PHYSICAL REVIEW E 89, 012111 (2014)

Rank distributions: A panoramic macroscopic outlook Iddo I. Eliazar1,* and Morrel H. Cohen2,† 1

School of Chemistry, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel 2 Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA and Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA (Received 7 September 2013; published 9 January 2014) This paper presents a panoramic macroscopic outlook of rank distributions. We establish a general framework for the analysis of rank distributions, which classifies them into five macroscopic “socioeconomic” states: monarchy, oligarchy-feudalism, criticality, socialism-capitalism, and communism. Oligarchy-feudalism is shown to be characterized by discrete macroscopic rank distributions, and socialism-capitalism is shown to be characterized by continuous macroscopic size distributions. Criticality is a transition state between oligarchy-feudalism and socialism-capitalism, which can manifest allometric scaling with multifractal spectra. Monarchy and communism are extreme forms of oligarchy-feudalism and socialism-capitalism, respectively, in which the intrinsic randomness vanishes. The general framework is applied to three different models of rank distributions—top-down, bottom-up, and global—and unveils each model’s macroscopic universality and versatility. The global model yields a macroscopic classification of the generalized Zipf law, an omnipresent form of rank distributions observed across the sciences. An amalgamation of the three models establishes a universal rank-distribution explanation for the macroscopic emergence of a prevalent class of continuous size distributions, ones governed by unimodal densities with both Pareto and inverse-Pareto power-law tails. DOI: 10.1103/PhysRevE.89.012111

PACS number(s): 02.50.−r, 89.65.−s

I. INTRODUCTION

The measurement of positive-valued quantities is perhaps one of the most common features of all quantitative sciences; examples include populations of human settlements [1–3], occurrences of words in published texts [4–6], publications of scientists [7–9], wealth and income of citizens [10–12], sizes of firms and of firm bankruptcies [13–15], in degrees and out degrees of network nodes [16–18], and first-passage and search times [19,20]. In what follows we consider as given a dataset of positive-valued quantities. In a rank-distribution representation, the entries of the dataset are ordered monotonically according to their sizes, using a rank variable. Typically, the ordering is from largest to smallest, so the largest entry is ranked No. 1, the second largest entry is ranked No. 2, etc. The quintessential example of rank distributions is Zipf’s law [21,22],1 in which the magnitudes of the ranked entries follow a power law in the rank variable [23–27]. On a log-log scale Zipf’s law is characterized by a straight line with a negative slope. In recent years a series of research papers established that a generalized Zipf law—with one power-law behavior governing the high ranks and yet another power-law behavior governing the low ranks—is omnipresent across the sciences [28–32]. In a size-distribution representation, the entries of the dataset are quantified statistically via a probability density function. This density governs the probability distribution of a randomly sampled entry. A prevalent form of size distributions

*

[email protected] [email protected] 1 Although named after the American linguist George Kingsley Zipf, this law was first discovered in demography by Auerbach [1], in linguistics by Estoup [4], and in scientific productivity by Lotka [7]. †

1539-3755/2014/89(1)/012111(12)

observed across the sciences is the “unimodal power law” (UPL): size distributions governed by unimodal densities with power-law tails both at zero and at infinity [10–12,33–35]. In other words, a UPL density follows a skewed bell-shape curve (over the positive half line) with inverse-Pareto asymptotics at zero and with Pareto asymptotics at infinity. The bulk region of UPL size distributions is often approximated as log-normal. The distributions of income and wealth in human societies are well-known UPL examples. Transient generation of UPL size distributions with lognormal bulk is attained by running geometric Brownian motion for a deterministic time period and then stopping it after an additional exponentially distributed time period [35–37]. Equilibrium generation of UPL size distributions with lognormal bulk is attained by geometric Langevin equations with sigmoidal underlying forces [38–41]. More generally, UPL size distributions can be universally generated by geometric Langevin equations with U-shaped underlying potentials that are “on the edge of convexity” [42]. The goals of this paper are to present a panoramic macroscopic overview of rank distributions and to establish a universal rank-distribution explanation for the macroscopic emergence of UPL size distributions. To that end we introduce (in Sec. II) a general framework for the analysis of rank distributions which classifies them into five macroscopic “socioeconomic” states or phases: monarchy, oligarchyfeudalism, criticality, socialism-capitalism, and communism. Oligarchy-feudalism is shown to be characterized by discrete macroscopic rank distributions, and socialism-capitalism is shown to be characterized by continuous macroscopic size distributions. Criticality is a transition state between the phases of oligarchy-feudalism and socialism-capitalism that can manifest allometric scaling. Monarchy and communism are extreme forms of oligarchy-feudalism and socialismcapitalism, respectively, in which the intrinsic statistical variability vanishes and “solidifies” into deterministic states.

