Explosive Contagion in Networks.

The spread of social phenomena such as behaviors, ideas or products is an ubiquitous but remarkably complex phenomenon. A successful avenue to study the spread of social phenomena relies on epidemic models by establishing analogies between the transmission of social phenomena and infectious diseases. Such models typically assume simple social interactions restricted to pairs of individuals; effects of the context are often neglected. Here we show that local synergistic effects associated with acquaintances of pairs of individuals can have striking consequences on the spread of social phenomena at large scales. The most interesting predictions are found for a scenario in which the contagion ability of a spreader decreases with the number of ignorant individuals surrounding the target ignorant. This mechanism mimics ubiquitous situations in which the willingness of individuals to adopt a new product depends not only on the intrinsic value of the product but also on whether his acquaintances will adopt this product or not. In these situations, we show that the typically smooth (second order) transitions towards large social contagion become explosive (first order). The proposed synergistic mechanisms therefore explain why ideas, rumours or products can suddenly and sometimes unexpectedly catch on.

Prominent effect of soil network heterogeneity on microbial invasion.

Using a network representation for real soil samples and mathematical models for microbial spread, we show that the structural heterogeneity of the soil habitat may have a very significant influence on the size of microbial invasions of the soil pore space. In particular, neglecting the soil structural heterogeneity may lead to a substantial underestimation of microbial invasion. Such effects are explained in terms of a crucial interplay between heterogeneity in microbial spread and heterogeneity in the topology of soil networks. The main influence of network topology on invasion is linked to the existence of long channels in soil networks that may act as bridges for transmission of microorganisms between distant parts of soil.

Epidemics in networks of spatially correlated three-dimensional root-branching structures.

Using digitized images of the three-dimensional, branching structures for root systems of bean seedlings, together with analytical and numerical methods that map a common susceptible-infected-recovered ('SIR') epidemiological model onto the bond percolation problem, we show how the spatially correlated branching structures of plant roots affect transmission efficiencies, and hence the invasion criterion, for a soil-borne pathogen as it spreads through ensembles of morphologically complex hosts. We conclude that the inherent heterogeneities in transmissibilities arising from correlations in the degrees of overlap between neighbouring plants render a population of root systems less susceptible to epidemic invasion than a corresponding homogeneous system. Several components of morphological complexity are analysed that contribute to disorder and heterogeneities in the transmissibility of infection. Anisotropy in root shape is shown to increase resilience to epidemic invasion, while increasing the degree of branching enhances the spread of epidemics in the population of roots. Some extension of the methods for other epidemiological systems are discussed.

Assortative mating and mutation diffusion in spatial evolutionary systems.

The influence of spatial structure on the equilibrium properties of a sexual population model defined on networks is studied numerically. Using a small-world-like topology of the networks as an investigative tool, the contributions to the fitness of assortative mating and of global mutant spread properties are considered. Simple measures of nearest-neighbor correlations and speed of spread of mutants through the system have been used to confirm that both of these dynamics are important contributory factors to the fitness. It is found that assortative mating increases the fitness of populations. Quick global spread of favorable mutations is shown to be a key factor increasing the equilibrium fitness of populations.

Scaling behavior of the disordered contact process.

The one-dimensional contact process with weak to intermediate quenched disorder in its transmission rates is investigated via quasistationary Monte Carlo simulation. We address the contested questions of both the nature of dynamical scaling, conventional or activated, as well as of universality of critical exponents by employing a scaling analysis of the distribution of lifetimes and the quasistationary density of infection. We find activated scaling to be the appropriate description for intermediate to strong disorder. Critical exponents taken at face value are disorder dependent and approach the values expected for the limit of strong disorder as predicted by strong-disorder renormalization-group analysis of the process. However, no definitive conclusion about the nature of exponents is possible from this numerical approach on its own.

Contact process in disordered and periodic binary two-dimensional lattices.

