Properties of highly clustered networks

We propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient. Among other things, our results indicate that increased clustering leads to a decrease in the size of the giant component of the network. We also study SIR-type epidemic processes within the model and find that clustering decreases the size of epidemics, but also decreases the epidemic threshold, making it easier for diseases to spread. In addition, clustering causes epidemics to saturate sooner, meaning that they infect a near-maximal fraction of the network for quite low transmission rates.Comment: 7 pages, 2 figures, 1 tabl

A Simple Model of Epidemics with Pathogen Mutation

We study how the interplay between the memory immune response and pathogen mutation affects epidemic dynamics in two related models. The first explicitly models pathogen mutation and individual memory immune responses, with contacted individuals becoming infected only if they are exposed to strains that are significantly different from other strains in their memory repertoire. The second model is a reduction of the first to a system of difference equations. In this case, individuals spend a fixed amount of time in a generalized immune class. In both models, we observe four fundamentally different types of behavior, depending on parameters: (1) pathogen extinction due to lack of contact between individuals, (2) endemic infection (3) periodic epidemic outbreaks, and (4) one or more outbreaks followed by extinction of the epidemic due to extremely low minima in the oscillations. We analyze both models to determine the location of each transition. Our main result is that pathogens in highly connected populations must mutate rapidly in order to remain viable.Comment: 9 pages, 11 figure

The spread of epidemic disease on networks

The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are non-uniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.Comment: 12 pages, 3 figure

Space as a low-temperature regime of graphs

I define a statistical model of graphs in which 2-dimensional spaces arise at low temperature. The configurations are given by graphs with a fixed number of edges and the Hamiltonian is a simple, local function of the graphs. Simulations show that there is a transition between a low-temperature regime in which the graphs form triangulations of 2-dimensional surfaces and a high-temperature regime, where the surfaces disappear. I use data for the specific heat and other observables to discuss whether this is a phase transition. The surface states are analyzed with regard to topology and defects.Comment: 22 pages, 12 figures; v3: published version; J.Stat.Phys. 201

Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang

We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics and Fourier analysis.Comment: 25 page

Mixing patterns in networks

We study assortative mixing in networks, the tendency for vertices in networks to be connected to other vertices that are like (or unlike) them in some way. We consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age. As a special example of the latter we consider mixing according to vertex degree, i.e., according to the number of connections vertices have to other vertices: do gregarious people tend to associate with other gregarious people? We propose a number of measures of assortative mixing appropriate to the various mixing types, and apply them to a variety of real-world networks, showing that assortative mixing is a pervasive phenomenon found in many networks. We also propose several models of assortatively mixed networks, both analytic ones based on generating function methods, and numerical ones based on Monte Carlo graph generation techniques. We use these models to probe the properties of networks as their level of assortativity is varied. In the particular case of mixing by degree, we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices.Comment: 14 pages, 2 tables, 4 figures, some additions and corrections in this versio

From dynamical scaling to local scale-invariance: a tutorial

Dynamical scaling arises naturally in various many-body systems far from equilibrium. After a short historical overview, the elements of possible extensions of dynamical scaling to a local scale-invariance will be introduced. Schr\"odinger-invariance, the most simple example of local scale-invariance, will be introduced as a dynamical symmetry in the Edwards-Wilkinson universality class of interface growth. The Lie algebra construction, its representations and the Bargman superselection rules will be combined with non-equilibrium Janssen-de Dominicis field-theory to produce explicit predictions for responses and correlators, which can be compared to the results of explicit model studies. At the next level, the study of non-stationary states requires to go over, from Schr\"odinger-invariance, to ageing-invariance. The ageing algebra admits new representations, which acts as dynamical symmetries on more general equations, and imply that each non-equilibrium scaling operator is characterised by two distinct, independent scaling dimensions. Tests of ageing-invariance are described, in the Glauber-Ising and spherical models of a phase-ordering ferromagnet and the Arcetri model of interface growth.Comment: 1+ 23 pages, 2 figures, final for