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The general framework is then exemplified (in Sec. III) and applied (in Sec. IV) to three different models of rank distributions: a top-down model [43], a bottom-up model [44], and a global model [45]. In the top-down model, the macroscopic rank distributions are versatile, whereas the macroscopic size distributions are universally governed by Pareto statistics. In the bottom-up model, the macroscopic rank distributions are universally governed by exponential statistics, and the macroscopic size distributions are universally governed by inverse-Pareto statistics. In the global model, the macroscopic rank distributions are universally governed by power-law statistics, whereas the macroscopic size distributions are versatile. Moreover, at criticality, both the top-down model and the global model display allometric scaling governed by power-law multifractal spectra. We “close the circle” (in Sec. V) by addressing the generalized Zipf law and UPL size distributions. The global model yields a macroscopic classification of the generalized Zipf law, based on the law’s pair of exponents, one governing the behavior of the high ranks and the other governing the behavior of the low ranks. An amalgamation of the three models provides a universal rank-distribution explanation for the macroscopic emergence of UPL size distributions. We further describe (in Sec. V) the “energy landscapes” underlying UPL size distributions in particular and, in general, underlying the macroscopic size distributions of the global model. In this paper we set off from the analytic methodologies introduced in [43–45] and expand them into a general framework. Thus, we transcend from specific methodologies tailored for specific models to a general framework applicable to rank-distribution models at large. This transcendence enables us to attain the goals of this paper, which are unattainable from the narrow perspective of any of the specific rank-distribution models (top-down, bottom-up, and global). For the sake of illustration and with no loss of generality— we henceforth consider the entries of the dataset to represent the wealth of individuals in a given human society. Also, φ (x) ≈ ψ (x) is a shorthand notation for the asymptotic equivalence of the functions φ (x) and ψ (x) as the variable x tends to a limit l (specifically, limx→l [φ (x) /ψ (x)] = c, where c is some positive constant). II. GENERAL FRAMEWORK

Consider a human society of size s, whose members are ranked in decreasing order of their wealth: r = 1 labeling the richest member, with wealth Ws (1); r = 2 labeling the second richest member, with wealth Ws (2); and r = s labeling the poorest member, with wealth Ws (s). The sequence of wealth values satisfies Ws (1)  Ws (2)  · · ·  Ws (s)  0. An illustrative example is the Forbes 2012 list of richest Americans [46]; the size of this list is s = 400, Bill Gates is ranked first with Ws (1) = 66B, Warren Buffett is ranked second with Ws (2) = 46B, Larry Ellison is ranked third with Ws (3) = 41B, etc. In what follows, we denote by As (k) = Ws (1) + Ws (2) + · · · + Ws (k) the aggregate wealth of the society’s k richest members (k = 1,2, . . . ,s); the society’s overall wealth is thus given by As (s). Our goal is to explore the macroscopic behavior of the distribution of wealth in large human societies. To that end we

henceforth focus on the infinite-size society limit s → ∞ and apply asymptotic analyses. A. Macroscopic classification

In this section, we introduce a macroscopic socioeconomic classification of the distribution of wealth which is based on two proportions. The first is the proportion of wealth held by the society’s k richest members: As (k) s→∞ As (s)

R (k) = lim

(1)

(k = 1,2,3, . . .). The second is the proportion of wealth held by a nontrivial proportion θ of the society’s richest members: L (θ ) = lim

s→∞

As (θ s) As (s)

(2)

(0 < θ < 1). For example, in the case of America R (3) is the proportion of wealth held by Bill Gates, Warren Buffett, and Larry Ellison (according to the aforementioned Forbes 2012 list), and L (0.1) is the proportion of wealth held by the 10% of richest Americans. The proportion L (θ )—as a function of the variable θ —is termed the Lorenz curve [47] and is widely applied in economics [48–50]. In recent years applications of the Lorenz curve have expanded well beyond the social sciences; see [51] and references therein. The classification is composed of five macroscopic socioeconomic states which are characterized by the rich proportion R (k) (k = 1,2,3, . . .) and by the Lorenz proportion L (θ ) (0 < θ < 1) as follows: (i) Monarchy. The society’s overall wealth is held by one single absolute monarch: R (k) = 1. (ii) Oligarchy-feudalism. The society’s overall wealth is held by a countable collection of oligarchs-lords: 0 < R (k) < 1. (iii) Criticality. Any finite collection the society’s richest members holds a zero proportion of the society’s overall wealth, but yet the society’s overall wealth is held by a zero proportion of the society’s richest members: R (k) = 0 and L (θ ) = 1. (iv) Socialism-capitalism. Any nontrivial proportion of the society’s overall wealth is held by a nontrivial proportion of the society’s richest members: θ < L (θ ) < 1. (v) Communism. The society’s overall wealth is equally distributed among all its members: L (θ ) = θ . Monarchy and communism are the boundaries of our “socioeconomic spectrum.” They can be viewed as extreme forms of oligarchy-feudalism and socialism-capitalism, respectively. We have used the term socialism-capitalism to label that socioeconomic system in which “any nontrivial proportion of the society’s wealth is held by a nontrivial proportion of the society’s richest members.” This definition implies that in socialism-capitalism the society’s overall wealth reaches all segments of the populations, albeit not uniformly. Our focus here is on the classification of the wealth distributions of societies according to their mathematical properties and not on the sociopolitical systems affecting them . These classes are sharply defined mathematically, whereas the idealized labels we chose for them are not sharply defined in their common usage. In the real world there is no socialism without markets

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(black, gray, or authorized), no capitalism without government intervention, no communism without economic inequity, no monarchism without dispersion of wealth, and no system without contravening corruption and crime. Our classification stands on its own, indifferent to these imperfections and to any particular underlying sociopolitical system, be it, for example, “socialism” or “capitalism.” We now examine oligarchy-feudalism, socialism-capitalism, and criticality in further detail. B. Oligarchy-feudalism

In the state of oligarchy-feudalism the society’s overall wealth is distributed among a countable collection of oligarchs-lords. The proportion of wealth held by the rth richest oligarch-lord is given by P (r) = R (r) − R (r − 1)

(3)

(r = 1,2,3, . . .), where we set R (0) = 0. In other words, if we pick at random one dollar of the society’s overall wealth, then the probability that this dollar belongs to the rth richest oligarch-lord is P (r). Thus, the society’s macroscopic statistics are discrete and are quantified by the macroscopic rank distribution of Eq. (3). Monarchy is an extreme form of oligarchy-feudalism, characterized by the totally asymmetric macroscopic rank distribution P (1) = 1 and P (r) = 0 (r = 2,3, . . .). Note that in the passage from oligarchy-feudalism to monarchy the inherent statistical variability vanishes and “solidifies” into a deterministic state. C. Socialism-capitalism

