Critical random graphs: limiting constructions and distributional properties

We consider the Erdos-Renyi random graph G(n,p) inside the critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous paper (arXiv:0903.4730) that considering the connected components of G(n,p) as a sequence of metric spaces with the graph distance rescaled by n^{-1/3} and letting n go to infinity yields a non-trivial sequence of limit metric spaces C = (C_1, C_2, ...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R_+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak, Pittel and Wierman by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component.Comment: 30 pages, 4 figure

Critical random graphs : limiting constructions and distributional properties

We consider the Erdos-Renyi random graph G(n, p) inside the critical window, where p = 1/n + lambda n(-4/3) for some lambda is an element of R. We proved in Addario-Berry et al. [2009+] that considering the connected components of G(n, p) as a sequence of metric spaces with the graph distance rescaled by n(-1/3) and letting n -> infinity yields a non-trivial sequence of limit metric spaces C = (C-1, C-2,...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak et al. [1994] by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component

Singular inextensible limit in the vibrations of post-buckled rods: Analytical derivation and role of boundary conditions

In-plane vibrations of an elastic rod clamped at both extremities are studied. The rod is modeled as an extensible planar Kirchhoff elastic rod under large displacements and rotations. Equilibrium configurations and vibrations around these configurations are computed analytically in the incipient post-buckling regime. Of particular interest is the variation of the first mode frequency as the load is increased through the buckling threshold. The loading type is found to have a crucial importance as the first mode frequency is shown to behave singularly in the zero thickness limit in the case of prescribed axial displacement, whereas a regular behavior is found in the case of prescribed axial load.This publication is based in part upon work supported by Award no. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST) (A.G.). A.G. is a Wolfson/Royal Society Merit Award holder. Support from the Royal Society, through the International Exchanges Scheme (Grant IE120203), is also acknowledge

Acyclic dominating partitions,

Abstract Given a graph G = (V, E), let P be a partition of V . We say that P is dominating if, for each part P of P, the set V \ P is a dominating set in G (equivalently, if every vertex has a neighbour of a different colour from its own). We say that P is acyclic if for any parts P, P of P, the bipartite subgraph G [P, P ] consisting of the edges between P and P in P contains no cycles. The acyclic dominating number ad(G) of G is the least number of parts in any partition of V that is both acyclic and dominating; and we shall denote by ad(d) the maximum over all graphs G of maximum degree at most d of ad(G). In this paper, we prove that a

Statistical Mechanics of Steiner trees

The Minimum Weight Steiner Tree (MST) is an important combinatorial optimization problem over networks that has applications in a wide range of fields. Here we discuss a general technique to translate the imposed global connectivity constrain into many local ones that can be analyzed with cavity equation techniques. This approach leads to a new optimization algorithm for MST and allows to analyze the statistical mechanics properties of MST on random graphs of various types

Exact asymptotics of the freezing transition of a logarithmically correlated random energy model

We consider a logarithmically correlated random energy model, namely a model for directed polymers on a Cayley tree, which was introduced by Derrida and Spohn. We prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the KPP-equation - thus translating Bramson's work on the KPP-equation into a discrete time case. We also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point

Slowdown for time inhomogeneous branching Brownian motion

We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but rather proportional to $T^{1/3}$. We conjecture that this is the worse case correction possible

Comparison of phylogenetic trees through alignment of embedded evolutionary distances

<p>Abstract</p> <p>Background</p> <p>The understanding of evolutionary relationships is a fundamental aspect of modern biology, with the phylogenetic tree being a primary tool for describing these associations. However, comparison of trees for the purpose of assessing similarity and the quantification of various biological processes remains a significant challenge.</p> <p>Results</p> <p>We describe a novel approach for the comparison of phylogenetic distance information based on the alignment of representative high-dimensional embeddings (xCEED: Comparison of Embedded Evolutionary Distances). The xCEED methodology, which utilizes multidimensional scaling and Procrustes-related superimposition approaches, provides the ability to measure the global similarity between trees as well as incongruities between them. We demonstrate the application of this approach to the prediction of coevolving protein interactions and demonstrate its improved performance over the mirrortree, tol-mirrortree, phylogenetic vector projection, and partial correlation approaches. Furthermore, we show its applicability to both the detection of horizontal gene transfer events as well as its potential use in the prediction of interaction specificity between a pair of multigene families.</p> <p>Conclusions</p> <p>These approaches provide additional tools for the study of phylogenetic trees and associated evolutionary processes. Source code is available at <url>http://gomezlab.bme.unc.edu/tools</url>.</p