A single-walker approach for studying quasi-nonergodic systems

The jump-walking Monte-Carlo algorithm is revisited and updated to study the equilibrium properties of systems exhibiting quasi-nonergodicity. It is designed for a single processing thread as opposed to currently predominant algorithms for large parallel processing systems. The updated algorithm is tested on the Ising model and applied to the lattice-gas model for sorption in aerogel at low temperatures, when dynamics of the system is critically slowed down. It is demonstrated that the updated jump-walking simulations are able to produce equilibrium isotherms which are typically hidden by the hysteresis effect characteristic of the standard single-flip simulations.

Effects of local and global network connectivity on synergistic epidemics

Epidemics in networks can be affected by cooperation in transmission of infection and also connectivity between nodes. An interplay between these two properties and their influence on epidemic spread are addressed in the paper. A particular type of cooperative effects (called synergy effects) is considered, where the transmission rate between a pair of nodes depends on the number of infected neighbors. The connectivity effects are studied by constructing networks of different topology, starting with lattices with only local connectivity and then with networks that have both local and global connectivity obtained by random bond-rewiring to nodes within a certain distance. The susceptible-infected-removed epidemics were found to exhibit several interesting effects: (i) for epidemics with strong constructive synergy spreading in networks with high local connectivity, the bond rewiring has a negative role in epidemic spread, i.e., it reduces invasion probability; (ii) in contrast, for epidemics with destructive or weak constructive synergy spreading on networks of arbitrary local connectivity, rewiring helps epidemics to spread; (iii) and, finally, rewiring always enhances the spread of epidemics, independent of synergy, if the local connectivity is low.

Effect of disorder on condensation in the lattice gas model on a random graph

The lattice gas model of condensation in a heterogeneous pore system, represented by a random graph of cells, is studied using an exact analytical solution. A binary mixture of pore cells with different coordination numbers is shown to exhibit two phase transitions as a function of chemical potential in a certain temperature range. Heterogeneity in interaction strengths is demonstrated to reduce the critical temperature and, for large-enough degreeS of disorder, divides the cells into ones which are either on average occupied or unoccupied. Despite treating the pore space loops in a simplified manner, the random-graph model provides a good description of condensation in porous structures containing loops. This is illustrated by considering capillary condensation in a structural model of mesoporous silica SBA-15.

Effects of variable-state neighborhoods for spreading synergystic processes on lattices

A theoretical framework for the description of susceptible-infected-removed (SIR) spreading processes with synergistic transmission of infection on a lattice is developed. The model incorporates explicitly the effects of time-dependence of the state of the hosts in the neighborhood of transmission events. Exact solution of the model shows that time-dependence of the state of nearest neighbors of recipient hosts is a key factor for synergistic spreading processes. It is demonstrated that the higher the connectivity of a lattice, the more prominent is the effect of synergy on spread.

Zero-temperature random-field Ising model on a bilayered Bethe lattice

The zero-temperature random-field Ising model is solved analytically for magnetization versus external field for a bilayered Bethe lattice. The mechanisms of infinite avalanches which are observed for small values of disorder are established. The effects of variable interlayer interaction strengths on infinite avalanches are investigated. The spin-field correlation length is calculated and its critical behavior is discussed. Direct Monte Carlo simulations of spin-flip dynamics are shown to support the analytical findings. We find, paradoxically, that a reduction of the interlayer bond strength can cause a phase transition from a regime with continuous magnetization reversal to a regime where magnetization exhibits a discontinuity associated with an infinite avalanche. This effect is understood in terms of the proposed mechanisms for the infinite avalanche.

Capillary condensation in one-dimensional irregular confinement

A lattice-gas model with heterogeneity is developed for the description of fluid condensation in finite sized one-dimensional pores of arbitrary shape. Mapping to the random-field Ising model allows an exact solution of the model to be obtained at zero-temperature, reproducing the experimentally observed dependence of the amount of fluid adsorbed in the pore on external pressure. It is demonstrated that the disorder controls the sorption for long pores and can result in H2-type hysteresis. Finite-temperature Metropolis dynamics simulations support analytical findings in the limit of low temperatures. The proposed framework is viewed as a fundamental building block of the theory of capillary condensation necessary for reliable structural analysis of complex porous media from adsorption-desorption data.