Consider a human society whose distribution of wealth has mean m, and is governed by the tail distribution function F> (v) (v > 0). Namely, m is the average wealth of the society’s members, and F> (v) is the proportion of the society’s members whose wealth is greater than the value v. In terms of the mean m and the quintile function Q (u) (0 < u < 1)—the inverse function of the tail distribution function F> (v)—the society’s Lorenz curve admits the following Pietra representation [51,52]:  1 θ L (θ ) = Q (u) du (4) m 0 (0 < θ < 1). In turn, differentiating Eq. (4) implies that the quintile function is given by Q (u) = mL (u) (0 < u < 1). Socialism-capitalism was defined by θ < L (θ ) < 1. Hence, in the state of socialism-capitalism, inverting the quintile function Q (u) = mL (u), yields a nondegenerate tail distribution function F> (v). In other words, if we pick at random a member of the society, then the probability that her/his wealth is greater than the value v is F> (v). Thus, the society’s macroscopic statistics are continuous and are quantified by a finite-mean macroscopic size distribution with quintile function Q (u) = mL (u). Communism is an extreme form of socialism-capitalism that is characterized by a degenerate macroscopic size distribution (quantified by a δ-function underlying density). Note that in the passage from socialism-capitalism to communism the inherent statistical variability vanishes and “solidifies” into a deterministic state.

PHYSICAL REVIEW E 89, 012111 (2014) D. Criticality

We established that oligarchy-feudalism and socialismcapitalism have inherent statistical structures which are discrete and continuous, respectively. Consequently, criticality is the boundary between discrete and continuous statistics. Oligarchy-feudalism and socialism-capitalism can be regarded as socioeconomic phases in analogy with the thermodynamic phases of matter. Continuing the analogy, criticality is their common phase boundary, marking the phase transition between them. Let Ns (p) denote the number of the society’s richest members that collectively hold a nontrivial proportion p of the society’s overall wealth (0 < p < 1). In the case of criticality the following two limits must hold: (i) lims→∞ Ns (p) = ∞, in order to assure the rich proportion R (k) = 0; and (ii) lims→∞ [Ns (p) /s] = 0, in order to assure the Lorenz proportion L (θ ) = 1. Namely, as a function of the size variable s, the number Ns (p) must diverge sublinearly in the limit s → ∞. An allometric scaling [53] of the rich with respect to the overall population yields such a sublinear divergence, Ns (p) = cs η(p) ,

(5)

where c is a positive coefficient and where the exponent η (p) takes values in the unit interval [0 < η (p) < 1]. The exponent η (p), as a function of the proportion p, quantifies an underlying multifractal spectrum [54]. As in the statistical physics of critical phenomena [55], we see that power laws— manifested here by the allometric scaling of Eq. (5)—are intimately related to the socioeconomic phase transition between oligarchy-feudalism and socialism-capitalism.

E. Hierarchical classifications

The socioeconomic classification is not all encompassing, as it does not take into account the scenario in which the macroscopic rank distribution—in the context  of either monarchy or oligarchy-feudalism—is improper: ∞ r=1 P (r) < 1. Indeed, in the socioeconomic classification it is implicitly assumed that in the monarchic state, as well as in the oligarchicfeudal state,  the macroscopic rank distribution is always proper: ∞ r=1 P (r) = 1. The “improper scenario” calls for hierarchical socioeconomic classifications which are beyond the scope of this paper. As a simple example of the improper scenario consider an enlightened monarch holding a nontrivial fraction f of the society’s overall wealth (0 < f < 1), and distributing the reminder of the wealth equally among all her subjects. This hierarchical distribution of wealth results in the Lorenz curve, L1 (θ ) = f + (1 − f ) L2 (θ )

(6)

(0 < θ < 1), where L2 (θ ) = θ is the communist Lorenz curve. Equation (6) well manifests the two-level hierarchical structure of the “enlightened monarch” example: the Lorenz curve L1 (θ ) quantifying the level including the monarch and the Lorenz curve L2 (θ ) quantifying the level excluding the monarch.

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III. EXAMPLES

To demonstrate the potency of the general framework we present in this section five different examples in which the macroscopic socioeconomic classification can be applied: networks, porous media, matrices, time series, and iterated maps. A. Networks

Networks are omnipresent objects of fundamental importance in almost all fields of science [16–18]. Consider a given vast network with nodes and edges, focus on the network’s connected components, and set the “wealth” of each connected component to be its size (e.g., number of nodes, physical mass, physical volume, etc.). The general framework yields a socioeconomic classification of the network’s macroscopic topology, ranging from the extreme “monarchic scenario” of total connectedness, to the extreme “communist scenario” of total disconnectedness. Indeed, monarchy manifests a macroscopic topology in which one giant connected component dominates the entire network, whereas communism manifests a macroscopic topology in which the network is fragmented into a vast collection of isolated microscopic islands. In the state of oligarchy-feudalism the macroscopic discrete rank distribution quantifies the probability that a randomly chosen node belongs to one of the large connected components dominating the network, and in the state of socialism-capitalism the macroscopic continuous size distribution quantifies the size of a randomly chosen connected component. To be more detailed, consider a specific parameter of the connectedness, e.g., the mean number of edges emerging from each node. Initiating from communism and increasing the connectedness, the state of the network changes to socialism-capitalism, as the microscopic islands coalesce into mesoscopic clusters quantified by the continuous size distribution emanating from the Lorenz curve of Eq. (4). On the other hand, initiating from monarchy and decreasing the connectedness, the state of the network changes to oligarchyfeudalism, as the giant connected component breaks down into multiple macroscopic clusters quantified by the discrete rank distribution of Eq. (3). At a critical value of the connectedness, often termed the percolation threshold, there is a phase transition between the socialism-capitalism phase below the threshold and the oligarchy-feudalism phase above it. In the state of criticality at the phase boundary, the distribution of cluster sizes often follows the allometric scaling of Eq. (5), called finite-size scaling in the statistical physics literature [55]. B. Porous media

