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We simulate spreads of diseases for the susceptible–infected–recovered (SIR) model on contact networks with a particular focus on incorporated protective measures such as face masks. We consider the small-world network model. By using a large-deviation approach, in particular the 1 / t Wang–Landau algorithm, we obtained the full probability density function (pdf) of the cumulative number C of infected people over the full range of its support. In this way we are able to reach probabilities as small as 10−50. We obtain distinct characteristics in the pdf such as non-analyticities induced by the onset of the protective measures. Still, the results indicate that the mathematical large-deviation principle also holds for this extended SIR model, meaning that the size-dependence enters in P (C) in a simple fashion and the distribution is determined by the so-called rate function. We observe different phases in the pdf, which we investigate by analyzing the corresponding infection courses, i.e. time series, and and their correlations to the observed values of C. [ABSTRACT FROM AUTHOR]

We study the vertex-cover problem, which is a nondeterministic polynomial-time hard optimization problem and a prototypical model exhibiting phase transitions on random graphs, such as Erdôs-Rényi (ER) random graphs. These phase transitions coincide with changes of the solution space structure, e.g., for the ER ensemble at connectivity c = e «s 2.7183 from replica symmetric to replica-symmetry broken. For the vertex-cover problem, the typical complexity of exact branch-and-bound algorithms, which proceed by exploring the landscape of feasible configurations, also changes close to this phase transition from "easy" to "hard." In this work, we consider an algorithm which has a completely different strategy: The problem is mapped onto a linear programming problem augmented by a cutting-plane approach; hence the algorithm operates in a space outside the space of feasible configurations until the final step, where a solution is found. Here we show that this type of algorithm also exhibits an easy-hard transition around c -- e, which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem. [ABSTRACT FROM AUTHOR]

Domain walls and droplet-like excitation of the random-field Ising magnet are studied in d = {3,4,5,6,7} dimensions by means of exact numerical ground-state calculations. They are obtained using the established mapping to the graph-theoretical maximum-flow problem. This allows us to study large system sizes of more than 5 x 106 spins in exact thermal equilibrium. All simulations are carried out at the critical point for the strength h of the random fields, h = hc(d). Using finite-size scaling, energetic and geometric properties like stiffness exponents and fractal dimensions are calculated. Using these results, we test (hyper)scaling relations, which seem to be fulfilled below the upper critical dimension du = 6. Also, for d < du, the stiffness exponent can be obtained from the scaling of the ground-state energy. [ABSTRACT FROM AUTHOR]

Abstract: The concept of replica symmetry breaking found in the solution of the mean-field Sherrington-Kirkpatrick spinglass model has been applied to a variety of problems in science ranging from biological to computational and even financial analysis. Thus it is of paramount importance to understand which predictions of the mean-field solution apply to non-mean-field systems, such as realistic short-range spin-glass models. The one-dimensional spin glass with random power-law interactions promises to be an ideal test-bed to answer this question: Not only can large system sizes—which are usually a shortcoming in simulations of high-dimensional short-range system—be studied, by tuning the power-law exponent of the interactions the universality class of the model can be continuously tuned from the mean-field to the short-range universality class. We present details of the model, as well as recent applications to some questions of the physics of spin glasses. First, we study the existence of a spin-glass state in an external field. In addition, we discuss the existence of ultrametricity in short-range spin glasses. Finally, because the range of interactions can be changed, the model is a formidable test-bed for optimization algorithms. [ABSTRACT FROM AUTHOR]

