Prediction of depinning transitions in interface models using Gini and Kolkata indices.

The intermittent dynamics of driven interfaces through disordered media and its subsequent depinning for large enough driving force is a common feature for a myriad of diverse systems, starting from mode-I fracture, vortex lines in superconductors, and magnetic domain walls to invading fluid in a porous medium, to name a few. In this work, we outline a framework that can give a precursory signal of the imminent depinning transition by monitoring the variations in sizes or the inequality of the intermittent responses of a system that are seen prior to the depinning point. In particular, we use measures traditionally used to quantify economic inequality, i.e., the Gini index and the Kolkata index, for the case of the unequal responses of precritical systems. The crossing point of these two indices serves as a precursor to imminent depinning. Given a scale-free size distribution of the responses, we calculate the expressions for these indices, evaluate their crossing points, and give a recipe for forecasting depinning transitions. We apply this method to the Edwards-Wilkinson, Kardar-Parisi-Zhang, and fiber bundle model interface with variable interaction strengths and quenched disorder. The results are applicable for any interface dynamics undergoing a depinning transition. The results also explain previously observed near-universal values of Gini and Kolkata indices in self-organized critical systems.

Finding critical points and correlation length exponents using finite size scaling of Gini index.

The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through the Gini index (g) therefore leads to a scaling form g=G[|F-F_{c}|N^{1/dν}], where F denotes the driving parameter for the transition (e.g., temperature T for ferromagnetic to paramagnetic transition, or lattice occupation probability p in percolation), N is the system size, d is the spatial dimension and ν is the correlation length exponent. We demonstrate the scaling for the Ising model in two and three dimensions, site percolation on square lattice, and the fiber bundle model of fracture.

Critical Scaling through Gini Index.

In the systems showing critical behavior, various response functions have a singularity at the critical point. Therefore, as the driving field is tuned toward its critical value, the response functions change drastically, typically diverging with universal critical exponents. In this Letter, we quantify the inequality of response functions with measures traditionally used in economics, namely by constructing a Lorenz curve and calculating the corresponding Gini index. The scaling of such a response function, when written in terms of the Gini index, shows singularity at a point that is at least as universal as the corresponding critical exponent. The critical scaling, therefore, becomes a single parameter fit, which is a considerable simplification from the usual form where the critical point and critical exponents are independent. We also show that another measure of inequality, the Kolkata index, crosses the Gini index at a point just prior to the critical point. Therefore, monitoring these two inequality indices for a system where the critical point is not known can produce a precursory signal for the imminent criticality. This could be useful in many systems, including that in condensed matter, bio- and geophysics to atmospheric physics. The generality and numerical validity of the calculations are shown with the Monte Carlo simulations of the two dimensional Ising model, site percolation on square lattice, and the fiber bundle model of fracture.

Inequality of avalanche sizes in models of fracture.

Prediction of an imminent catastrophic event in a driven disordered system is of paramount importance-from the laboratory scale controlled fracture experiment to the largest scale of mechanical failure, i.e., earthquakes. It has long been conjectured that the statistical regularities in the energy emission time series mirror the "health" of such driven systems and hence have the potential for forecasting imminent catastrophe. Among other statistical regularities, a measure of how unequal avalanche sizes are is potentially a crucial indicator of imminent failure. The inequalities of avalanche sizes are quantified using inequality indices traditionally used in socioeconomic systems: the Gini index g, the Hirsch index h, and the Kolkata index k. It is shown analytically (for the mean-field case) and numerically (for the non-mean-field case) with models of quasi-brittle materials that the indices show universal behavior near the breaking points in such models and hence could serve as indicators of imminent breakdown of stressed disordered systems.

Sandpile Universality in Social Inequality: Gini and Kolkata Measures.