The connected components of networks naturally lead us to porous media. Consider a continuous two-dimensional 2D/3D space of given area/volume, cut holes in the space that divide it into contiguous domains whose combined area/volume is a fraction f (0  f  1) of the total area/volume, and set the wealth of each individual domain to be its area/volume. As the fraction f decreases from 1 to 0, the same sequence of network connectedness phases is traversed: monarchy,

oligarchy-feudalism, criticality at a percolation threshold f∗ , socialism-capitalism, and communism. An illustrative classroom demonstration of 2D porousmedia percolation is constructed as follows. The medium consists of an electrically conducting sheet placed on an insulating plate; holes are punched in the sheet randomly, turning it porous. Specifically, the experiment initiates from an unpunctured sheet (monarchy, f = 1), the fraction f decreases as more and more holes are punched, and eventually the sheet is effectively removed (communism, f = 0). Clearly, the resistivity of the medium increases as the fraction f decreases, and at the percolation threshold (criticality, f = f∗ ) the medium turns from conducting (oligarchy-feudalism, f > f∗ ) to insulating (socialism-capitalism, f < f∗ ). Granular materials have a grain space occupied by solid, impenetrable grains with an interstitial pore space. The pore space can be represented by a network of nodes within pores and edges through channels between them. When the pore space is occupied by a fluid, it can be induced to flow only if the network of pores is above its percolation threshold. Oil, natural gas, or water thus cannot be induced to flow through an underground reservoir that is either in the socialist-capitalist or communist states. To extract those resources, a phase transformation to either the oligarchic-feudal or monarchic states must first be effected, e.g., by fracking. C. Matrices

As networks, also diagonalizable matrices are ubiquitous objects of prime importance in almost all fields of science. Consider a given diagonalizable s × s matrix with nonnegative eigenvalues, and set the wealth of each eigenvalue to be its value. Such matrices commonly appear in the form M  M, where M is a data matrix with real-valued entries and s columns, and where M  is the transposition of M. The general framework yields a socioeconomic classification of the matrix’s macroscopic structure—in the limit s → ∞— ranging from the extreme monarchic scenario of reduction to a single dimension, to the extreme “communist scenario” of the identity transformation. Indeed, monarchy manifests the macroscopic structure of a single significant eigenvalue (all other being zero), whereas communism manifests the macroscopic structure of identical eigenvalues (which can be normalized to one, thus yielding the identity transformation). The state of oligarchy-feudalism quantifies the scenario of principle component analysis (PCA), where the underlying high-dimensional data can be effectively projected onto a lowdimensional linear subspace [56,57]. The state of socialismcapitalism quantifies the scenario of a macroscopic continuous size distribution governing the eigenvalues, the quintessential example of such eigenvalue distributions being Wigner’s semicircle law [58,59].2 D. Time series

Time series analysis is applied in every field of science involving dynamic evolution [60,61]. Consider a dynamical 2

In the setting of Wigner’s semicircle law the matrices are Hermitian, and their eigenvalues are real valued.

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system sampled at times t = 0,1, . . . ,s, yielding a vector of real-valued measurements per each sample time t. Focus on the fluctuations of the time series—the fluctuation corresponding to time t being the difference between the measurements at times t and t − 1 (t = 1,2, . . . ,s)—and set the wealth of each fluctuation to be its Euclidean norm. The general framework yields a socioeconomic classification of the time series’ macroscopic fluctuations, ranging from the extreme monarchic scenario of a single huge rare-event fluctuation (dwarfing all other fluctuations), to the extreme communist scenario of identical fluctuations. The state of oligarchy-feudalism manifests the scenario of a few macroscopically large spikes dominating the “fluctuation landscape,” and the state of socialism-capitalism manifests the scenario of a fluctuation landscape looking like a lawn. Using the terminology of Mandelbrot [62], the state of oligarchy-feudalism characterizes “wild fluctuations,” and the state of socialism-capitalism characterizes “mild fluctuations”; for recent studies of mild and wild fluctuations, the readers are referred to [42] and to [63,64]. E. Iterated maps

Chaos theory attracted great interest, both scientific and popular, in recent decades [65]. One of the foundational models of chaos theory is iterated maps, 1D deterministic dynamic systems, generated by the iteration of an underlying nonlinear map. Iterated maps are capable of producing remarkably complex dynamical behaviors, and their quintessential example is the iconic logistic map [66–68]. Consider a given nonlinear map defined on the unit interval, focus on its long-term orbits, digitize the unit interval into s bins (of equal length 1/s), and define the wealth of each bin to be the fraction of time the orbits spends in the bin. The general framework yields a socioeconomic classification of the long-term dynamics of the iterated map, ranging from the extreme monarchic scenario of a single attracting fixed point, to the extreme communist scenario of a uniform stationary distribution over the unit interval. Indeed, monarchy manifests a long-term scenario in which the orbits are attracted to a global fixed point, whereas communism manifests a long-term scenario in which the orbits fluctuate randomly across the unit interval and “visit” all bins in an equiprobable fashion. The state of oligarchy-feudalism corresponds to periodic oscillations and is quantified by a macroscopic discrete rank distribution which is uniform over the periodic states; for example, if the long-term orbits alternate periodically between n different states, then the macroscopic discrete rank distribution is given by P (r) = 1/n for r  n and P (r) = 0 for r > n. The state of socialism-capitalism corresponds to chaotic dynamics, and its macroscopic continuous size distribution is contingent on the orbits’ stationary distribution. Interestingly, the state of criticality characterizes the onset of chaos. IV. MODELS

With the general framework of Sec. II at hand, we now apply it to three specific models of rank distributions: a top-down model [43], a bottom-up model [44], and a global model [45]. First, however, we review the notion of regular variation [69], which is used in all the three models.

