Showing papers in "Random Structures and Algorithms in 2011"
TL;DR: In this paper, a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors is described.
Abstract: We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this survey we discuss some of them. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
103 citations
TL;DR: Bollobas, Janson and Riordan as mentioned in this paper introduced a family of random graph models producing inhomogeneous graphs with n vertices and Θ(n) edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function κ: [ 0, 1]2 → [0, ∞).
Abstract: Recently, Bollobas, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with n vertices and Θ(n) edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function κ: [0, 1]2 → [0, ∞). To understand these models, we should like to know when different kernels κ give rise to “similar” graphs, and, given a real-world network, how “similar” is it to a typical graph G(n, κ) derived from a given kernel κ.
The analogous questions for dense graphs, with Θ(n2) edges, are answered by recent results of Borgs, Chayes, Lovasz, Sos, Szegedy and Vesztergombi, who showed that several natural metrics on graphs are equivalent, and moreover that any sequence of graphs converges in each metric to a graphon, i.e., a kernel taking values in [0, 1].
Possible generalizations of these results to graphs with o(n2) but ω(n) edges are discussed in a companion article [Bollobas and Riordan, London Math Soc Lecture Note Series 365 (2009), 211–287]; here we focus only on graphs with Θ(n) edges, which turn out to be much harder to handle. Many new phenomena occur, and there are a host of plausible metrics to consider; many of these metrics suggest new random graph models and vice versa. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 1-38, 2011 © 2011 Wiley Periodicals, Inc.
84 citations
TL;DR: It is proved that for fixed integer D and positive reals α and γ, there exists a constant C0 such that for all p satisfying p(n) ≥ C0/n, the random graph G(n,p) asymptotically almost surely contains a copy of every tree with maximum degree at most D and at most (1 - α)n vertices.
Abstract: We prove that for fixed integer D and positive reals α and γ, there exists a constant C0 such that for all p satisfying p(n) ≥ C0/n, the random graph G(n,p) asymptotically almost surely contains a copy of every tree with maximum degree at most D and at most (1 - α)n vertices, even after we delete a (1/2 - γ)-fraction of the edges incident to each vertex. The proof uses Szemeredi's regularity lemma for sparse graphs and a bipartite variant of the theorem of Friedman and Pippenger on embedding bounded degree trees into expanding graphs. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
82 citations
TL;DR: It is proved that for densities well below the satisfiability threshold, the solution space of random k‐SAT decomposes into an exponential number of connected components and give quantitative bounds for the diameter, volume, and number.
Abstract: For various random constraint satisfaction problems there is a significant gap between the largest constraint density for which solutions exist and the largest density for which any polynomial time algorithm is known to find solutions. Examples of this phenomenon include random k-SAT, random graph coloring, and a number of other random constraint satisfaction problems. To understand this gap, we study the structure of the solution space of random k-SAT (i.e., the set of all satisfying assignments viewed as a subgraph of the Hamming cube). We prove that for densities well below the satisfiability threshold, the solution space decomposes into an exponential number of connected components and give quantitative bounds for the diameter, volume, and number.© 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 251–268, 2011 © 2011 Wiley Periodicals, Inc.
77 citations
TL;DR: The aim of this article is to present an extension of this model of sparse random graphs in which the edges are far from independent, and to prove several results about this extension.
Abstract: In 2007, we introduced a general model of sparse random graphs with (conditional) independence between the edges. The aim of this article is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with (conditionally) independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non-Poisson) multi-type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with power-law degree sequences with a wide range of degree exponents and clustering coefficients. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 269–323, 2011 © 2011 Wiley Periodicals, Inc.
73 citations
TL;DR: The giant component of the Erdős-Renyi random graph is described and the asymptotics of the diameter of the giant component for any rate of decay of e are inferred to determine fundamental characteristics of the supercritical random graph.