The determination of the parameters of cylindrical optical waveguides, e.g. the diameters d → = (d 1 , ... , d r) of r layers of (semi-) transparent optical fibres, can be executed by inverse evaluation of the scattering intensities that emerge under monochromatic illumination. The inverse problem can be solved by optimising the mismatch R (d →) between the measured and simulated scattering patterns. The global optimum corresponds to the correct parameter values. The mismatch R (d →) can be seen as an energy landscape as a function of the diameters. In this work, we study the structure of the energy landscape for different values of the complex refractive indices n → , for r = 1 and r = 2 layers. We find that for both values of r, depending on the values of n → , two very different types of energy landscapes exist, respectively. One type is dominated by one global minimum and the other type exhibits a multitude of local minima. From an algorithmic viewpoint, this corresponds to easy and hard phases, respectively. Our results indicate that the two phases are separated by sharp phase-transition lines and that the shape of these lines can be described by one "critical" exponent b, which depends slightly on r. Interestingly, the same exponent also describes the dependence of the number of local minima on the diameters. Thus, our findings are comparable to previous theoretical studies on easy-hard transitions in basic combinatorial optimisation or decision problems like Travelling Salesperson and Satisfiability. To our knowledge our results are the first indicating the existence of easy-hard transitions for a real-world optimisation problem of technological relevance. [ABSTRACT FROM AUTHOR]

Efficiently identifying the most important communities and key transition nodes in weighted and unweighted networks is a prevalent problem in a wide range of disciplines. Here, we focus on the optimal clustering using variational kinetic parameters, linked to Markov processes defined on the underlying networks, namely, the slowest relaxation time and the Kemeny constant. We derive novel relations in terms of mean first passage times for optimizing clustering via the Kemeny constant and show that the optimal clustering boundaries have equal round-trip times to the clusters they separate. We also propose an efficient method that first projects the network nodes onto a 1D reaction coordinate and subsequently performs a variational boundary search using a parallel tempering algorithm, where the variational kinetic parameters act as an energy function to be extremized. We find that maximization of the Kemeny constant is effective in detecting communities, while the slowest relaxation time allows for detection of transition nodes. We demonstrate the validity of our method on several test systems, including synthetic networks generated from the stochastic block model and real world networks (Santa Fe Institute collaboration network, a network of co-purchased political books, and a street network of multiple cities in Luxembourg). Our approach is compared with existing clustering algorithms based on modularity and the robust Perron cluster analysis, and the identified transition nodes are compared with different notions of node centrality. [ABSTRACT FROM AUTHOR]

The growing integration of renewable energy sources (RES) into the power system causes congestion to occur more frequently. In order to reduce congestion in the short term and to make the utilization of the power system more efficient in the long term, power flow control (PFC) in the transmission system has been proposed. However, exemplary studies show that congestion will increase also in the distribution system if the transmission system is expanded. For this reason, the potential of PFC to reduce congestion in a model of a real 110 kV distribution system is investigated. Several Unified Power Flow Controller (UPFC) devices are optimized in terms of their number and placement in the power system, their size, control parameters, and costs, by using a Parallel Tempering approach as well as a greedy algorithm. Two optimization variants are considered, one reducing the number of degrees of freedom by integrating system knowledge while the other does not. It is found that near a critical grid state and disregarding costs, PFC can reduce congestion significantly (99.13%). When costs of the UPFCs are taken into account, PFC can reduce congestion by 73.2%. A basic economic analysis of the costs reveals that the usage of UPFCs is profitable. Furthermore, it is found that the reduction in the solution space of the optimization problem leads to better results faster and that, contrary to expectations, the optimization problem is simple to solve. The developed methods allow not only for the determination of the optimal use of UPFCs to minimize congestion, but also to estimate their profitability. [ABSTRACT FROM AUTHOR]