PHYSICAL REVIEW E 89, 012111 (2014) A. Regular variation

Given a monotone increasing and non-negative valued function φ (x) (x > a, where a is a non-negative lower bound), consider the limit ω (θ ) = lim

x→∞

φ (θ x) , φ (x)

(7)

where 0 < θ < 1. If the limit ω (θ ) exists and is positive, then the function φ (x) is said to be regularly varying [69]. It is straightforward to observe that regular variation implies that the multiplicative structure ω (θ1 θ2 ) = ω (θ1 ) ω (θ2 ) holds for all 0 < θ1 ,θ2 < 1. In turn, this observation implies that the limit ω (θ ) must be a power law, ω (θ ) = θ 

(8)

(0 < θ < 1), where  is a non-negative regular-variation exponent [the non-negativity follows from the monotone increasing structure of the function φ (x)]. In the special case  = 0 the function φ (x) is termed slowly varying, and the corresponding limit is identically one: ω (θ ) = 1. The class of slowly varying functions includes asymptotically constant functions, logarithmic functions, powers of slowly varying functions, and logarithms of slowly varying functions. Moreover, regularly varying functions are based on slowly varying functions as follows: A function φ (x) is regularly varying with exponent  if and only if it admits the representation φ (x) = x  ψ (x), where the function ψ (x) is slowly varying. The class of regularly varying functions is a generalization of the class of asymptotically power-law functions, and it plays a focal role in various fields of mathematical analysis and probability theory [69]. An additional extension is allowing the regular-variation exponent to be infinite. In the special case  = ∞ the function φ (x) is termed rapidly varying, and the corresponding limit is identically zero: ω (θ ) = 0. So, in conclusion, we have the following regular-variation classification of the function φ (x) based on the limit ω (θ ): (i) slow variation if ω (θ ) = 1; (ii) “standard” regular variation if 0 < ω (θ ) < 1, in which case ω (θ ) = θ  with positive exponent ; (iii) rapid variation if ω (θ ) = 0. Armed with this asymptotic classification, we are now in position to analyze the top-down, bottom-up, and global models. B. The top-down model

In the top-down model the wealth values of the society members are considered to be the first s entries of an infinite monotone decreasing sequence with positive values [43]. To that end we set off from a smooth aggregate function A (x) (x > 0) which is positive valued, monotone increasing, and concave. The entries of the infinite monotone decreasing sequence are set to be the unit increments of the aggregate function, and the wealth of the rth richest member is thus given by Ws (r) = A (r) − A (r − 1) ∼ = A (r)

(9)

(r = 1,2, . . . ,s). In turn, the aggregate wealth of the society’s k richest members is As (k) = A (k) (k = 1,2, . . . ,s).

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The rich proportion R (k) is given by R (k) =

A (k) A (∞)

with exponent 0 < β < 1, in which case the allometric scaling is governed by the multifractal spectrum (10)

(k = 1,2,3, . . .), where A(∞) = limx→∞ A(x). Since the aggregate function is monotone increasing the limit A(∞) is always well defined and is either finite or infinite. Evidently, (i) R(k) = 1 if and only if A(k) = A(∞) < ∞; (ii) 0 < R(k) < 1 if and only if A(k) < A(∞) < ∞; and (iii) R(k) = 0 if and only if A(∞) = ∞. The Lorenz proportion L (θ ) is given by L (θ ) = lim

s→∞

A (θ s) = θα A (s)

(11)

(0 < θ < 1). Equation (11) implies that the proportion L (θ ) exists if and only if the aggregate function A (x) is regularly varying, in which case the proportion L (θ ) is given by the right-hand side of Eq. (11), where α is the regularvariation exponent of A (x). Since the aggregate function A (x) is monotone increasing and concave, its regular-variation exponent is restricted to the range 0  α  1. Consequently, (i) L (θ ) = 1 if and only if α = 0; (ii) θ < L (θ ) < 1 if and only if 0 < α < 1; and (iii) L (θ ) = θ if and only if α = 1. The macroscopic socioeconomic classification of the topdown model is thus characterized as follows: (i) Monarchy: The aggregate function A (x) is constant. (ii) Oligarchy-feudalism: The aggregate function A (x) is nonconstant and bounded. (iii) Criticality: The aggregate function A (x) is unbounded and slowly varying. (iv) Socialism-capitalism: The aggregate function A (x) is regularly varying with exponent 0 < α < 1. (v) Communism: The aggregate function A (x) is regularly varying with exponent α = 1. In the state of oligarchy-feudalism the model displays macroscopic versatility, as Eq. (10) establishes that all macroscopic rank distributions [recall Eq. (3)] are attainable. On the other hand, in the state of socialism-capitalism the model displays macroscopic universality, as Eq. (16) asserts that socialism-capitalism is characterized by the universal Lorenz curve L (θ ) = θ α . The Pietra inversion of this Lorenz curve yields the Pareto tail distribution function   mα 1/(1−α) F> (v) = (12) v (mα < v < ∞).3 Thus, in the state of socialism-capitalism, the macroscopic size distribution is universally governed by Pareto statistics. Note that the Pareto mean m, the Pareto lowerbound cutoff c = mα, and the Pareto exponent  = 1/ (1 − α) are coupled by the relation c = m ( − 1). Also note that the Pareto exponent is in the finite-mean range 1 <  < ∞. With regard to the state of criticality, we show in the Appendix that the allometric scaling of Eq. (5) holds if and only if the function B (x) = A[exp (x)] (x  0) is regularly varying

3 Indeed, Eq. (11) implies that Q (u) = mL (u) = mαuα−1 (0 < u < 1), and the inversion of the quintile function Q (u) yields the tail distribution function of Eq. (12).