Abstract: We provide a complete description of the giant component of the Erdős-Renyi random graph \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{G}}(n,p)\end{align*} \end{document} **image** as soon as it emerges from the scaling window, i.e., for p = (1+e)/n where e3n →∞ and e = o(1). Our description is particularly simple for e = o(n-1/4), where the giant component \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{C}_1}\end{align*} \end{document} **image** is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{C}_1}\end{align*} \end{document} **image** ). Let Z be normal with mean \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\frac{2}{3} \varepsilon^3 n\end{align*} \end{document} **image** and variance e3n, and let \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{K}\end{align*} \end{document} **image** be a random 3-regular graph on \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}2\left\lfloor Z\right\rfloor\end{align*} \end{document} **image** vertices. Replace each edge of \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{K}\end{align*} \end{document} **image** by a path, where the path lengths are i.i.d. geometric with mean 1/e. Finally, attach an independent Poisson( 1-e )-Galton-Watson tree to each vertex. A similar picture is obtained for larger e = o(1), in which case the random 3-regular graph is replaced by a random graph with Nk vertices of degree k for k ≥ 3, where Nk has mean and variance of order ekn. This description enables us to determine fundamental characteristics of the supercritical random graph. Namely, we can infer the asymptotics of the diameter of the giant component for any rate of decay of e, as well as the mixing time of the random walk on \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{C}_1}\end{align*} \end{document} **image** . © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 139–178, 2011 © 2011 Wiley Periodicals, Inc.
70 citations
TL;DR: A variant of a Johnson‐Lindenstrauss lemma for matrices with circulant structure is proved, which allows to minimize the randomness used, is easy to implement and provides good running times.
Abstract: We prove a variant of a Johnson-Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimize the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space k = O(e−2 log3 n) instead of the classical bound k = O(e−2 log n). © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011
(Supported by DFG Heisenberg (HI 584/3-2); FWF START-Preis Sparse Approximation and Optimization in High Dimensions (Y 432-N15).)
62 citations
TL;DR: If the perturbation (noise) is random and the matrix has low rank, then better estimates can be obtained, and this method relies on high dimensional geometry and is different from those used an earlier papers.
Abstract: Computing the first few singular vectors of a large matrix is a problem that frequently comes up in statistics and numerical analysis. Given the presence of noise, an exact calculation is hard to achieve, and the following problem is of importance:
How much does a small perturbation to the matrix change the singular vectors?
Answering this question, classical theorems, such as those of Davis-Kahan and Wedin, give tight estimates for the worst-case scenario. In this paper, we show that if the perturbation (noise) is random and our matrix has low rank, then better estimates can be obtained. Our method relies on high dimensional geometry and is different from those used in earlier papers. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
55 citations
TL;DR: The main theorem asserts that there is a constant c such that for almost every graph G, each eigenfunction of G has at most two large nodal domains, and in addition at most c exceptional vertices outside these primary domains.
Abstract: We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our main focus in this article is on the nodal domains associated with the different eigenfunctions. In the analogous realm of Laplacians of Riemannian manifolds, nodal domains have been the subject of intensive research for well over a hundred years. Graphical nodal domains turn out to have interesting and unexpected properties. Our main theorem asserts that there is a constant c such that for almost every graph G, each eigenfunction of G has at most two large nodal domains, and in addition at most c exceptional vertices outside these primary domains. We also discuss variations of these questions and briefly report on some numerical experiments which, in particular, suggest that almost surely there are just two nodal domains and no exceptional vertices. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 39–58, 2011
46 citations
TL;DR: In this article, the authors studied the asymptotic growth of the diameter of a graph obtained by adding sparse "long" edges to a square box and showed that the graph diameter for the graph reduced to a box of side L scales like (log L)Δ+o(1) where Δ−1 := log2(2d/s).
Abstract: We study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} ${\mathbb Z}^d$ \end{document} . We focus on the cases when an edge between x and y is added with probability decaying with the Euclidean distance as |x − y|−s+o(1) when |x − y| → ∞. For s ∈ (d, 2d) we show that the graph diameter for the graph reduced to a box of side L scales like (log L)Δ+o(1) where Δ−1 := log2(2d/s). In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance L. We also show that a ball of radius r in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius exp{r1/Δ+o(1)} in the Euclidean metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 210-227, 2011
(Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.)
44 citations
TL;DR: In this article, auteur traite de la signification et de l'usage du concept d'ethos, which reste un concept heuristique for saisir and interpreter les recurrences de comportements qui peuvent s’y observer.
