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

Connectivity of sparse Bluetooth networks

Consider a random geometric graph defined on n vertices uniformly distributed in the d-dimensional unit torus. Two vertices are connected if their distance is less than a “visibility radius ” rn. We consider Bluetooth networks that are locally sparsified random geometric graphs. Each vertex selects c of its neighbors in the random geometric graph at random and connects only to the selected points. We show that if the visibility radius is at least of the order of n−(1−δ)/d for some δ> 0, then a constant value of c is sufficient for the graph to be connected, with high probability. It suffices to take c ≥ √ (1 + ɛ)/δ + K for any positive ɛ where K is a constant depending on d only. On the other hand, with c ≤ √ (1 − ɛ)/δ, the graph is disconnected, with high probability. 1 Introduction an

Long and short paths in uniform random recursive dags

In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant \sigma such that S_n/log(n) tends to \sigma in probability as n tends to infinity. We also show that max_{1 \le i \le n} S_i/log(n) tends to \sigma in probability.Comment: 16 page

Environmental Exposures and Invasive Meningococcal Disease: An Evaluation of Effects on Varying Time Scales

Invasive meningococcal disease (IMD) is an important cause of meningitis and bacteremia worldwide. Seasonal variation in IMD incidence has long been recognized, but mechanisms responsible for this phenomenon remain poorly understood. The authors sought to evaluate the effect of environmental factors on IMD risk in Philadelphia, Pennsylvania, a major urban center. Associations between monthly weather patterns and IMD incidence were evaluated using multivariable Poisson regression models controlling for seasonal oscillation. Short-term weather effects were identified using a case-crossover approach. Both study designs control for seasonal factors that might otherwise confound the relation between environment and IMD. Incidence displayed significant wintertime seasonality (for oscillation, P < 0.001), and Poisson regression identified elevated monthly risk with increasing relative humidity (per 1% increase, incidence rate ratio = 1.04, 95% confidence interval: 1.004, 1.08). Case-crossover methods identified an inverse relation between ultraviolet B radiation index 1–4 days prior to onset and disease risk (odds ratio = 0.54, 95% confidence interval: 0.34, 0.85). Extended periods of high humidity and acute changes in ambient ultraviolet B radiation predict IMD occurrence in Philadelphia. The latter effect may be due to decreased pathogen survival or virulence and may explain the wintertime seasonality of IMD in temperate regions of North America

Landscape Diversity Related to Buruli Ulcer Disease in Côte d'Ivoire

Buruli ulcer (BU) is one of the most neglected but treatable tropical diseases. The causative organism, Mycobacterium ulcerans, is from the family of bacteria that causes tuberculosis and leprosy. This severe skin disease leads to long-term functional disability if not treated. BU has been reported in over 30 countries mainly with tropical and subtropical climates, but Côte d'Ivoire is one of the most affected countries. M. ulcerans is an environmental bacterium and its mode of transmission to humans is still unclear, such that the disease is often referred to as the “mysterious disease” or the “new leprosy”. Here, we explored the relationship between environmental and socioeconomic factors and BU cases on a nationwide scale. We found that irrigated rice field cultures areas, and, to a lesser extent, banana fields as well as areas in the vicinity of dams used for irrigation and aquaculture purposes, represent high risk zones for the human population to contract BU in Côte d'Ivoire. This work identifies high-risk areas for BU in Côte d'Ivoire and deserves to be extended to different countries. We need now to obtain a global vision and understanding of the route of transmission of M. ulcerans to humans in order to better implement control strategies

Search for the standard model Higgs boson in the H to ZZ to 2l 2nu channel in pp collisions at sqrt(s) = 7 TeV

A search for the standard model Higgs boson in the H to ZZ to 2l 2nu decay channel, where l = e or mu, in pp collisions at a center-of-mass energy of 7 TeV is presented. The data were collected at the LHC, with the CMS detector, and correspond to an integrated luminosity of 4.6 inverse femtobarns. No significant excess is observed above the background expectation, and upper limits are set on the Higgs boson production cross section. The presence of the standard model Higgs boson with a mass in the 270-440 GeV range is excluded at 95% confidence level.Comment: Submitted to JHE

Search for New Physics with Jets and Missing Transverse Momentum in pp collisions at sqrt(s) = 7 TeV

A search for new physics is presented based on an event signature of at least three jets accompanied by large missing transverse momentum, using a data sample corresponding to an integrated luminosity of 36 inverse picobarns collected in proton--proton collisions at sqrt(s)=7 TeV with the CMS detector at the LHC. No excess of events is observed above the expected standard model backgrounds, which are all estimated from the data. Exclusion limits are presented for the constrained minimal supersymmetric extension of the standard model. Cross section limits are also presented using simplified models with new particles decaying to an undetected particle and one or two jets

Search for anomalous t t-bar production in the highly-boosted all-hadronic final state

A search is presented for a massive particle, generically referred to as a Z', decaying into a t t-bar pair. The search focuses on Z' resonances that are sufficiently massive to produce highly Lorentz-boosted top quarks, which yield collimated decay products that are partially or fully merged into single jets. The analysis uses new methods to analyze jet substructure, providing suppression of the non-top multijet backgrounds. The analysis is based on a data sample of proton-proton collisions at a center-of-mass energy of 7 TeV, corresponding to an integrated luminosity of 5 inverse femtobarns. Upper limits in the range of 1 pb are set on the product of the production cross section and branching fraction for a topcolor Z' modeled for several widths, as well as for a Randall--Sundrum Kaluza--Klein gluon. In addition, the results constrain any enhancement in t t-bar production beyond expectations of the standard model for t t-bar invariant masses larger than 1 TeV.Comment: Submitted to the Journal of High Energy Physics; this version includes a minor typo correction that will be submitted as an erratu