The superior olivary complex (SOC) is an essential auditory brainstem relay involved in sound localization. To identify the genetic program underlying its maturation, we profiled the rat SOC transcriptome at postnatal days 0, 4, 16, and 25 (P0, P4, P16, and P25, respectively), using genome-wide microarrays (41,012 oligonucleotides (oligos)). Differences in gene expression between two consecutive stages were highest between P4 and P16 (3.6%) and dropped to 0.06% between P16 and P25. To identify SOC-related genetic programs, we also profiled the entire brain at P4 and P25. The number of differentially expressed oligonucleotides between SOC and brain almost doubled from P4 to P25 (4.4% versus 7.6%). These data demonstrate considerable molecular specification around hearing onset, which is rapidly finalized. Prior to hearing onset, several transcription factors associated with the peripheral auditory system were up-regulated, probably coordinating the development of the auditory system. Additionally, crystallin-γ subunits and serotonin-related genes were highly expressed. The molecular repertoire of mature neurons was sculpted by SOC-related up- and down-regulation of voltage-gated channels and G-proteins. Comparison with the brain revealed a significant enrichment of hearing impairment-related oligos in the SOC (26 in the SOC, only 11 in the brain). Furthermore, 29 of 453 SOC-related oligos mapped within 19 genetic intervals associated with hearing impairment. Together, we identified sequential genetic programs in the SOC, thereby pinpointing candidates that may guide its development and ensure proper function. The enrichment of hearing impairment-related genes in the SOC may have implications for restoring hearing because central auditory structures might be more severely affected than previously appreciated. [ABSTRACT FROM AUTHOR]

In this paper, we study a simple model of a diffusive particle on a line, undergoing a stochastic resetting with rate r, via rescaling its current position by a factor a, which can be either positive or negative. For |a|<1, the position distribution becomes stationary at long times and we compute this limiting distribution exactly for all |a|<1. This symmetric distribution has a Gaussian shape near its peak at x=0, but decays exponentially for large |x|. We also studied the mean first-passage time (MFPT) T(0) to a target located at a distance L from the initial position (the origin) of the particle. As a function of the initial position x, the MFPT T(x) satisfies a nonlocal second order differential equation and we have solved it explicitly for 0≤a<1. For -1

A simple zipper model is introduced, representing in a simplified way, e.g., the folded DNA double helix or hairpin structures in RNA. The double stranded hairpin is connected to a heat bath at temperature T and subject to an external force f, which couples to the free length L of the unzipped sequence. The leftmost zipped position can be seen as the position of a random walker in a special external field. Increasing the force leads to a zipping-unzipping first-order phase transition at a critical force f&#95;{c}(T) in the thermodynamic limit of a very large chain. We compute analytically, as a function of temperature T and force f, the full distribution P(L) of free lengths in the thermodynamic limit and show that it is qualitatively very different for ff&#95;{c}. Next we consider quasistatic work processes where the force is incremented according to a linear protocol. Having obtained P(L) already allows us to derive an analytical expression for the work distribution P(W) in the zipped phase f

The Higgs RNA model with an added term for a coupling to an external force is studied in regard to finite-time force-driving protocols with a minimal-work requirement. In this paper, RNA sequences which at low temperature exhibit hairpins are considered, which are often cited as typical template systems in stochastic thermodynamics. The optimized work protocols for this glassy many-particle system are determined numerically using the parallel tempering method. The protocols show distinct jumps at the beginning and end, which have been observed for single-particle systems and are proven to be optimal in the fast protocol limit generally. Optimality seems to be achieved by staying close to the equilibrium unfolding transition point, in agreement with experimental and theoretical observations. The change of work distributions, compared to those resulting from a naive linear driving protocol, are discussed generally and in terms of free energy estimation as well as the effect of optimized protocols on rare work process starting conditions.

We consider a discrete-time random walk on a one-dimensional lattice with space- and time-dependent random jump probabilities, known as the beta random walk. We are interested in the probability that, for a given realization of the jump probabilities (a sample), a walker starting at the origin at time t=0 is at position beyond ξsqrt[T/2] at time T. This probability fluctuates from sample to sample and we study the large-deviation rate function, which characterizes the tails of its distribution at large time T≫1. It is argued that, up to a simple rescaling, this rate function is identical to the one recently obtained exactly by two of the authors for the continuum version of the model. That continuum model also appears in the macroscopic fluctuation theory of a class of lattice gases, e.g., in the so-called KMP model of heat transfer. An extensive numerical simulation of the beta random walk, based on an importance sampling algorithm, is found in good agreement with the detailed analytical predictions. A first-order transition in the tilted measure, predicted to occur in the continuum model, is also observed in the numerics.

