergraphs      Michael Behrisch  Amin Coja‐Oghlan  Mihyun Kang 《Random Structures and Algorithms》2010,36(2):149-184 We establish central and local limit theorems for the number of vertices in the largest component of a random d‐uniform hypergraph Hd(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      13. Critical behavior in inhomogeneous random graphs      Remco van der Hofstad 《Random Structures and Algorithms》2013,42(4):480-508 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 wi, where wi 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 n2/3, 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      14. Improved random graph isomorphism      Tomek Czajka  Gopal Pandurangan   《Journal of Discrete Algorithms》2008,6(1):85-92 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)].      15. Distribution of the number of spanning regular subgraphs in random graphs      Pu Gao 《Random Structures and Algorithms》2013,43(3):338-353 In this paper, we examine the moments of the number of d ‐factors in \begin{align*}\mathcal{ G}(n,p)\end{align*} for all p and d satisfying d3 = o(p2n). We also determine the limiting distribution of the number of d ‐factors inside this range with further restriction that \begin{align*}(1-p)\sqrt{dn}\to\infty\end{align*} as n →∞.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013      16. Corrigendum: The cover time of the giant component of a random graph,Random Structures and Algorithms 32 (2008), 401–439      Colin Cooper  Alan Frieze 《Random Structures and Algorithms》2009,34(2):300-304 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      17. A law of the iterated logarithm for random walk in random scenery with deterministic normalizers      Thomas M. Lewis 《Journal of Theoretical Probability》1993,6(2):209-230 LetX, X i ,i≥1, be a sequence of independent and identically distributed ? d -valued random vectors. LetS o=0 and \(S_n = \sum\nolimits_{i = 1}^n {X_i } \) forn≤1. Furthermore letY, Y(α), α∈? d , be independent and identically distributed ?-valued random variables, which are independent of theX i . Let \(Z_n = \sum\nolimits_{i = 0}^n {Y(S_i )} \) . We will call (Z n ) arandom walk in random scenery. In this paper, we consider the law of the iterated logarithm for random walk in random scenery where deterministic normalizers are utilized. For example, we show that if (S n ) is simple, symmetric random walk in the plane,E[Y]=0 andE[Y 2]=1, then $$\mathop {\overline {\lim } }\limits_{n \to \infty } \frac{{Z_n }}{{\sqrt {2n\log (n)\log (\log (n))} }} = \sqrt {\frac{2}{\pi }} a.s.$$       18. An estimate of the speed of convergence in the multidmensional central limit theorem without moment hypotheses      L. V. Rozovskii 《Mathematical Notes》1978,23(4):343-351 Letx 1, ...,x n , (n ? 1) be independent random vectors in Rd and b be a vector in Rd. For an arbitrary Borel set A in Rd we set $$\begin{gathered} P_n \left( A \right) = P\left\{ {X_1 + ... + X_n - b \in A} \right\}, \hfill \\ \Delta _n \left( A \right) = \left| {P_n \left( A \right) - \Phi \left( A \right)} \right|, \hfill \\ \end{gathered} $$ where Φ(A) is the probability function of a standard normal vector in Rd. In this note are obtained estimates for Δn(A), where A belongs to the class of convex Borel sets in Rd.      19. On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series      Matthias Beck  Alan Stapledon 《Mathematische Zeitschrift》2010,264(1):195-207 For every positive integer n, consider the linear operator U n on polynomials of degree at most d with integer coefficients defined as follows: if we write ${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}For every positive integer n, consider the linear operator U n on polynomials of degree at most d with integer coefficients defined as follows: if we write \frach(t)(1 - t)d + 1=?m 3 0 g(m)  tm{\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}} , for some polynomial g(m) with rational coefficients, then \fracUnh(t)(1- t)d+1 = ?m 3 0g(nm)  tm{\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}} . We show that there exists a positive integer n d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n d , U n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–Macauley graded rings.      20. A note on Hamiltonicity of uniform random intersection graphs      Mindaugas Bloznelis  Irmantas Radavičius 《Lithuanian Mathematical Journal》2011,51(2):155-161 We consider a collection of n independent random subsets of [m] = {1, 2, . . . , m} that are uniformly distributed in the class of subsets of size d, and call any two subsets adjacent whenever they intersect. This adjacency relation defines a graph called the uniform random intersection graph and denoted by G n,m,d . We fix d = 2, 3, . . . and study when, as n,m → ∞, the graph G n,m,d contains a Hamilton cycle (the event denoted \( {G_{n,m,d}} \in \mathcal{H} \)). We show that \( {\mathbf{P}}\left( {{G_{n,m,d}} \in \mathcal{H}} \right) = o(1) \) for d 2 nm ?1 ? lnm ? 2 ln lnm → ?∞ and \( {\mathbf{P}}\left( {{G_{n,m,d}} \in \mathcal{H}} \right) = 1 - o(1) \) for 2nm ?1 ? lnm ? ln lnm → +∞.

