We study the inequality of citations received for different publications of various researchers and Nobel laureates in Physics, Chemistry, Medicine and Economics using Google Scholar data from 2012 to 2024. Citation distributions are found to be highly unequal, with even greater disparity among Nobel laureates. Measures of inequality, such as the Gini and Kolkata indices, emerge as useful indicators for distinguishing Nobel laureates from others. Such high inequality corresponds to growing critical fluctuations, suggesting that excellence aligns with an imminent (self-organized dynamical) critical point. Additionally, Nobel laureates exhibit systematically lower values of the Tsallis–Pareto parameter b and Shannon entropy, indicating more structured citation distributions. We also analyze the inequality in Olympic medal tallies across countries and find similar levels of disparity. Our results suggest that inequality measures can serve as proxies for competitiveness and excellence.

We study the inequality of citations received for different publications of various researchers and Nobel laureates in Physics, Chemistry, Medicine and Economics usingtheir Google Scholar data for the period 2012-2024. Our findings reveal that citationdistributions are highly unequal, with even greater disparity among the Nobel laureates. We then show that measures of inequality, such as Gini and Kolkata indices, couldbe useful indicators for distinguishing the Nobel laureates from the others. It may benoted, such a high level of inequality corresponds to the growing critical fluctuations,indicating that excellence corresponds to an imminent (self-organized dynamical) critical point. We also analyze the inequality in the medal tally of different countries in thesummer and winter Olympic games over the years, and find that a similar level of highinequality exists there as well. Our results indicate that inequality measures can serveas proxies for competitiveness and excellence.

We show that the inequality in the divergent acoustic energy release rate in quasistatically compressed nanoporous materials can be used as a precursor to failure. A quantification of the inequality in the evolution of the energy release rate using social inequality (such as Gini and Kolkata) indices can predict large bursts of energy release. We also verify similar behavior for simulations of viscoelastic fiber bundle models that mimic the strain-hardening dynamics of the samples. The results demonstrate experimental applicability of the precursory signal for fracture with a diverging energy release rate using social inequality indices.

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-Fc|N1/d?], 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 ? 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.

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.

We study the temporal evolution of avalanches in the fiber bundle model of disordered solids, when the model is gradually driven towards the critical breakdown point. We use two types of loading protocols: (i) quasistatic loading and (ii) loading by a discrete amount. In the quasistatic loading, where the load is increased by the minimum amount needed to initiate an avalanche, the temporal shapes of avalanches are asymmetric away from the critical point and become symmetric as the critical point is approached. A measure of asymmetry (A) follows a universal form A?(?-?c)?, with ??0.25, where ? is the load per fiber and ?c is the critical load per fiber. This behavior is independent of the disorder present in the system in terms of the individual failure threshold values. Thus it is possible to use this asymmetry measure as a precursor to imminent failure. For the case of discrete loading, the load is always increased by a fixed amount. The dynamics of the model in this case can be solved in the mean field limit. It shows that the avalanche shapes always remain asymmetric. We also present a variable range load sharing version of this case, where the results remain qualitatively similar. © 2024 American Physical Society.

We have numerically studied a mean-field fiber bundle model of fracture at a non-zero temperature and acted upon by a constant external tensile stress. The individual fibers fail due to creep-like dynamics that lead up to a catastrophic breakdown. We quantify the variations in sizes of the resulting avalanches by calculating the Lorenz function and two inequality indices – Gini (g) and Kolkata (k) indices – derived from the Lorenz function. We show that the two indices cross just prior to the failure point when the dynamics goes through intermittent avalanches. For a continuous failure dynamics (finite numbers of fibers breaking at each time step), the crossing does not happen. However, in that phase, the usual prediction method i.e., linear relation between the time of minimum strain-rate (time at which rate of fiber breaking is the minimum) and failure time, holds. The boundary between continuous and intermittent dynamics is very close to the boundary between crossing and non-crossing of the two indices in the temperature-stress phase space, both drawn from independent analytical calculations and are verified by numerical simulations

Failure statistics of banks in the US show that their sizes are highly unequal (ranging from a few tens of thousands to over a billion dollars) and also, they come in “waves” of intermittent activities. This motivates a self-organized critical picture for the interconnected banking network. For such dynamics, recent developments in studying the inequality of the events, measured through the well-known Gini index and the more recently introduced Kolkata index, have been proved to be fruitful in anticipating large catastrophic events. In this chapter we review such developments for catastrophic failures using a simple model called the fiber bundle model. We then analyse the failure data of banks in terms of the inequality indices and study a simple variant of the fiber bundle model to analyse the same. It appears, both from the data and the model, that coincidence of these two indices signal a systemic risk in the network.