Social inequalities are ubiquitous and evolve towards a universal limit. Herein, we extensively review the values of inequality measures, namely the Gini (g) index and the Kolkata (k) index, two standard measures of inequality used in the analysis of various social sectors through data analysis. The Kolkata index, denoted as k, indicates the proportion of the 'wealth' owned by (1-k) fraction of the 'people'. Our findings suggest that both the Gini index and the Kolkata index tend to converge to similar values (around g=k≈0.87, starting from the point of perfect equality, where g=0 and k=0.5) as competition increases in different social institutions, such as markets, movies, elections, universities, prize winning, battle fields, sports (Olympics), etc., under conditions of unrestricted competition (no social welfare or support mechanism). In this review, we present the concept of a generalized form of Pareto's 80/20 law (k=0.80), where the coincidence of inequality indices is observed. The observation of this coincidence is consistent with the precursor values of the g and k indices for the self-organized critical (SOC) state in self-tuned physical systems such as sand piles. These results provide quantitative support for the view that interacting socioeconomic systems can be understood within the framework of SOC, which has been hypothesized for many years. These findings suggest that the SOC model can be extended to capture the dynamics of complex socioeconomic systems and help us better understand their behavior.

Prediction of imminent failure using supervised learning in a fiber bundle model.

Prediction of a breakdown in disordered solids under external loading is a question of paramount importance. Here we use a fiber bundle model for disordered solids and record the time series of the avalanche sizes and energy bursts. The time series contain statistical regularities that not only signify universality in the critical behavior of the process of fracture, but also reflect signals of proximity to a catastrophic failure. A systematic analysis of these series using supervised machine learning can predict the time to failure. Different features of the time series become important in different variants of training samples. We explain the reasons for such a switch over of importance among different features. We show that inequality measures for avalanche time series play a crucial role in imminent failure predictions, especially for imperfect training sets, i.e., when simulation parameters of training samples differ considerably from those of the testing samples. We also show the variation of predictability of the system as the interaction range and strengths of disorders are varied in the samples, varying the failure mode from brittle to quasibrittle (with interaction range) and from nucleation to percolation (with disorder strength). The effectiveness of the supervised learning is best when the samples just enter the quasibrittle mode of failure showing scale-free avalanche size distributions.

Opinion dynamics: public and private.

We study here the dynamics of opinion formation in a society where we take into account the internally held beliefs and externally expressed opinions of the individuals, which are not necessarily the same at all times. While these two components can influence one another, their difference, both in dynamics and in the steady state, poses interesting scenarios in terms of the transition to consensus in the society and characterizations of such consensus. Here we study this public and private opinion dynamics and the critical behaviour of the consensus forming transitions, using a kinetic exchange model. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

Kinetic exchange models of societies and economies.

The statistical nature of collective human behaviour in a society is a topic of broad current interest. From formation of consensus through exchange of ideas, distributing wealth through exchanges of money, traffic flows, growth of cities to spread of infectious diseases, the application range of such collective responses cuts across multiple disciplines. Kinetic models have been an elegant and powerful tool to explain such collective phenomena in a myriad of human interaction-based problems, where an energy consideration for dynamics is generally inaccessible. Nonetheless, in this age of Big Data, seeking empirical regularities emerging out of collective responses is a prominent and essential approach, much like the empirical thermodynamic principles preceding quantitative foundations of statistical mechanics. In this introductory article of the theme issue, we will provide an overview of the field of applications of kinetic theories in different socio-economic contexts and its recent boosting topics. Moreover, we will put the contributions to the theme issue in an appropriate perspective. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

Machine learning predictions of COVID-19 second wave end-times in Indian states.

The estimate of the remaining time of an ongoing wave of epidemic spreading is a critical issue. Due to the variations of a wide range of parameters in an epidemic, for simple models such as Susceptible-Infected-Removed (SIR) model, it is difficult to estimate such a time scale. On the other hand, multidimensional data with a large set attributes are precisely what one can use in statistical learning algorithms to make predictions. Here we show, how the predictability of the SIR model changes with various parameters using a supervised learning algorithm. We then estimate the condition in which the model gives the least error in predicting the duration of the first wave of the COVID-19 pandemic in different states in India. Finally, we use the SIR model with the above mentioned optimal conditions to generate a training data set and use it in the supervised learning algorithm to estimate the end-time of the ongoing second wave of the pandemic in different states in India.