Mechanisms of evolution of avalanches in regular graphs

A mapping of avalanches occurring in the zero-temperature random-field Ising model to life periods of a population experiencing immigration is established. Such a mapping allows the microscopic criteria for the occurrence of an infinite avalanche in a q-regular graph to be determined. A key factor for an avalanche of spin flips to become infinite is that it interacts in an optimal way with previously flipped spins. Based on these criteria, we explain why an infinite avalanche can occur in q-regular graphs only for q>3 and suggest that this criterion might be relevant for other systems. The generating function techniques developed for branching processes are applied to obtain analytical expressions for the durations, pulse shapes, and power spectra of the avalanches. The results show that only very long avalanches exhibit a significant degree of universality.

Prediction of invasion from the early stage of an epidemic

Predictability of undesired events is a question of great interest in many scientific disciplines including seismology, economy and epidemiology. Here, we focus on the predictability of invasion of a broad class of epidemics caused by diseases that lead to permanent immunity of infected hosts after recovery or death. We approach the problem from the perspective of the science of complexity by proposing and testing several strategies for the estimation of important characteristics of epidemics, such as the probability of invasion. Our results suggest that parsimonious approximate methodologies may lead to the most reliable and robust predictions. The proposed methodologies are first applied to analysis of experimentally observed epidemics: invasion of the fungal plant pathogen Rhizoctonia solani in replicated host microcosms. We then consider numerical experiments of the susceptible-infected-removed model to investigate the performance of the proposed methods in further detail. The suggested framework can be used as a valuable tool for quick assessment of epidemic threat at the stage when epidemics only start developing. Moreover, our work amplifies the significance of the small-scale and finite-time microcosm realizations of epidemics revealing their predictive power.

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.

The effect of heterogeneity on invasion in spatial epidemics: from theory to experimental evidence in a model system

Heterogeneity in host populations is an important factor affecting the ability of a pathogen to invade, yet the quantitative investigation of its effects on epidemic spread is still an open problem. In this paper, we test recent theoretical results, which extend the established "percolation paradigm" to the spread of a pathogen in discrete heterogeneous host populations. In particular, we test the hypothesis that the probability of epidemic invasion decreases when host heterogeneity is increased. We use replicated experimental microcosms, in which the ubiquitous pathogenic fungus Rhizoctonia solani grows through a population of discrete nutrient sites on a lattice, with nutrient sites representing hosts. The degree of host heterogeneity within different populations is adjusted by changing the proportion and the nutrient concentration of nutrient sites. The experimental data are analysed via Bayesian inference methods, estimating pathogen transmission parameters for each individual population. We find a significant, negative correlation between heterogeneity and the probability of pathogen invasion, thereby validating the theory. The value of the correlation is also in remarkably good agreement with the theoretical predictions. We briefly discuss how our results can be exploited in the design and implementation of disease control strategies.

Synergy in spreading processes: from exploitative to explorative foraging strategies

An epidemiological model which incorporates synergistic effects that allow the infectivity and/or susceptibility of hosts to be dependent on the number of infected neighbors is proposed. Constructive synergy induces an exploitative behavior which results in a rapid invasion that infects a large number of hosts. Interfering synergy leads to a slower and sparser explorative foraging strategy that traverses larger distances by infecting fewer hosts. The model can be mapped to a dynamical bond percolation with spatial correlations that affect the mechanism of spread but do not influence the critical behavior of epidemics.

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.