Abstract: Dans cet article, l’auteur traite de la signification et de l’usage du concept d’ethos. Dans la sociologie classique, ce dernier a ete utilise pour comprendre et qualifier la rationalite socialement et ethiquement encastree des conduites sociales. Exprimant l’interiorisation d’un principe organisateur de pratiques, dessinant une matrice globale des comportements, il est porteur d’une vision qui parait datee a l’heure du “brouillage des classes sociales”, de la “modernite liquide” et de “l’homme pluriel”. Neanmoins, si l’on reconnait l’existence d’un univers social non entierement liquefie, c’est-a-dire ou continuent a exister des milieux sociaux sedimentes qui impregnent avec plus ou moins de profondeur le systeme de personnalite de celui qui s’y insere durablement, l’ethos reste un concept heuristique pour saisir et interpreter les recurrences de comportements qui peuvent s’y observer. Permettant de comprendre ce que les espaces d’insertion “font aux individus” ainsi que la facon dont les principes organisateurs de pratiques entrent en transaction (en consonance, en dissonance ou en concurrence), il garde toute sa pertinence dans la boite a outils theoriques du sociologue et de l’anthropologue d’aujourd’hui.
TL;DR: In this paper, it was shown that hypergraphs with sufficient regularity and no atypical vertices behave as if they were complete for the purpose of embedding bounded degree hypergraph.
Abstract: We obtain a hypergraph generalisation of the graph blow-up lemma proved by Komlos, Sarkozy and Szemeredi, showing that hypergraphs with sufficient regularity and no atypical vertices behave as if they were complete for the purpose of embedding bounded degree hypergraphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 39, 275–376, 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: A general framework is described and algorithms where this kind of problems can be fitted and solved are described and provided, which allow for effective simulation of desired events of probability s.
Abstract: Let s∈(0,1) be uniquely determined but only its approximations can be obtained with a finite computational effort. Assume one aims to simulate an event of probability s. Such settings are often encountered in statistical simulations. We consider two specific examples. First, the exact simulation of non-linear diffusions ([3]). Second, the celebrated Bernoulli factory problem ([10, 13]) of generating an f(p)-coin given a sequence X1,X2,… of independent tosses of a p-coin (with known f and unknown p). We describe a general framework and provide algorithms where this kind of problems can be fitted and solved. The algorithms are straightforward to implement and thus allow for effective simulation of desired events of probability s. Our methodology links the simulation problem to existence and construction of unbiased estimators. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 441–452, 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: Using the Lefthanded Local Lemma the authors prove that πch(Pn) ≤ 4 for every n, inspired by a striking result of Thue asserting that the path Pn on n vertices has a nonrepetitive three-colouring, no matter how large n is.
Abstract: A vertex colouring c of a graph G is called nonrepetitive if for every integer r ≥ 1 and every path P = (v1,v2,…,v2r) in G, the first half of P is coloured differently from the second half of P, that is, c(vj)≠c(vr+j) for some j = 1,2,…,r. This notion was inspired by a striking result of Thue asserting that the path Pn on n vertices has a nonrepetitive three-colouring, no matter how large n is. A k-list assignment of a graph G is a mapping L which assigns a set L(v) of k permissible colours to each vertex v of G. The Thue choice number of G, denoted by πch(G), is the least integer k such that for every k-list assignment L there is a nonrepetitive colouring c of G satisfying c(v) ∈ L(v) for every vertex v of G. Using the Lefthanded Local Lemma we prove that πch(Pn) ≤ 4 for every n. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: In this paper, it was shown that a uniform, rooted unordered binary tree (also known as rooted, binary polya tree) with n leaves has the Brownian continuum random tree as its scaling limit for the Gromov-Hausdorff topology.
Abstract: We prove that a uniform, rooted unordered binary tree (also known as rooted, binary Polya tree) with n leaves has the Brownian continuum random tree as its scaling limit for the Gromov-Hausdorff topology. The limit is thus, up to a constant factor, the same as that of uniform plane trees or labeled trees. Our analysis rests on a combinatorial and probabilistic study of appropriate trimming procedures of trees. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 467–501, 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: In this article, game-theoretic versions of several classical results on nonrepetitive sequences were proved using an extension of the Lovasz Local Lemma, which can dramatically reduce the number of edges needed in a dependency graph when there is an ordering underlying the significant dependencies of events.
