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

Immune Boosting Explains Regime-Shifts in Prevaccine-Era Pertussis Dynamics

Understanding the biological mechanisms underlying episodic outbreaks of infectious diseases is one of mathematical epidemiology’s major goals. Historic records are an invaluable source of information in this enterprise. Pertussis (whooping cough) is a re-emerging infection whose intermittent bouts of large multiannual epidemics interspersed between periods of smaller-amplitude cycles remain an enigma. It has been suggested that recent increases in pertussis incidence and shifts in the age-distribution of cases may be due to diminished natural immune boosting. Here we show that a model that incorporates this mechanism can account for a unique set of pre-vaccine-era data from Copenhagen. Under this model, immune boosting induces transient bursts of large amplitude outbreaks. In the face of mass vaccination, the boosting model predicts larger and more frequent outbreaks than do models with permanent or passively-waning immunity. Our results emphasize the importance of understanding the mechanisms responsible for maintaining immune memory fo

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

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

Azimuthal anisotropy of charged particles at high transverse momenta in PbPb collisions at sqrt(s[NN]) = 2.76 TeV

The azimuthal anisotropy of charged particles in PbPb collisions at nucleon-nucleon center-of-mass energy of 2.76 TeV is measured with the CMS detector at the LHC over an extended transverse momentum (pt) range up to approximately 60 GeV. The data cover both the low-pt region associated with hydrodynamic flow phenomena and the high-pt region where the anisotropies may reflect the path-length dependence of parton energy loss in the created medium. The anisotropy parameter (v2) of the particles is extracted by correlating charged tracks with respect to the event-plane reconstructed by using the energy deposited in forward-angle calorimeters. For the six bins of collision centrality studied, spanning the range of 0-60% most-central events, the observed v2 values are found to first increase with pt, reaching a maximum around pt = 3 GeV, and then to gradually decrease to almost zero, with the decline persisting up to at least pt = 40 GeV over the full centrality range measured.Comment: Replaced with published version. Added journal reference and DO

Measurement of the t t-bar production cross section in the dilepton channel in pp collisions at sqrt(s) = 7 TeV

The t t-bar production cross section (sigma[t t-bar]) is measured in proton-proton collisions at sqrt(s) = 7 TeV in data collected by the CMS experiment, corresponding to an integrated luminosity of 2.3 inverse femtobarns. The measurement is performed in events with two leptons (electrons or muons) in the final state, at least two jets identified as jets originating from b quarks, and the presence of an imbalance in transverse momentum. The measured value of sigma[t t-bar] for a top-quark mass of 172.5 GeV is 161.9 +/- 2.5 (stat.) +5.1/-5.0 (syst.) +/- 3.6(lumi.) pb, consistent with the prediction of the standard model.Comment: Replaced with published version. Included journal reference and DO

Performance of CMS muon reconstruction in pp collision events at sqrt(s) = 7 TeV

The performance of muon reconstruction, identification, and triggering in CMS has been studied using 40 inverse picobarns of data collected in pp collisions at sqrt(s) = 7 TeV at the LHC in 2010. A few benchmark sets of selection criteria covering a wide range of physics analysis needs have been examined. For all considered selections, the efficiency to reconstruct and identify a muon with a transverse momentum pT larger than a few GeV is above 95% over the whole region of pseudorapidity covered by the CMS muon system, abs(eta) < 2.4, while the probability to misidentify a hadron as a muon is well below 1%. The efficiency to trigger on single muons with pT above a few GeV is higher than 90% over the full eta range, and typically substantially better. The overall momentum scale is measured to a precision of 0.2% with muons from Z decays. The transverse momentum resolution varies from 1% to 6% depending on pseudorapidity for muons with pT below 100 GeV and, using cosmic rays, it is shown to be better than 10% in the central region up to pT = 1 TeV. Observed distributions of all quantities are well reproduced by the Monte Carlo simulation.Comment: Replaced with published version. Added journal reference and DO