Heterogeneity in susceptible-infected-removed (SIR) epidemics on lattices

The percolation paradigm is widely used in spatially explicit epidemic models where disease spreads between neighbouring hosts. It has been successful in identifying epidemic thresholds for invasion, separating non-invasive regimes, where the disease never invades the system, from invasive regimes where the probability of invasion is positive. However, its power is mainly limited to homogeneous systems. When heterogeneity (environmental stochasticity) is introduced, the value of the epidemic threshold is, in general, not predictable without numerical simulations. Here, we analyse the role of heterogeneity in a stochastic susceptible-infected-removed epidemic model on a two-dimensional lattice. In the homogeneous case, equivalent to bond percolation, the probability of invasion is controlled by a single parameter, the transmissibility of the pathogen between neighbouring hosts. In the heterogeneous model, the transmissibility becomes a random variable drawn from a probability distribution. We investigate how heterogeneity in transmissibility influences the value of the invasion threshold, and find that the resilience of the system to invasion can be suitably described by two control parameters, the mean and variance of the transmissibility. We analyse a two-dimensional phase diagram, where the threshold is represented by a phase boundary separating an invasive regime in the high-mean, low-variance region from a non-invasive regime in the low-mean, high-variance region of the parameter space. We thus show that the percolation paradigm can be extended to the heterogeneous case. Our results have practical implications for the analysis of disease control strategies in realistic heterogeneous epidemic systems.

Unveiling the neuromorphological space

This article proposes the concept of neuromorphological space as the multidimensional space defined by a set of measurements of the morphology of a representative set of almost 6000 biological neurons available from the NeuroMorpho database. For the first time, we analyze such a large database in order to find the general distribution of the geometrical features. We resort to McGhee's biological shape space concept in order to formalize our analysis, allowing for comparison between the geometrically possible tree-like shapes, obtained by using a simple reference model, and real neuronal shapes. Two optimal types of projections, namely, principal component analysis and canonical analysis, are used in order to visualize the originally 20-D neuron distribution into 2-D morphological spaces. These projections allow the most important features to be identified. A data density analysis is also performed in the original 20-D feature space in order to corroborate the clustering structure. Several interesting results are reported, including the fact that real neurons occupy only a small region within the geometrically possible space and that two principal variables are enough to account for about half of the overall data variability. Most of the measurements have been found to be important in representing the morphological variability of the real neurons.

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.

Complexity and anisotropy in host morphology make populations less susceptible to epidemic outbreaks

One of the challenges in epidemiology is to account for the complex morphological structure of hosts such as plant roots, crop fields, farms, cells, animal habitats and social networks, when the transmission of infection occurs between contiguous hosts. Morphological complexity brings an inherent heterogeneity in populations and affects the dynamics of pathogen spread in such systems. We have analysed the influence of realistically complex host morphology on the threshold for invasion and epidemic outbreak in an SIR (susceptible-infected-recovered) epidemiological model. We show that disorder expressed in the host morphology and anisotropy reduces the probability of epidemic outbreak and thus makes the system more resistant to epidemic outbreaks. We obtain general analytical estimates for minimally safe bounds for an invasion threshold and then illustrate their validity by considering an example of host data for branching hosts (salamander retinal ganglion cells). Several spatial arrangements of hosts with different degrees of heterogeneity have been considered in order to separately analyse the role of shape complexity and anisotropy in the host population. The estimates for invasion threshold are linked to morphological characteristics of the hosts that can be used for determining the threshold for invasion in practical applications.

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.

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.

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.

Establishment of facultative sexuals

The existence of sex is one of the major unsolved problems in biology. We use computer simulations to model conditions in which sex may first become established. We develop an individual-based population model and show that a hypothetical facultative sex gene can fix, provided that the initial cost is low. It is demonstrated that the equilibrium fitness in the population increases with increasing population size and decreasing mutation rate. The probability of the establishment of the sex gene is found not to be directly related to the fitness difference between the asexual and sexual populations. This change in fitness on changing the parameters of the model is investigated.

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.

Fast time-evolution method for dynamical systems

A fast time-evolution method is developed for systems for which the dynamical behavior can be reduced to the eigenvector/eigenvalue problem. The method does not use the eigenvectors/eigenvalues themselves and is based on a polynominal expansion of the formal operator solution in the eigenfrequency domain. It is complementary to the standard time-integration approaches and allows one to calculate or simulate the state of a system at arbitrary times. The time evolution of, e.g., classical harmonic atomic systems and quantum systems described by linear Hamiltonians can be treated by this method.