The critical behavior of the contact process (CP) in disordered and periodic binary two-dimensional (2D) lattices is investigated numerically by means of Monte Carlo simulations as well as via an analytical approximation and standard mean field theory. Phase-separation lines calculated numerically are found to agree well with analytical predictions around the homogeneous point. For the disordered case, values of static scaling exponents obtained via quasistationary simulations are found to change with disorder strength. In particular, the finite-size scaling exponent of the density of infected sites approaches a value consistent with the existence of an infinite-randomness fixed point as conjectured before for the 2D disordered CP. At the same time, both dynamical and static scaling exponents are found to coincide with the values established for the homogeneous case thus confirming that the contact process in a heterogeneous environment belongs to the directed percolation universality class.

Simulating the contact process in heterogeneous environments.

The one-dimensional contact process (CP) in a heterogeneous environment-a binary chain consisting of two types of site with different recovery rates-is investigated. It is argued that the commonly used random-sequential Monte Carlo simulation method which employs a discrete notion of time is not faithful to the rates of the contact process in a heterogeneous environment. Therefore, a modification of this algorithm along with two alternative continuous-time implementations are analyzed. The latter two are an adapted version of the n -fold way used in Ising model simulations and a method based on a modified priority queue. It is demonstrated that the commonly used (but incorrect as we believe) discrete-time method yields a different critical threshold from all other algorithms considered. Finite-size scaling of the lowest gap in the spectrum of the Liouville time-evolution operator for the CP gives an estimate of the critical rate which supports these findings. Further, a performance test indicates an advantage in using the continuous-time methods in systems with heterogeneous rates. This result promises to help in the analysis of the CP in disordered systems with heterogeneous rates in which simulation is a challenging task due to very long relaxation times.

Supercritical series expansion for the contact process in heterogeneous and disordered environments.

The supercritical series expansion of the survival probability for the one-dimensional contact process in heterogeneous and disordered lattices is used for the evaluation of the loci of critical points and critical exponents beta . The heterogeneity and disorder are modeled by considering binary regular and irregular lattices of nodes characterized by different recovery rates and identical transmission rates. Two analytical approaches based on nested Padé approximants and partial differential approximants were used in the case of expansions with respect to two variables (two recovery rates) for the evaluation of the critical values and critical exponents. The critical exponents in heterogeneous systems are very close to those for the homogeneous contact process thus confirming that the contact process in periodic heterogeneous environment belongs to the directed percolation universality class. The disordered systems, in contrast, seem to have continuously varying critical exponents.

Temporal and dimensional effects in evolutionary graph theory.

The spread in time of a mutation through a population is studied analytically and computationally in fully connected networks and on spatial lattices. The time t* for a favorable mutation to dominate scales with the population size N as N(D+1)/D in D-dimensional hypercubic lattices and as NlnN in fully-connected graphs. It is shown that the surface of the interface between mutants and nonmutants is crucial in predicting the dynamics of the system. Network topology has a significant effect on the equilibrium fitness of a simple population model incorporating multiple mutations and sexual reproduction.

Infrared absorption in glasses and their crystalline counterparts.

It is demonstrated by means of numerical analysis that absorption of light in glasses in the IR region, [Formula: see text], can be understood in terms of light absorption in their crystalline counterparts. This signifies the important role of local structural motifs (units) present both in glasses and crystals. Decomposition of the coupling coefficient into coherent and incoherent contributions gives insight into the origin of the spectral features of IR absorption. The analysis has been undertaken for classical molecular dynamics models of vitreous silica and its crystalline counterpart, α-cristobalite.

Contact process in heterogeneous and weakly disordered systems.

The critical behavior of the contact process (CP) in heterogeneous periodic and weakly disordered environments is investigated using the supercritical series expansion and Monte Carlo (MC) simulations. Phase-separation lines and critical exponents beta (from series expansion) and eta (from MC simulations) are calculated. A general analytical expression for the locus of critical points is suggested for the weak-disorder limit and confirmed by the series expansion analysis and the MC simulations. Our results for the critical exponents show that the CP in heterogeneous environments remains in the directed percolation universality class, while for environments with quenched disorder, the data are compatible with the scenario of continuously changing critical exponents.