We study the probability distribution P(A) of the area A=∫&#95;{0}^{T}x(t)dt swept under fractional Brownian motion (fBm) x(t) until its first passage time T to the origin. The process starts at t=0 from a specified point x=L. We show that P(A) obeys exact scaling relation P(A)=D^{1/2H}/L^{1+1/H}Φ&#95;{H}(D^{1/2H}A/L^{1+1/H}), where 0

We numerically simulated the spread of disease for a Susceptible-Infected-Recovered (SIR) model on contact networks drawn from a small-world ensemble. We investigated the impact of two types of vaccination strategies, namely random vaccination and high-degree heuristics, on the probability density function (pdf) of the cumulative number C of infected people over a large range of its support. To obtain the pdf even in the range of probabilities as small as 10-80, we applied a large-deviation approach, in particular the 1/t Wang-Landau algorithm. To study the size-dependence of the pdfs within the framework of large-deviation theory, we analyzed the empirical rate function. To find out how typical as well as extreme mild or extreme severe infection courses arise, we investigated the structures of the time series conditioned to the observed values of C., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2023 Feld, Hartmann. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

We consider a system of noninteracting particles on a line with initial positions distributed uniformly with density ρ on the negative half-line. We consider two different models: (i) Each particle performs independent Brownian motion with stochastic resetting to its initial position with rate r and (ii) each particle performs run-and-tumble motion, and with rate r its position gets reset to its initial value and simultaneously its velocity gets randomized. We study the effects of resetting on the distribution P(Q,t) of the integrated particle current Q up to time t through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution P&#95;{an}(Q,t) approaches its stationary limit uniformly for all Q. In contrast, the quenched distribution P&#95;{qu}(Q,t) attains its stationary form for QQ&#95;{crit}(t). We show that Q&#95;{crit}(t) increases linearly with t for large t. On the scale where Q∼Q&#95;{crit}(t), we show that P&#95;{qu}(Q,t) has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as 10^{-14000}, we were able to compute numerically this nonanalytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.

We numerically study the dynamics of the SIR disease model on small-world networks by using a large-deviation approach. This allows us to obtain the probability density function of the total fraction of infected nodes and of the maximum fraction of simultaneously infected nodes down to very small probability densities like 10^{-2500}. We analyze the structure of the disease dynamics and observed three regimes in all probability density functions, which correspond to quick mild, quick extremely severe, and sustained severe dynamical evolutions, respectively. Furthermore, the mathematical rate functions of the densities are investigated. The results indicate that the so-called large-deviation property holds for the SIR model. Finally, we measured correlations with other quantities like the duration of an outbreak or the peak position of the fraction of infections, also in the rare regions which are not accessible by standard simulation techniques.

Energy grids play an important role in modern society. In recent years, there was a shift from using few central power sources to using many small power sources, due to efforts to increase the percentage of renewable energies. Therefore, the properties of extremely stable and unstable networks are of interest. In this paper, distributions of the basin stability, a nonlinear measure to quantify the ability of a power grid to recover from perturbations, and its correlations with other measurable quantities, namely, diameter, flow backup capacity, power-sign ratio, universal order parameter, biconnected component, clustering coefficient, two core, and leafs, are studied. The energy grids are modeled by an Erdős-Rényi random graph ensemble and a small-world graph ensemble, where the latter is defined in such a way that it does not exhibit dead ends. Using large-deviation techniques, we reach very improbable power grids that are extremely stable as well as ones that are extremely unstable. The 1/t-algorithm, a variation of Wang-Landau, which does not suffer from error saturation, and additional entropic sampling are used to achieve good precision even for very small probabilities ranging over eight decades.