Social inequalities are ubiquitous, and here we show that the values of the Gini (g) and Kolkata (k) indices, two generic inequality indices, approach each other (starting from g=0 and k=0.5 for equality) as the competitions grow in various social institutions like markets, universities and elections. It is further shown that these two indices become equal and stabilize at a value (at g=kâ 0.87) under unrestricted competitions. We propose to view this coincidence of inequality indices as a generalized version of the (more than a) century old 80-20 law of Pareto. Furthermore, the coincidence of the inequality indices noted here is very similar to the ones seen before for self-organized critical (SOC) systems. The observations here, therefore, stand as a quantitative support toward viewing interacting socio-economic systems in the framework of SOC, an idea conjectured for years.

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 (Formula presented.) fraction of the ‘people’. Our findings suggest that both the Gini index and the Kolkata index tend to converge to similar values (around (Formula presented.), starting from the point of perfect equality, where (Formula presented.) and (Formula presented.)) 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 ((Formula presented.)), 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 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.

This review presents an overview of the current research in kinetic exchange models for opinion formation in a society. The review begins with a brief introduction to previous models and subsequently provides an in-depth discussion of the progress achieved in the Biswas-Chatterjee-Sen model proposed in 2012, also known as the BChS model in some later research publications. The unique feature of the model is its inclusion of negative interaction between agents. The review covers various topics, including phase transitions between different opinion states, critical behavior dependent on various parameters, and applications in realistic scenarios such as the United States presidential election and Brexit.

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.

A popular measure for citation inequalities of individual scientists has been the Hirsch index (h). If for any scientist the number n of citations is plotted against the serial number n of the papers having those many citations (when the papers are ordered from the highest cited to the lowest), then h corresponds to the nearest lower integer value of n below the fixed point of the non-linear citation function (or given by n = h = n if both n and n are a dense set of integers near the h value). The same index can be estimated (from h = s = n) for the avalanche or cluster of size (s) distributions (n) in the elastic fiber bundle or percolation models. Another such inequality index called the Kolkata index (k) says that (1 ? k) fraction of papers attract k fraction of citations (k = 0.80 corresponds to the 80–20 law of Pareto). We find, for stress (?), the lattice occupation probability (p) or the Kolkata Index (k) near the bundle failure threshold (?) or percolation threshold (p) or the critical value of the Kolkata Index k a good fit to Widom–Stauffer like scaling (Formula presented.) = (Formula presented.), (Formula presented.) or (Formula presented.), respectively, with the asymptotically defined scaling function f, for systems of size N (total number of fibers or lattice sites) or N (total number of citations), and ? denoting the appropriate scaling exponent. We also show that if the number (N) of members of parliaments or national assemblies of different countries (with population N) is identified as their respective h ? indexes, then the data fit the scaling relation (Formula presented.), resolving a major recent controversy.

Statistical physicists and social scientists both extensively study some characteristic features of the unequal distributions of energy, cluster, or avalanche sizes and of income, wealth, etc., among the particles (or sites) and population, respectively. While physicists concentrate on the self-similar (fractal) structure (and the characteristic exponents) of the largest (percolating) cluster or avalanche, social scientists study the inequality indices such as Gini and Kolkata, given by the non-linearity of the Lorenz function representing the cumulative fraction of the wealth possessed by different fractions of the population. Here, using results from earlier publications and some new numerical and analytical results, we reviewed how the above-mentioned social inequality indices, when extracted from the unequal distributions of energy (in kinetic exchange models), cluster sizes (in percolation models), or avalanche sizes (in self-organized critical or fiber bundle models) can help in a major way in providing precursor signals for an approaching critical point or imminent failure point. Extensive numerical and some analytical results have been discussed.

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.

We have studied few social inequality measures associated with the sub-critical dynamical features (measured in terms of the avalanche size distributions) of four self-organized critical models while the corresponding systems approach their respective stationary critical states. It has been observed that these inequality measures (specifically the Gini and Kolkata indices) exhibit nearly universal values though the models studied here are widely different, namely the Bak–Tang–Wiesenfeld sandpile, the Manna sandpile and the quenched Edwards–Wilkinson interface, and the fiber bundle interface. These observations suggest that the self-organized critical systems have broad similarity in terms of these inequality measures. A comparison with similar earlier observations in the data of socio-economic systems with unrestricted competitions suggest the emergent inequality as a result of the possible proximity to the self-organized critical states.

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'.