Universal features of terahertz absorption in disordered materials.

Using an analytical theory, experimental terahertz time-domain spectroscopy data, and numerical evidence, we demonstrate that the frequency dependence of the absorption coupling coefficient between far-infrared photons and atomic vibrations in disordered materials has the universal functional form, C(omega)=A+Bomega(2), where the material-specific constants A and B are related to the distributions of fluctuating charges obeying global and local charge neutrality, respectively.

Spectral properties of disordered fully connected graphs.

The spectral properties of disordered fully connected graphs with a special type of node-node interactions are investigated. The approximate analytical expression for the ensemble-averaged spectral density for the Hamiltonian defined on the fully connected graph is derived and analyzed for both the electronic and vibrational problems which can be related to the contact process and to the problem of stochastic diffusion, respectively. It is demonstrated how to evaluate the extreme eigenvalues and use them for finding the lower-bound estimates of the critical parameter for the contact process on the disordered fully connected graphs.

Extinction of epidemics in lattice models with quenched disorder.

The extinction of the contact process for epidemics in lattice models with quenched disorder is analyzed in the limit of small density of infected sites. It is shown that the problem in such a regime can be mapped to the quantum-mechanical one characterized by the Anderson Hamiltonian for an electron in a random lattice. It is demonstrated both analytically (self-consistent mean field) and numerically (by direct diagonalization of the Hamiltonian and by means of cellular automata simulations) that disorder enhances the contact process, given the mean values of random parameters are not influenced by disorder.

Instantaneous frequency and amplitude identification using wavelets: application to glass structure.

This paper describes a method for extracting rapidly varying, superimposed amplitude-modulated and frequency-modulated signal components. The method is based upon the continuous wavelet transform (CWT) and uses a new wavelet that is a modification to the well-known Morlet wavelet to allow analysis at high resolution. In order to interpret the CWT of a signal correctly, an approximate analytic expression for the CWT of an oscillatory signal is examined via a stationary-phase approximation. This analysis is specialized for the new wavelet and the results are used to construct expressions for the amplitude and frequency modulations of the components in a signal from the transform of the signal. The method is tested on a representative, variable-frequency signal as an example before being applied to a function of interest in our subject area-a structural correlation function of a disordered material-which immediately reveals previously undetected features.

Spatial decay of the single-particle density matrix in insulators: analytic results in two and three dimensions.

Analytic results for the asymptotic decay of the electron density matrix in insulators have been obtained in all three dimensions (D = 1,2,3) for a tight-binding model defined on a simple cubic lattice. The anisotropic decay length is shown to be dependent on the energy parameters of the model. The existence of the power-law prefactor, proportional, variant r(-D/2), is demonstrated.

Fast Chebyshev-polynomial method for simulating the time evolution of linear dynamical systems.

We present a fast method for simulating the time evolution of any linear dynamical system possessing eigenmodes. This method does not require an explicit calculation of the eigenvectors and eigenfrequencies, and is based on a Chebyshev polynomial expansion of the formal operator matrix solution in the eigenfrequency domain. It does not suffer from the limitations of ordinary time-integration methods, and can be made accurate to almost machine precision. Among its possible applications are harmonic classical mechanical systems, quantum diffusion, and stochastic transport theory. An example of its use is given for the problem of vibrational wave-packet propagation in a disordered lattice.

Origin of the boson peak in systems with lattice disorder.

The origin of the boson peak in models with force-constant disorder has been established by calculations using the coherent potential approximation. The analytical results obtained are supported by precise numerical solutions. The boson peak in the disordered system is associated with the lowest van Hove singularity in the spectrum of the reference crystalline system, pushed down in frequency by disorder-induced level-repelling and hybridization effects.