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     

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     

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)].     

Distribution of the number of spanning regular subgraphs in random graphs

In this paper, we examine the moments of the number of d ‐factors in \begin{align*}\mathcal{ G}(n,p)\end{align*} for all p and d satisfying d³ = o(p²n). We also determine the limiting distribution of the number of d ‐factors inside this range with further restriction that \begin{align*}(1-p)\sqrt{dn}\to\infty\end{align*} as n →∞.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

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     

A law of the iterated logarithm for random walk in random scenery with deterministic normalizers

LetX, X _i ,i≥1, be a sequence of independent and identically distributed ? ^d -valued random vectors. LetS _o=0 and \(S_n = \sum\nolimits_{i = 1}^n {X_i } \) forn≤1. Furthermore letY, Y(α), α∈? ^d , be independent and identically distributed ?-valued random variables, which are independent of theX _i . Let \(Z_n = \sum\nolimits_{i = 0}^n {Y(S_i )} \) . We will call (Z _n ) arandom walk in random scenery. In this paper, we consider the law of the iterated logarithm for random walk in random scenery where deterministic normalizers are utilized. For example, we show that if (S _n ) is simple, symmetric random walk in the plane,E[Y]=0 andE[Y ²]=1, then $$\mathop {\overline {\lim } }\limits_{n \to \infty } \frac{{Z_n }}{{\sqrt {2n\log (n)\log (\log (n))} }} = \sqrt {\frac{2}{\pi }} a.s.$$      

An estimate of the speed of convergence in the multidmensional central limit theorem without moment hypotheses

Letx ₁, ...,x _n , (n ? 1) be independent random vectors in R_d and b be a vector in R_d. For an arbitrary Borel set A in Rd we set $$\begin{gathered} P_n \left( A \right) = P\left\{ {X_1 + ... + X_n - b \in A} \right\}, \hfill \\ \Delta _n \left( A \right) = \left| {P_n \left( A \right) - \Phi \left( A \right)} \right|, \hfill \\ \end{gathered} $$ where Φ(A) is the probability function of a standard normal vector in R_d. In this note are obtained estimates for Δ_n(A), where A belongs to the class of convex Borel sets in R_d.     

On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series

For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write ${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write \frach(t)(1 - t)^{d + 1}=?_{m 3 0} g(m)  t^m{\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}} , for some polynomial g(m) with rational coefficients, then \fracU_nh(t)(1- t)^d+1 = ?_{m 3 0}g(nm)  t^m{\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}} . We show that there exists a positive integer n _d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n _d , U _n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–Macauley graded rings.     

A note on Hamiltonicity of uniform random intersection graphs

We consider a collection of n independent random subsets of [m] = {1, 2, . . . , m} that are uniformly distributed in the class of subsets of size d, and call any two subsets adjacent whenever they intersect. This adjacency relation defines a graph called the uniform random intersection graph and denoted by G _n,m,d . We fix d = 2, 3, . . . and study when, as n,m → ∞, the graph G _n,m,d contains a Hamilton cycle (the event denoted \( {G_{n,m,d}} \in \mathcal{H} \)). We show that \( {\mathbf{P}}\left( {{G_{n,m,d}} \in \mathcal{H}} \right) = o(1) \) for d ² nm ^?1 ? lnm ? 2 ln lnm → ?∞ and \( {\mathbf{P}}\left( {{G_{n,m,d}} \in \mathcal{H}} \right) = 1 - o(1) \) for 2nm ^?1 ? lnm ? ln lnm → +∞.