Social inequality analysis of fiber bundle model statistics and prediction of materials failure.

Inequalities are abundant in a society with a number of agents competing for a limited amount of resources. Statistics on such social inequalities are usually represented by the Lorenz function L(p), where p fraction of the population possesses L(p) fraction of the total wealth, when the population is arranged in ascending order of their wealth. Similarly, in scientometrics, such inequalities can be represented by a plot of the citation count versus the respective number of papers by a scientist, again arranged in ascending order of their citation counts. Quantitatively, these inequalities are captured by the corresponding inequality indices, namely, the Kolkata k and the Hirsch h indices, given by the fixed points of these nonlinear (Lorenz and citation) functions. In statistical physics of criticality, the fixed points of the renormalization group generator functions are studied in their self-similar limit, where their (fractal) structure converges to a unique form (macroscopic in size and lone). The statistical indices in social science, however, correspond to the fixed points where the values of the generator function (wealth or citation sizes) are commensurately abundant in fractions or numbers (of persons or papers). It has already been shown that under extreme competitions in markets or at universities, the k index approaches a universal limiting value, as the dynamics of competition progresses. We introduce and study these indices for the inequalities of (prefailure) avalanches, given by their nonlinear size distributions in fiber bundle models of nonbrittle materials. We show how prior knowledge of the terminal and (almost) universal value of the k index for a wide range of disorder parameters can help in predicting an imminent catastrophic breakdown in the model. This observation is also complemented by noting a similar (but not identical) behavior of the Hirsch index (h), redefined for such avalanche statistics.

Optimization strategies of human mobility during the COVID-19 pandemic: A review.

The impact of the ongoing COVID-19 pandemic is being felt in all spheres of our lives - cutting across the boundaries of nation, wealth, religions or race. From the time of the first detection of infection among the public, the virus spread though almost all the countries in the world in a short period of time. With humans as the carrier of the virus, the spreading process necessarily depends on the their mobility after being infected. Not only in the primary spreading process, but also in the subsequent spreading of the mutant variants, human mobility plays a central role in the dynamics. Therefore, on one hand travel restrictions of varying degree were imposed and are still being imposed, by various countries both nationally and internationally. On the other hand, these restrictions have severe fall outs in businesses and livelihood in general. Therefore, it is an optimization process, exercised on a global scale, with multiple changing variables. Here we review the techniques and their effects on optimization or proposed optimizations of human mobility in different scales, carried out by data driven, machine learning and model approaches.

Parallel Minority Game and it's application in movement optimization during an epidemic.

We introduce a version of the Minority Game where the total number of available choices is D > 2 , but the agents only have two available choices to switch. For all agents at an instant in any given choice, therefore, the other choice is distributed between the remaining D - 1 options. This brings in the added complexity in reaching a state with the maximum resource utilization, in the sense that the game is essentially a set of MG that are coupled and played in parallel. We show that a stochastic strategy, used in the MG, works well here too. We discuss the limits in which the model reduces to other known models. Finally, we study an application of the model in the context of population movement between various states within a country during an ongoing epidemic. we show that the total infected population in the country could be as low as that achieved with a complete stoppage of inter-region movements for a prolonged period, provided that the agents instead follow the above mentioned stochastic strategy for their movement decisions between their two choices. The objective for an agent is to stay in the lower infected state between their two choices. We further show that it is the agents moving once between any two states, following the stochastic strategy, who are less likely to be infected than those not having (or not opting for) such a movement choice, when the risk of getting infected during the travel is not considered. This shows the incentive for the moving agents to follow the stochastic strategy.

Failure processes of cemented granular materials.