η (p) = p 1/β

(13)

(0 < p < 1). A specific example of an aggregate function yielding allometric scaling is given by the asymptotics A (x) ≈ [ln (x)]β (as x → ∞). C. The bottom-up model

In the bottom-up model the wealth values of the society members are considered to be the first s entries of an infinite monotone increasing sequence with positive values [44]. To that end we set off from a smooth aggregate function A (x) (x > 0) which is positive valued, monotone increasing, and convex. The entries of the infinite monotone increasing sequence are set to be the unit increments of the aggregate function, and the wealth of the rth richest member is thus given by Ws (r) = A (s − r + 1) − A (s − r) ∼ = A (s − r)

(14)

(r = 1,2, . . . ,s). In turn, the aggregate wealth of the society’s k richest members is As (k) = A (s) − A (s − k) (k = 1,2, . . . ,s). The rich proportion R (k) is given by A (s − k) A (s) B[exp (s) exp (−k)] = 1 − lim s→∞ B[exp (s)] B[x exp (−k)] = 1 − lim x→∞ B[x] = 1 − exp (−βk)

R (k) = 1 − lim

s→∞

(15)

(k = 1,2,3, . . .), where B (x) = A [ln (x)] (x  1). Namely, Eq. (15) implies that the proportion R (k) exists if and only if the function B (x) is regularly varying, in which case the proportion R (k) is given by the right-hand side of Eq. (15), where β is the regular-variation exponent of B (x). Consequently, (i) R (k) = 1 if and only if β = ∞; (ii) 0 < R (k) < 1 if and only if 0 < β < ∞, in which case 1 − R (k) follows an exponential decay; and (iii) R (k) = 0 if and only if β = 0. The Lorenz proportion L (θ ) is given by L (θ ) = 1 − lim

s→∞

A [s (1 − θ )] = 1 − (1 − θ )α A (s)

(16)

(0 < θ < 1). Namely, Eq. (16) implies that the proportion L (θ ) exists if and only if the aggregate function A (x) is regularly varying, in which case the proportion L (θ ) is given by the right-hand side of Eq. (16), where α is the regularvariation exponent of A (x). Since the aggregate function A (x) is monotone increasing and convex its regular-variation exponent is restricted to the range 1  α  ∞. Consequently: (i) L (θ ) = 1 if and only if α = ∞; (ii) θ < L (θ ) < 1 if and only if 1 < α < ∞; (iii) L (θ ) = θ if and only if α = 1. The macroscopic socioeconomic classification of the bottom-up model is thus characterized as follows: (i) Monarchy: The function A [ln (x)] is rapidly varying.

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(ii) Oligarchy-feudalism: The function A [ln (x)] is regularly varying with exponent 0 < β < ∞. (iii) Criticality: The function A [ln (x)] is slowly varying and the aggregate function A (x) is rapidly varying. (iv) Socialism-capitalism: The aggregate function A (x) is regularly varying with exponent 1 < α < ∞. (v) Communism: The aggregate function A (x) is regularly varying with exponent α = 1. In both the states of oligarchy-feudalism and socialismcapitalism the model displays macroscopic universality. Indeed, Eq. (15) establishes that, in the state of oligarchyfeudalism, the macroscopic rank distribution is universally governed by exponential statistics. Also, Eq. (16) asserts that the state of socialism-capitalism is characterized by the Lorenz curve L (θ ) = 1 − (1 − θ )α . The Pietra inversion of this Lorenz curve yields the inverse-Pareto cumulative distribution function   v 1/(α−1) (17) F (v) = 1 − F> (v) = mα (0 < v < mα).4 Thus, in the state of socialism-capitalism, the macroscopic size distribution is universally governed by inverse-Pareto statistics. Note that the inverse-Pareto mean m, the inverse-Pareto upper-bound cutoff c = mα, and the inverse-Pareto exponent  = 1/ (α − 1) are coupled by the relation c = m ( + 1). D. The global model

In the global model the wealth values of the society members are considered to approximate the values, at the points {r/ (s + 1)}sr=1 , of a positive-valued and monotone decreasing function defined on the unit interval [45]. To that end we set off from a smooth aggregate function A (x) (0 < x < 1) which is positive valued, monotone decreasing, and convex. The wealth of the rth richest member is set to be given by      r +1 r Ws (r) = (s + 1) A −A s+1 s+1   r ∼ (18) = −A s+1 (r = 1,2, . . . ,s), and hence the positive-valued and monotone decreasing function which the wealth-values approximate is  −A (x) (0 < x  1). In turn, the aggregate wealth of the society’s k richest members is given by      k+1 1 As (k) = (s + 1) A −A (19) s+1 s+1 (k = 1,2, . . . ,s). The analysis of the global model requires the notion of regular variation at zero, which is analogous to the notion of regular variation at infinity, described in Sec. IV A. Given a monotone decreasing and non-negative valued function φ (x)

4 Indeed, Eq. (11) implies that Q (u) = mL (u) = mα (1 − u)α−1 (0 < u < 1), and the inversion of the quintile function Q (u) yields the cumulative distribution function of Eq. (17).