Abstract: We prove game-theoretic versions of several classical results on nonrepetitive sequences, showing the existence of winning strategies using an extension of the Lovasz Local Lemma which can dramatically reduce the number of edges needed in a dependency graph when there is an ordering underlying the significant dependencies of events. This appears to represent the first successful application of a Local Lemma to games. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: It is shown without using these lemmas that n0:= 2 × 108 is sufficient to show that graphs of order n ≥ n0, where n0 is a very large constant.
Abstract: In 1962 Posa conjectured that every graph G on n vertices with minimum degree \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb,amsfonts} \pagestyle{empty} \begin{document} \begin{align*}\delta(G)\ge \frac{2}{3}n\end{align*} \end{document} **image** contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of Posa's Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb,amsfonts} \pagestyle{empty} \begin{document} \begin{align*}\frac{2}{3}n\end{align*} \end{document} **image** . Still in 1996, Komlos, Sarkozy, and Szemeredi proved Posa's Conjecture, using the Regularity and Blow-up Lemmas, for graphs of order n ≥ n0, where n0 is a very large constant. Here we show without using these lemmas that n0:= 2 × 108 is sufficient. We are motivated by the recent work of Levitt, Sarkozy and Szemeredi, but our methods are based on techniques that were available in the 90's. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: The cover time of random geometric graph G = G(d, r, n) in d dimensions is studied, which is defined as follows: sample n points V independently and uniformly at random from I(d).
Abstract: We study the cover time of random geometric graphs. Let $I(d)=[0,1]^{d}$ **image** denote the unit torus in d dimensions. Let $D(x,r)$ **image** denote the ball (disc) of radius r. Let $\Upsilon_d$ **image** be the volume of the unit ball $D(0,1)$ **image** in d dimensions. A random geometric graph $G=G(d,r,n)$ **image** in d dimensions is defined as follows: Sample n points V independently and uniformly at random from $I(d)$ **image** . For each point x draw a ball $D(x,r)$ **image** of radius r about x. The vertex set $V(G)=V$ **image** and the edge set $E(G)=\{\{v,w\}: w
e v,\,w\in D(v,r)\}$ **image** . Let $G(d,r,n),\,d\geq 3$ **image** be a random geometric graph. Let $C_G$ **image** denote the cover time of a simple random walk on G. Let $c>1$ **image** be constant, and let $r=(c\log n/(\Upsilon_dn))^{1/d}$ **image** . Then whp the cover time satisfies $$C_G\sim c\log \left({{c}\over{c-1}}\right)n\log n.$$ © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 324–349, 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: This article studies the basic properties of stochastic Kronecker products based on an initiator matrix of size two and proves that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm.
Abstract: A random graph model based on Kronecker products of probability matrices has been recently proposed as a generative model for large-scale real-world networks such as the web. This model simultaneously captures several well-known properties of real-world networks; in particular, it gives rise to a heavy-tailed degree distribution, has a low diameter, and obeys the densification power law. Most properties of Kronecker products of graphs (such as connectivity and diameter) are only rigorously analyzed in the deterministic case. In this article, we study the basic properties of stochastic Kronecker products based on an initiator matrix of size two (which is the case that is shown to provide the best fit to many real-world networks). We will show a phase transition for the emergence of the giant component and another phase transition for connectivity, and prove that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 453–466, 2011
(Work performed in part while visiting Yahoo! Research.)
TL;DR: In this paper, the authors show equivalence of sharp threshold functions of a random intersection graph with m ≥ n3 and a graph G(n,p) with independent edges under some additional assumptions.
Abstract: We solve the conjecture of Fill, Scheinerman and Singer-Cohen (Random Struct Algorithms 16 (2000), 156–176) and show equivalence of sharp threshold functions of a random intersection graph ${\cal g}$ **image** (n,m,p) with m ≥ n3 and a graph G(n,p) with independent edges. Moreover we prove sharper equivalence results under some additional assumptions. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: The concept of H-free to induced subgraph containment is strengthened, proving that if H has coloring number k + 1 then almost all H- free graphs can be covered by k graphs that are cliques or independent sets if and only if H is in some well-defined sense critical.