Nonperturbative renormalization group approach to frustrated magnets

This article is devoted to the study of the critical properties of classical XY and Heisenberg frustrated magnets in three dimensions. We first analyze the experimental and numerical situations. We show that the unusual behaviors encountered in these systems, typically nonuniversal scaling, are hardly compatible with the hypothesis of a second order phase transition. We then review the various perturbative and early nonperturbative approaches used to investigate these systems. We argue that none of them provides a completely satisfactory description of the three-dimensional critical behavior. We then recall the principles of the nonperturbative approach - the effective average action method - that we have used to investigate the physics of frustrated magnets. First, we recall the treatment of the unfrustrated - O(N) - case with this method. This allows to introduce its technical aspects. Then, we show how this method unables to clarify most of the problems encountered in the previous theoretical descriptions of frustrated magnets. Firstly, we get an explanation of the long-standing mismatch between different perturbative approaches which consists in a nonperturbative mechanism of annihilation of fixed points between two and three dimensions. Secondly, we get a coherent picture of the physics of frustrated magnets in qualitative and (semi-) quantitative agreement with the numerical and experimental results. The central feature that emerges from our approach is the existence of scaling behaviors without fixed or pseudo-fixed point and that relies on a slowing-down of the renormalization group flow in a whole region in the coupling constants space. This phenomenon allows to explain the occurence of generic weak first order behaviors and to understand the absence of universality in the critical behavior of frustrated magnets.Comment: 58 pages, 15 PS figure

Search for 14.4 keV Solar Axions from M1 Transition of 57Fe with CUORE Crystals

We report the results of a search for axions from the 14.4 keV M1 transition from 57Fe in the core of the sun using the axio-electric effect in TeO2bolometers. The detectors are 5 × 5 × 5 cm3 crystals operated at about 10 mK in a facility used to test bolometers for the CUORE experiment at the Laboratori Nazionali del Gran Sasso in Italy. An analysis of 43.65 kg⋅d of data was made using a newly developed low energy trigger which was optimized to reduce the energy threshold of the detector. An upper limit of 0.58 c⋅kg−1⋅d−1 is established at 95% C.L., which translates into lower bounds fA ≥ 3.12 × 105 GeV 95% C.L. (DFSZ model) and fA ≥ 2.41 × 104 GeV 95% C.L. (KSVZ model) on the Peccei-Quinn symmetry-breaking scale, for a value of S = 0.5 of the flavor-singlet axial vector matrix element. These bounds can be expressed in terms of axion masses as mA ≤ 19.2 eV and mA ≤ 250 eV at 95% C.L. in the DFSZ and KSVZ models respectively. Bounds are given also for the interval 0.35 ≤ S ≤ 0.55

Search for 14.4 keV Solar Axions from M1 Transition of 57Fe with CUORE Crystals

We report the results of a search for axions from the 14.4 keV M1 transition from 57Fe in the core of the sun using the axio-electric effect in TeO2bolometers. The detectors are 5 × 5 × 5 cm3 crystals operated at about 10 mK in a facility used to test bolometers for the CUORE experiment at the Laboratori Nazionali del Gran Sasso in Italy. An analysis of 43.65 kg⋅d of data was made using a newly developed low energy trigger which was optimized to reduce the energy threshold of the detector. An upper limit of 0.58 c⋅kg−1⋅d−1 is established at 95% C.L., which translates into lower bounds fA ≥ 3.12 × 105 GeV 95% C.L. (DFSZ model) and fA ≥ 2.41 × 104 GeV 95% C.L. (KSVZ model) on the Peccei-Quinn symmetry-breaking scale, for a value of S = 0.5 of the flavor-singlet axial vector matrix element. These bounds can be expressed in terms of axion masses as mA ≤ 19.2 eV and mA ≤ 250 eV at 95% C.L. in the DFSZ and KSVZ models respectively. Bounds are given also for the interval 0.35 ≤ S ≤ 0.55