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(0 < x < b, where b is a positive upper bound), consider the limit ω (θ ) = limx→0 [φ (θ x) /φ (x)], where θ > 1. The asymptotic regular-variation classification (at zero) of the function φ (x) based on the limit ω (θ ) is as follows [69]: (i) slow variation if ω (θ ) = 1; (ii) “standard” regular variation if 0 < ω (θ ) < 1, in which case ω (θ ) = 1/θ  with positive exponent ; and (iii) rapid variation if ω (θ ) = 0. As in Sec. IV A slow variation and rapid variations correspond, respectively, to the regular-variation exponents  = 0 and  = ∞. The rich proportion R (k) is given by

 A k+1 A [(k + 1) x]  = 1 − lim R (k) = 1 − lim s+1 1 s→∞ A x→0 A (x) s+1 = 1−

1 (k + 1)α

(20)

(k = 1,2,3, . . .). Namely, Eq. (20) implies that the proportion R (k) exists if and only if the function A (x) is regularly varying at zero, in which case the proportion R (k) is given by the right-hand side of Eq. (20), where α is the regular-variation exponent of A (x). Consequently, (i) R (k) = 1 if and only if α = ∞; (ii) 0 < R (k) < 1 if and only if 0 < α < ∞, in which case 1 − R (k) follows a power-law decay; and (iii) R (k) = 0 if and only if α = ∞. The Lorenz proportion L (θ ) is given by 

A θs+1 A (θ ) s+1

1  =1− L (θ ) = 1 − lim (21) s→∞ A A (0) s+1 (0 < θ < 1), where A (0) = limx→0 A (x). Since the aggregate function is monotone decreasing the limit A (0) is always well defined and is either finite or infinite. Evidently: (i) L (θ ) = 1 if and only if A (0) = ∞; (ii) θ < L (θ ) < 1 if and only if A (0) < ∞ and A (x) is nonlinear; and (iii) L (θ ) = θ if and only if A (x) is linear [in which case A (x) = c (1 − x), where c is a positive coefficient]. The macroscopic socioeconomic classification of the global model is thus characterized as follows: (i) Monarchy: The aggregate function A (x) is rapidly varying at zero. (ii) Oligarchy-feudalism: The aggregate function A (x) is regularly varying at zero with exponent 0 < α < ∞. (iii) Criticality: The aggregate function A (x) is unbounded and slowly varying at zero. (iv) Socialism-capitalism: The aggregate function A (x) is bounded and nonlinear. (v) Communism: The aggregate function A (x) is linear. In the state of oligarchy-feudalism the model displays macroscopic universality, as Eq. (20) establishes that the macroscopic rank distribution is universally governed by power-law statistics. On the other hand, in the state of socialism-capitalism the model displays macroscopic versatility, as Eq. (21) asserts that all Lorenz curves are attainable, and hence, via the Pietra inversion, all finite-mean macroscopic size distributions are attainable. Moreover, equating Eqs. (4) and (21) implies that in the state of socialism-capitalism the mean of the macroscopic size distribution is m = A (0), and the quintile function of the macroscopic size distribution is given by Q (u) = −A (u) (0 < u < 1). Thus, we arrive at the

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TABLE I. Summary of the classification of the top-down, bottom-up, and global models into the five macroscopic socioeconomic states: monarchy, oligarchy-feudalism, criticality, socialism-capitalism, and communism. In the top-down and bottom-up models the variation is at infinity (x → ∞), whereas in the global model the variation is at zero (x → 0). Top-down

Bottom-up

Global

A(x) positive, monotone increasing, and concave on (0,∞)

A(x) positive, monotone increasing, and convex on (0,∞)

A(x) positive, monotone decreasing, and convex on (0,1)

Wealth

Ws (r) ∼ = A (r)

Ws (r) ∼ = A (s − r)

r Ws (r) ∼ ) = −A ( s+1

Monarchy

A(x) constant

A[ln(x)] rapidly varying

A(x) rapidly varying

Oligarchyfeudalism

A(x) nonconstant and bounded

A[ln(x)] regularly varying with exponent 0 < β < ∞

A(x) regularly varying with exponent 0 < α < ∞

Criticality

A(x) unbounded and slowly varying

A[ln(x)] slowly varying and A(x) rapidly varying

A(x) unbounded and slowly varying

Socialismcapitalism

A(x) regularly varying with exponent 0 < α < 1

A(x) regularly varying with exponent 1 < α < ∞

A(x) bounded and nonlinear

Communism

A(x) regularly varying with exponent α = 1

A(x) regularly varying with exponent α = 1

A(x) linear

Aggregate function

following representation of the model’s wealth values in the state of socialism-capitalism:   r (22) Ws (r) ∼ =Q s+1 (r = 1,2, . . . ,s). With regard to the state of criticality, we show in the Appendix that the allometric scaling of Eq. (5) holds if and  only if the function B (x) = A exp (−x) (x  0) is regularly varying (at infinity) with exponent 0 < β < 1, in which case the allometric scaling is governed by the multifractal spectrum η (p) = 1 − (1 − p)1/β

(23)

(0 < p < 1). A specific example of an aggregate function yielding allometric scaling is given by the asymptotics A (x) ≈ [− ln (x)]β (as x → 0).