Abstract: Determining the cardinality and describing the structure of H-free graphs is well-investigated for many graphs H. In the nineties, Promel and Steger proved that for a graph H with chromatic number k + 1 almost all graphs not containing H as a subgraph are k-colorable if and only if H contains a color-critical edge. We strengthen the concept of H-free to induced subgraph containment, proving that if H has coloring number k + 1 then almost all H-free graphs can be covered by k graphs that are cliques or independent sets if and only if H is in some well-defined sense critical. The family of critical graphs includes C4 and C2k+1 for all k ≥ 3. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: In this article, the authors consider a variant of the problem with a Ramsey flavor, where each random edge that arrives in a sequence of rounds must be colored with one of r colors, and the goal can be either to create a giant component in every color class, or alternatively to avoid it in every colour.
Abstract: The classical result in the theory of random graphs, proved by Erdős and Renyi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in a sequence of rounds must be colored with one of r colors. The goal can be either to create a giant component in every color class, or alternatively, to avoid it in every color. One can analyze the offline or online setting for this problem. In this paper, we consider all these variants and provide nontrivial upper and lower bounds; in certain cases (like online avoidance) the obtained bounds are asymptotically tight. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: Using an associated branching process as the basis of the approximation, it is shown that typical inter‐point distances in a multi‐type random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions.
Abstract: Using an associated branching process as the basis of our approximation, we show that typical inter-point distances in a multi-type random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 179–209, 2011
(Supported by Schweizer Nationalfonds Projekt (No. 20-117625/1); Institut Mittag–Leffler, Djursholm, Sweden; EPSRC and BBSRC through OCISB.)
TL;DR: In this article, the authors give an asymptotically sharp estimate for 𝔼(XH(i)), for every \documentclass{article} \usepackage{mathrsfs} \package{amsmath} \pagestyle{empty}
Abstract: Consider the triangle-free process, which is defined as follows. Start with G(0), an empty graph on n vertices. Given G(i - 1), let G(i) = G(i - 1) ∪{g(i)}, where g(i) is an edge that is chosen uniformly at random from the set of edges that are not in G(i − 1) and can be added to G(i - 1) without creating a triangle. The process ends once a maximal triangle-free graph has been created. Let H be a fixed triangle-free graph and let XH(i) count the number of copies of H in G(i). We give an asymptotically sharp estimate for 𝔼(XH(i)), for every \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}1 \ll i \le 2^{-5} n^{3/2} \sqrt{\ln n}\end{align*} \end{document} **image** , at the limit as n →∞. Moreover, we provide conditions that guarantee that a.a.s. XH(i) = 0, and that XH(i) is concentrated around its mean.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: In this article, the authors consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or equivalently, the path length of a given length.
Abstract: We consider a conditioned Galton–Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length.
We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis.
Moreover, the latter proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet–Melou and Janson (Bousquet-Melou and Janson, Ann Appl Probab 16 (2006) 1597–1632), saying that the vertical profile of a randomly labelled conditioned Galton–Watson tree converges in distribution, after suitable normalization, to the density of ISE (Integrated Superbrownian Excursion). © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 381–395, 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: It is proved that for every positive integer r, there is a function dr such that no r-uniform simple hypergraph with average degree at least dr(s) is s-list-colorable.
Abstract: A hypergraph is simple if it has no two edges sharing more than a single vertex. It is s-list colorable (or s-choosable) if for any assignment of a list of s colors to each of its vertices, there is a vertex coloring assigning to each vertex a color from its list, so that no edge is monochromatic. We prove that for every positive integer r, there is a function dr(s) such that no r-uniform simple hypergraph with average degree at least dr(s) is s-list-colorable. This extends a similar result for graphs, due to the first author, but does not give as good estimates of dr(s) as are known for d2(s), since our proof only shows that for each fixed r ≥ 2, dr(s) ≤ 2**math-image** We use the result to prove that for any finite set of points X in the plane, and for any finite integer s, one can assign a list of s distinct colors to each point of the plane so that any coloring of the plane that colors each point by a color from its list contains a monochromatic isometric copy of X. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011
TL;DR: It is shown in this paper that large complete graphs have this property: the probability that the origin is blocked at distance n but not closer decreases only polynomially fast and not, as usual, exponentially.
