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Concept: Pareto distribution


The largest cities, the most frequently used words, the income of the richest countries, and the most wealthy billionaires, can be all described in terms of Zipf’s Law, a rank-size rule capturing the relation between the frequency of a set of objects or events and their size. It is assumed to be one of many manifestations of an underlying power law like Pareto’s or Benford’s, but contrary to popular belief, from a distribution of, say, city sizes and a simple random sampling, one does not obtain Zipf’s law for the largest cities. This pathology is reflected in the fact that Zipf’s Law has a functional form depending on the number of events N. This requires a fundamental property of the sample distribution which we call ‘coherence’ and it corresponds to a ‘screening’ between various elements of the set. We show how it should be accounted for when fitting Zipf’s Law.

Concepts: Sample, Sample size, Property, Zipf's law, Pareto distribution, Lorenz curve, Rank-size distribution


The world food system is globalized and inter-connected, in which trade plays an increasingly important role in facilitating food availability. In this paper, we present a novel application of network analysis to domestic food flows within the USA, a country with global importance as a major agricultural producer and trade power. We find normal node degree distributions and Weibull node strength and betweenness centrality distributions. An unassortative network structure with high clustering coefficients exists. These network properties indicate that the USA food flow network is highly social and well-mixed. However, a power law relationship between node betweenness centrality and node degree indicates potential network vulnerability to the disturbance of key nodes. We perform an equality analysis which serves as a benchmark for global food trade, where the Gini coefficient = 0.579, Lorenz asymmetry coefficient = 0.966, and Hoover index = 0.442. These findings shed insight into trade network scaling and proxy free trade and equitable network architectures.

Concepts: International trade, Social network, Welfare economics, Flow network, Pareto distribution, Income inequality metrics, Income distribution, Theil index



We propose a simple agent-based model on a network to conceptualize the allocation of limited wealth among more abundant expectations at the interplay of power, frustration, and initiative. Concepts imported from the statistical physics of frustrated systems in and out of equilibrium allow us to compare subjective measures of frustration and satisfaction to collective measures of fairness in wealth distribution, such as the Lorenz curve and the Gini index. We find that a completely libertarian, law-of-the-jungle setting, where every agent can acquire wealth from or lose wealth to anybody else invariably leads to a complete polarization of the distribution of wealth vs. opportunity. This picture is however dramatically ameliorated when hard constraints are imposed over agents in the form of a limiting network of transactions. There, an out of equilibrium dynamics of the networks, based on a competition between power and frustration in the decision-making of agents, leads to network coevolution. The ratio of power and frustration controls different dynamical regimes separated by kinetic transitions and characterized by drastically different values of equality. It also leads, for proper values of social initiative, to the emergence of three self-organized social classes, lower, middle, and upper class. Their dynamics, which appears mostly controlled by the middle class, drives a cyclical regime of dramatic social changes.

Concepts: Distribution, Economic inequality, Distribution of wealth, Social class, Pareto distribution, Lorenz curve, Working class, Wealth condensation


Understanding the exploration patterns of foragers in the wild provides fundamental insight into animal behavior. Recent experimental evidence has demonstrated that path lengths (distances between consecutive turns) taken by foragers are well fitted by a power law distribution. Numerous theoretical contributions have posited that “Lévy random walks”-which can produce power law path length distributions-are optimal for memoryless agents searching a sparse reward landscape. It is unclear, however, whether such a strategy is efficient for cognitively complex agents, from wild animals to humans. Here, we developed a model to explain the emergence of apparent power law path length distributions in animals that can learn about their environments. In our model, the agent’s goal during search is to build an internal model of the distribution of rewards in space that takes into account the cost of time to reach distant locations (i.e., temporally discounting rewards). For an agent with such a goal, we find that an optimal model of exploration in fact produces hyperbolic path lengths, which are well approximated by power laws. We then provide support for our model by showing that humans in a laboratory spatial exploration task search space systematically and modify their search patterns under a cost of time. In addition, we find that path length distributions in a large dataset obtained from free-ranging marine vertebrates are well described by our hyperbolic model. Thus, we provide a general theoretical framework for understanding spatial exploration patterns of cognitively complex foragers.

Concepts: Natural environment, Distance, Pareto distribution, Power law, Power laws


The distribution of firms' growth and firms' sizes is a topic under intense scrutiny. In this paper, we show that a thermodynamic model based on the maximum entropy principle, with dynamical prior information, can be constructed that adequately describes the dynamics and distribution of firms' growth. Our theoretical framework is tested against a comprehensive database of Spanish firms, which covers, to a very large extent, Spain’s economic activity, with a total of 1 155 142 firms evolving along a full decade. We show that the empirical exponent of Pareto’s law, a rule often observed in the rank distribution of large-size firms, is explained by the capacity of economic system for creating/destroying firms, and that can be used to measure the health of a capitalist-based economy. Indeed, our model predicts that when the exponent is larger than 1, creation of firms is favoured; when it is smaller than 1, destruction of firms is favoured instead; and when it equals 1 (matching Zipf’s law), the system is in a full macroeconomic equilibrium, entailing ‘free’ creation and/or destruction of firms. For medium and smaller firm sizes, the dynamical regime changes, the whole distribution can no longer be fitted to a single simple analytical form and numerical prediction is required. Our model constitutes the basis for a full predictive framework regarding the economic evolution of an ensemble of firms. Such a structure can be potentially used to develop simulations and test hypothetical scenarios, such as economic crisis or the response to specific policy measures.