We observe the failure process of a fiber bundle model with a variable stress release range, ?, and higher the value of ?, lower the stress release range. By tuning ? from low to high, it is possible to go from the mean-field (MF) limit of the model to the local load-sharing (LLS) limit where local stress concentration plays a crucial role. In the MF limit, individual avalanches (number of fibers breaking in going from one stable state to the next, s) and the corresponding energies E emitted during those avalanches have one-to-one linear correlation. This results in the same size distributions for both avalanches (P(s)) and energy bursts (Q(E)): a scale-free distribution with a universal exponent value of ?5/2. With increasing ?, the model enters the LLS limit beyond some ?. In this limit, due to the presence of local stress concentrations around a damaged region, such correlation C(?) between s and E decreases, i.e., a smaller avalanche can emit a large amount of energy or a large avalanche may emit a small amount of energy. The nature of the decrease in the correlation between s and E depends highly on the dimension of the bundle. In this work, we study the decrease in the correlation between avalanche size and the corresponding energy bursts with an increase in the load redistribution localization in the fiber bundle model in one and two dimensions. Additionally, we note that the energy size distribution remains scale-free for all values of ?, whereas the avalanche size distribution becomes exponential for ? > ?.

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’.

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.

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.

We show using scaling arguments and Monte Carlo simulations that a class of binary interacting models of opinion evolution belong to the Ising universality class in presence of an annealed noise term of finite amplitude. While the zero noise limit is known to show an active-absorbing transition, addition of annealed noise induces a continuous order–disorder transition with Ising universality class in the infinite-range (mean field) limit of the models.

The electoral college of voting system for the US presidential election is analogous to a coarse graining procedure commonly used to study phase transitions in physical systems. In a recent paper, opinion dynamics models manifesting a phase transition, were shown to be able to explain the cases when a candidate winning more number of popular votes could still lose the general election on the basis of the electoral college system. We explore the dependence of such possibilities on various factors like the number of states and total population (i.e., system sizes) and get an interesting scaling behavior. In comparison with the real data, it is shown that the probability of the minority win, calculated within the model assumptions, is indeed near the highest possible value. In addition, we also implement a two step coarse graining procedure, relevant for both opinion dynamics and information theory.

We study the local load sharing fiber bundle model and its energy burst statistics. While it is known that the avalanche size distribution of the model is exponential, we numerically show here that the avalanche size (s) and the corresponding average energy burst (?E?) in this version of the model have a non-linear relation (?E? ~ s). Numerical results indicate that ? ? 2.5 universally for different failure threshold distributions. With this numerical observation, it is then possible to show that the energy burst distribution is a power law, with a universal exponent value of ?(? + 1).

We discuss the cooperative failure dynamics in the fiber bundle model where the individual elements or fibers are Hookean springs that have identical spring constants but different breaking strengths. When the bundle is stressed or strained, especially in the equal-load-sharing scheme, the load supported by the failed fiber gets shared equally by the rest of the surviving fibers. This mean-field-type statistical feature (absence of fluctuations) in the load-sharing mechanism helped major analytical developments in the study of breaking dynamics in the model and precise comparisons with simulation results. We intend to present a brief review on these developments.

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 , where fraction of the population possesses 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 and the Hirsch 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 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 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 (), redefined for such avalanche statistics.

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.

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.

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 Z2-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 L2 or a long-lived metastable state (termed a "domain-wall state") for which formation of consensus takes a time scaling as L3.6. 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.

Dynamics of power outages remain an unpredictable hazard in spite of expensive consequences. While the operations of the components of power grids are well understood, the emergent complexity due to their interconnections gives rise to intermittent outages, and power-law statistics. Here we demonstrate that there are additional patterns in the outage size distributions that indicate the proximity of a grid to a catastrophic failure point. Specifically, the analysis of the data for the U.S. between 2002 and 2017 shows a significant anti-correlation between the exponent value of the power-law outage size distribution and the load carried by the grid. The observation is surprisingly similar to dependences noted for failure dynamics in other multi-component complex systems such as sheared granulates and earthquakes, albeit under much different physical conditions. This inspires a generic threshold-activated model, simulated in realistic network topologies, which can successfully reproduce the exponent variation in a similar range. Given sufficient data, the methods proposed here can be used to indicate proximity to failure points and forecast probabilities of major blackouts with a non-intrusive measurement of intermittent grid outages.

We show that the failure time ?f in the fiber bundle model, taken as a prototype of heterogeneous materials, depends crucially on the strength of the disorder ? and the stress release range R in the model. In the mean-field limit, the distribution of ?f is log-normal. In this limit, the average failure time shows the variation ?f?L?(?), where L is the system size. The exponent ? has a constant value above a critical disorder ?c (=1/6), while it is an increasing function of ? in the region ?<?c. On the other hand, in the limit where the local stress concentration plays a crucial role, we observe the scaling ?f?L?(?)?(R/L1-?(?)), where R is the stress release range. We find that the crossover length scale Rc, between the above two limiting cases, scales as Rc?L1-?(?).

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.