Abstract: Two infinite walks on the same finite graph are called compatible if it is possible to introduce delays into them in such a way that they never collide. Years ago, Peter Winkler asked the question: for which graphs are two independent random walks compatible with positive probability. Up to now, no such graphs were found. We show in this paper that large complete graphs have this property. The question is equivalent to a certain dependent percolation with a power-law behavior: the probability that the origin is blocked at distance n but not closer decreases only polynomially fast and not, as usual, exponentially. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: In this paper, the authors studied the minimal number τ(S,G) of translates of an arbitrary subset S of a group G needed to cover the group, and related notions of the efficiency of such coverings.
Abstract: In this paper we study the minimal number τ(S,G) of translates of an arbitrary subset S of a group G needed to cover the group, and related notions of the efficiency of such coverings. We focus mainly on finite subsets in discrete groups, reviewing the classical results in this area, and generalizing them to a much broader context. For example, the worst-case efficiency when S has k elements is of order 1/log k. We show that if n(k) grows at a suitable rate with k, then almost every k-subset of any given group with order n comes close to this worst-case bound. In contrast, if n(k) grows very rapidly, or if k is fixed and n →∞, then almost every k-subset of the cyclic group with order n comes close to the optimal efficiency. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: A scale-free random graph process in which the number of edges added at each step increases, which differs from the standard model in which a fixed number, m, of edges are added atEach step, is studied.
Abstract: We study a scale-free random graph process in which the number of edges added at each step increases. This differs from the standard model in which a fixed number, m, of edges are added at each step.
Let f(t) be the number of edges added at step t. In the standard scale-free model, f(t) = m constant, whereas in this paper we consider f(t) = [tc],c > 0. Such a graph process, in which the number of edges grows non-linearly with the number of vertices is said to have accelerating growth.
We analyze both an undirected and a directed process. The power law of the degree sequence of these processes exhibits widely differing behavior.
For the undirected process, the terminal vertex of each edge is chosen by preferential attachment based on vertex degree. When f(t) = m constant, this is the standard scale-free model, and the power law of the degree sequence is 3. When f(t) = [tc],c < 1, the degree sequence of the process exhibits a power law with parameter x = (3 − c)/(1 − c). As c → 0, x → 3, which gives a value of x = 3, as in standard scale-free model. Thus no more slowly growing monotone function f(t) alters the power law of this model away from x = 3. When c = 1, so that f(t) = t, the expected degree of all vertices is t, the vertex degree is concentrated, and the degree sequence does not have a power law.
For the directed process, the terminal vertex is chosen proportional to in-degree plus an additive constant, to allow the selection of vertices of in-degree zero. For this process when f(t) = m is constant, the power law of the degree sequence is x = 2 + 1/m. When f(t) = [tc], c > 0, the power law becomes x = 1 + 1/(1 + c), which naturally extends the power law to [1,2]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 396–421, 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: In this article, the methode des entretiens biographiques, a typologie interpretative, permet d'apprehender l'hete-rogeneite des experiences etudiantes ainsi que leur evolution dans le temps.
Abstract: Dans cet article, nous montrons en quoi la methode des entretiens biographiques, lorsqu’elle est agencee au concept theorique de carriere et a l’instrument d’analyse qu’est la typologie interpretative, permet d’apprehender l’heterogeneite des experiences etudiantes ainsi que leur evolution dans le temps. Nous exposons dans un premier temps la demarche de construction d’une typologie a six types de carrieres, lesquels permettent de comprendre l’heterogeneite des rapports au temps et aux etudes des etudiants. Dans un second temps, nous presentons en quoi la typologie n’est pas une fin en soi et qu’une fois construite, elle constitue un puissant instrument d’analyse des recits afin de saisir l’heterogeneite des logiques, cette fois non pas entre les etudiants, mais au fil du temps, au sein d’une meme biographie. L’article montre que la notion de carriere et la typologie sont des instruments permettant de contourner l’impression de fixation et de lissage du vecu induite a la fois par la reconstitution a posteriori de sa vie par l’enquete en situation d’entretien et par l’effort d’objectivation des carrieres par le chercheur. La demonstration s’appuie sur deux recits biographiques recoltes au cours d’une recherche portant sur les mobilites pour etudes d’etudiants universitaires francophones originaires d’un milieu minoritaire au Canada.