The mechanics of cohesive or cemented granular materials is complex, combining the heterogeneous responses of granular media, like force chains, with clearly defined material properties. Here we use a discrete element model simulation, consisting of an assemblage of elastic particles connected by softer but breakable elastic bonds, to explore how this class of material deforms and fails under uniaxial compression. We are particularly interested in the connection between the microscopic interactions among the grains or particles and the macroscopic material response. To this end, the properties of the particles and the stiffness of the bonds are matched to experimental measurements of a cohesive granular medium with tunable elasticity. The criterion for breaking a bond is also based on an explicit Griffith energy balance, with realistic surface energies. By varying the initial volume fraction of the particle assembles we show that this simple model reproduces a wide range of experimental behaviors, both in the elastic limit and beyond it. These include quantitative details of the distinct failure modes of shear-banding, ductile failure, and compaction banding or anticracks, as well as the transitions between these modes. The present work, therefore, provides a unified framework for understanding the failure of porous materials such as sandstone, marble, powder aggregates, snow, and foam.

Prediction of creep failure time using machine learning.

A subcritical load on a disordered material can induce creep damage. The creep rate in this case exhibits three temporal regimes viz. an initial decelerating regime followed by a steady-state regime and a stage of accelerating creep that ultimately leads to catastrophic breakdown. Due to the statistical regularities in the creep rate, the time evolution of creep rate has often been used to predict residual lifetime until catastrophic breakdown. However, in disordered samples, these efforts met with limited success. Nevertheless, it is clear that as the failure is approached, the damage become increasingly spatially correlated, and the spatio-temporal patterns of acoustic emission, which serve as a proxy for damage accumulation activity, are likely to mirror such correlations. However, due to the high dimensionality of the data and the complex nature of the correlations it is not straightforward to identify the said correlations and thereby the precursory signals of failure. Here we use supervised machine learning to estimate the remaining time to failure of samples of disordered materials. The machine learning algorithm uses as input the temporal signal provided by a mesoscale elastoplastic model for the evolution of creep damage in disordered solids. Machine learning algorithms are well-suited for assessing the proximity to failure from the time series of the acoustic emissions of sheared samples. We show that materials are relatively more predictable for higher disorder while are relatively less predictable for larger system sizes. We find that machine learning predictions, in the vast majority of cases, perform substantially better than other prediction approaches proposed in the literature.

Flory-like statistics of fracture in the fiber bundle model as obtained via Kolmogorov dispersion for turbulence: A conjecture.

It has long been conjectured that (rapid) fracture propagation dynamics in materials and turbulent motion of fluids are two manifestations of the same physical process. The universality class of turbulence (Kolmogorov dispersion, in particular) is conjectured to be identifiable with the Flory statistics for linear polymers (self-avoiding walks on lattices). These help us to relate fracture statistics to those of linear polymers (Flory statistics). The statistics of fracture in the fiber bundle model (FBM) are now well studied and many exact results are now available for the equal-load-sharing (ELS) scheme. Yet, the correlation length exponent in this model was missing and we show here how the correspondence between fracture statistics and the Flory mapping of Kolmogorov statistics for turbulence helps us to make a conjecture about the value of the correlation length exponent for fracture in the ELS limit of FBM and, also, about the upper critical dimension. In addition, the fracture avalanche size exponent values at lower dimensions (as estimated from such mapping to Flory statistics) also compare well with the observations.

Long route to consensus: Two-stage coarsening in a binary choice voting model.