Concepts: Scientific method, Economics, Thermodynamics, Prior probability, Economy, Zipf's law, Pareto distribution, Principle of maximum entropy


Using public data (Forbes Global 2000) we show that the asset sizes for the largest global firms follow a Pareto distribution in an intermediate range, that is “interrupted” by a sharp cut-off in its upper tail, where it is totally dominated by financial firms. This flattening of the distribution contrasts with a large body of empirical literature which finds a Pareto distribution for firm sizes both across countries and over time. Pareto distributions are generally traced back to a mechanism of proportional random growth, based on a regime of constant returns to scale. This makes our findings of an “interrupted” Pareto distribution all the more puzzling, because we provide evidence that financial firms in our sample should operate in such a regime. We claim that the missing mass from the upper tail of the asset size distribution is a consequence of shadow banking activity and that it provides an (upper) estimate of the size of the shadow banking system. This estimate-which we propose as a shadow banking index-compares well with estimates of the Financial Stability Board until 2009, but it shows a sharper rise in shadow banking activity after 2010. Finally, we propose a proportional random growth model that reproduces the observed distribution, thereby providing a quantitative estimate of the intensity of shadow banking activity.

Concepts: Economics, Finance, Zipf's law, Pareto distribution, Power law, Bank, Financial services, Pareto principle


MixP is an implementation that uses the Pareto principle to perform genomic prediction. This study was designed to develop two new computing strategies: one strategy for nonMCMC-based MixP (FMixP), and the other one for MCMC-based MixP (MMixP). The difference is that MMixP can estimate variances of SNP effects and the probability that a SNP has a large variance, but FMixP cannot. Simulated data from an international workshop and real data on large yellow croaker were used as the materials for the study. Four Bayesian methods, BayesA, BayesCπ, MMixP and FMixP, were used to compare the predictive results. The results show that BayesCπ, MMixP and FMixP perform better than BayesA for the simulated data, but all methods have very similar predictive abilities for the large yellow croaker. However, FMixP is computationally significantly faster than the MCMC-based methods. Our research may have a potential for the future applications in genomic prediction.

Concepts: Scientific method, Variance, Prediction, Futurology, Future, Forecasting, Bayesian probability, Pareto distribution


The temporal lobe is classically divided in two functional systems: the ventral visual pathway and the medial temporal memory system. However, their functional separation has been challenged by studies suggesting that the medial temporal lobe could be best understood as an extension of the hierarchically organized ventral visual pathway. Our purpose was to investigate (i) whether cerebral regions within the temporal lobe could be grouped into distinct functional assemblies, and (ii) which regions were central within these functional assemblies. We studied low intensity and low frequency electrical stimulations (0.5 mA, 1 Hz, 4 ms) performed during sixteen pre-surgical intracerebral EEG investigations in patients with medically intractable temporal or temporo-occipital lobe epilepsies. Eleven regions of interest were delineated per anatomical landmarks such as gyri and sulci. Effective connectivity based on electrophysiological feature (amplitude) of cortico-cortical evoked potentials (CCEPs) was evaluated and subjected to graph metrics. The amplitudes discriminated one medial module where the hippocampus could act as a signal amplifier. Mean amplitudes of CCEPs in regions of the temporal lobe showed a generalized Pareto distribution of probability suggesting neural synchronies to be self-organized critically. Our description of effective interactions within the temporal lobe provides a regional electrophysiological model of effective connectivity which is discussed in the context of the current hypothesis of pattern completion.

Concepts: Cerebral cortex, Temporal lobe, Cerebrum, Hippocampus, Superior temporal gyrus, Temporal lobe epilepsy, Lobe, Pareto distribution


The Gini index is a measure of the inequality of a distribution that can be derived from Lorenz curves. While commonly used in, e.g., economic research, it suffers from ambiguity via lack of Lorenz dominance preservation. Here, investigation of large sets of empirical distributions of incomes of the World’s countries over several years indicated firstly, that the Gini indices are centered on a value of 33.33% corresponding to the Gini index of the uniform distribution and secondly, that the Lorenz curves of these distributions are consistent with Lorenz curves of log-normal distributions. This can be employed to provide a Lorenz dominance preserving equivalent of the Gini index. Therefore, a modified measure based on log-normal approximation and standardization of Lorenz curves is proposed. The so-called UGini index provides a meaningful and intuitive standardization on the uniform distribution as this characterizes societies that provide equal chances. The novel UGini index preserves Lorenz dominance. Analysis of the probability density distributions of the UGini index of the World’s counties income data indicated multimodality in two independent data sets. Applying Bayesian statistics provided a data-based classification of the World’s countries' income distributions. The UGini index can be re-transferred into the classical index to preserve comparability with previous research.

Concepts: Probability theory, Normal distribution, Welfare economics, Pareto distribution, Lorenz curve, Income inequality metrics, Income distribution, Gini coefficient