Formation of consensus, in binary yes-no type of voting, is a well-defined process. However, even in presence of clear incentives, the dynamics involved can be incredibly complex. Specifically, formations of large groups of similarly opinionated individuals could create a condition of "support-bubbles" or spontaneous polarization that renders consensus virtually unattainable (e.g., the question of the UK exiting the EU). There have been earlier attempts in capturing the dynamics of consensus formation in societies through simple Z_{2}-symmetric models hoping to capture the essential dynamics of average behavior of a large number of individuals in a statistical sense. However, in absence of external noise, they tend to reach a frozen state with fragmented and polarized states, i.e., two or more groups of similarly opinionated groups with frozen dynamics. Here we show in a kinetic exchange opinion model considered on L×L square lattices, that while such frozen states could be avoided, an exponentially slow approach to consensus is manifested. Specifically, the system could either reach consensus in a time that scales as L^{2} or a long-lived metastable state (termed a "domain-wall state") for which formation of consensus takes a time scaling as L^{3.6}. The latter behavior is comparable to some voterlike models with intermediate states studied previously. The late-time anomaly in the timescale is reflected in the persistence probability of the model. Finally, the interval of zero crossing of the average opinion, i.e., the time interval over which the average opinion does not change sign, is shown to follow a scale-free distribution, which is compared with that seen in the opinion surveys regarding Brexit and associated issues since the late 1970s. The issue of minority spreading is also addressed by calculating the exit probability.

Avalanche dynamics in hierarchical fiber bundles.

Heterogeneous materials are often organized in a hierarchical manner, where a basic unit is repeated over multiple scales. The structure then acquires a self-similar pattern. Examples of such structure are found in various biological and synthetic materials. The hierarchical structure can have significant consequences for the failure strength and the mechanical response of such systems. Here we consider a fiber bundle model with hierarchical structure and study the avalanche dynamics exhibited by the model during the approach to failure. We show that the failure strength of the model generally decreases in a hierarchical structure, as opposed to the situation where no such hierarchy exists. However, we also report a special arrangement of the hierarchy for which the failure threshold could be substantially above that of a nonhierarchical reference structure.

Mapping heterogeneities through avalanche statistics.

Avalanche statistics of various threshold-activated dynamical systems are known to depend on the magnitude of the drive, or stress, on the system. Such dependences exist for earthquake size distributions, in sheared granular avalanches, laboratory-scale fracture and also in the outage statistics of power grids. In this work, we model threshold-activated avalanche dynamics and investigate the time required to detect local variations in the ability of model elements to bear stress. We show that the detection time follows a scaling law where the scaling exponents depend on whether the feature that is sought is either weaker, or stronger, than its surroundings. We then look at earthquake data from Sumatra and California, demonstrate the trade-off between the spatial resolution of a map of earthquake exponents (i.e. the b-values of the Gutenberg-Richter Law) and the accuracy of those exponents, and suggest a means to maximize both.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Statistical physics of fracture and earthquakes.

Manifestations of emergent properties in stressed disordered materials are often the result of an interplay between strong perturbations in the stress field around defects. The collective response of a long-ranged correlated multi-component system is an ideal playing field for statistical physics. Hence, many aspects of such collective responses in widely spread length and energy scales can be addressed by the tools of statistical physics. In this theme issue, some of these aspects are treated from various angles of experiments, simulations and analytical methods, and connected together by their common base of complex-system dynamics.This article is part of the theme issue 'Statistical physics of fracture and earthquakes' .

Record-breaking statistics near second-order phase transitions.

When a quantity reaches a value higher (or lower) than its value at any time before, it is said to have made a record. We numerically study the statistical properties of records in the time series of order parameters in different models near their critical points. Specifically, we choose the transversely driven Edwards-Wilkinson model for interface depinning in (1+1) dimensions and the Ising model in two dimensions, as paradigmatic and simple examples of nonequilibrium and equilibrium critical behaviors, respectively. The total number of record-breaking events in the time series of the order parameters of the models show maxima when the system is near criticality. The number of record-breaking events and associated quantities, such as the distribution of the waiting time between successive record events, show power-law scaling near the critical point. The exponent values are specific to the universality classes of the respective models. Such behaviors near criticality can be used as a precursor to imminent criticality, i.e., abrupt and catastrophic changes in the system. Due to the extreme nature of the records, its measurements are relatively free of detection errors and thus provide a clear signal regarding the state of the system in which they are measured.