E. Interim summary

The general framework of Sec. II, applied to the three models analyzed in this section, yields a panoramic macroscopic outlook of rank distributions. This outlook is summarized in Tables I–III. Table I presents the models’ basic details and macroscopic socioeconomic states. Table II presents each model’s macroscopic universality and macroscopic versatility in the states of oligarchy-feudalism and socialism-capitalism. Table III specifies the underlying model that gives rise to the statistics of a macroscopically observed discrete rank distribution or continuous size distribution, in the respective contexts of oligarchy-feudalism and socialism-capitalism; in other words, having observed a certain type of macroscopic statistics, Table III tells us what underlying model may be considered as admissible. We note that the macroscopic rank distributions emerging in the three models—see Eq. (10) in the context of the top-down model, Eq. (15) in the context of

TABLE II. Summary of the macroscopic versatility and the macroscopic universality of the top-down, bottom-up, and global models in the two stochastic states: oligarchy-feudalism, governed by discrete macroscopic rank distributions; and socialism-capitalism, governed by continuous macroscopic size distributions. Oligarchy-feudalism

Socialism-capitalism

Top-down

Macroscopic versatility: all macroscopic rank distributions are attainable

Macroscopic universality: macroscopic size distributions are universally governed by finite-mean Pareto statistics

Bottom-up

Macroscopic universality: macroscopic rank distributions are universally governed by exponential statistics

Macroscopic universality: macroscopic size distributions are universally governed by inversePareto statistics

Global

Macroscopic universality: macroscopic rank distributions are universally governed by power-law statistics

Macroscopic versatility: all finite-mean macroscopic size distributions are attainable

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PHYSICAL REVIEW E 89, 012111 (2014)

TABLE III. Summary of the macroscopic versatility and the macroscopic universality of the top-down, bottom-up, and global models in the two stochastic states: oligarchy-feudalism, governed by discrete macroscopic rank distributions; and socialism-capitalism, governed by continuous macroscopic size distributions. Macroscopically observed rank distribution

Macroscopically observed size distribution

General statistics

Top-down model

Global model

Exponential statistics

Top-down model Bottom-up model

Global model

Power-law statistics

Top-down model Global model

Top-down model Bottom-up model

the bottom-up model, and Eq. (20) in the context of the global model—are all “proper” in the sense discussed in Sec. II E.

(iv) Socialism-capitalism holds in the exponent range 0 < γ < 1, in which case the quintile function of the macroscopic size distribution is given by

V. CLOSING THE CIRCLE

We now turn to close the circle and illuminate the generalized Zipf law and UPL size distributions—from which we initiated in the Introduction—in light of the panoramic macroscopic outlook. Also, we describe the energy landscapes underlying UPL size distributions in particular, and, in general, underlying the macroscopic size distributions of the global model. A. The generalized Zipf law

As noted in the Introduction, the generalized Zipf law is empirically omnipresent across the sciences [28–32]. In our notation this empirical law is quantified by the beta-type form   δ  r s+1 γ ∼ 1− Ws (r) = c (24) r s+1 (r = 1,2, . . . ,s), where c is a positive constant and where γ and δ are positive exponents. Evidently, Eq. (18) implies that the generalized Zipf law is a special case of the global model with aggregate function  1 (1 − u)δ A (x) = c du (25) uγ x (0 < x  1), where c is a positive constant. Applying the analysis of Sec. IV D to the aggregate function of Eq. (25) yields the following macroscopic socioeconomic classification: (i) Monarchy is unattainable. (ii) Oligarchy-feudalism holds in the exponent range 1 < γ < ∞, in which case α = γ − 1 and hence the macroscopic rank distribution is governed by the power-law decay 1 1 − R (k) = (k + 1)γ −1

(26)

(k = 1,2,3, . . .). (iii) Criticality holds at the exponent value γ = 1, in which case β = 1 and hence the allometric scaling of the rich with respect to the overall population is governed by the linear multifractal spectrum η (p) = p (0 < p < 1).

(27)

Q (u) =

(1 − u)δ uγ

(28)

(0 < u < 1). (v) Communism is unattainable.5 In the case of socialism-capitalism the quintile function of Eq. (28) yields inverse-Pareto and Pareto asymptotics of the macroscopic size distribution at zero and at infinity, respectively. Indeed, a straightforward inversion of this quintile function implies that (i) the corresponding cumulative distribution function follows the asymptotics F (v) ≈ v 1/δ (as v → 0) and (ii) the corresponding tail distribution function follows the asymptotics F> (v) ≈ 1/v 1/γ (as v → ∞). This observation finally leads us to UPL size distributions, which are addressed in the next section. B. UPL size distributions

In the socioeconomic state of socialism-capitalism the macroscopic size distributions are universally Pareto in the top-down model, are universally inverse-Pareto in the bottomup model, and are versatile in the global model (with finite means in all three models). Since the Pareto statistics have a lower-bound cutoff and the inverse-Pareto statistics have an upper bound cutoff, the three models can be naturally intertwined into an amalgamated macroscopic size distribution with inverse-Pareto asymptotics at zero and with Pareto asymptotics at infinity. Specifically, the corresponding cumulative distribution function displays the asymptotic behavior F (v) ≈ v 0 , v→0

(29)

with exponent 0 < 0 < ∞, and the corresponding tail distribution function displays the asymptotic behavior 1 , (30) v ∞ with exponent 1 < ∞ < ∞. [The cumulative distribution function F (v) and the tail distribution function sum up to one: F (v) + F> (v) = 1.] F> (v) ≈

v→∞

5

In effect, if we allow the exponents γ and δ to be non-negative, then communism will hold at the exponent values γ = δ = 0.

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The most “streamlined” cumulative distribution function F (v) displaying the asymptotics of Eqs. (29) and (30) is concave when 0 < 0  1 and is sigmoidal when 1 < 0 < ∞. In turn, the underlying probability density function F (v) is: (i) monotone decreasing and unbounded in the exponent range 0 < 0 < 1; (ii) monotone decreasing and bounded at the exponent value 0 = 1; and (iii) unimodal in the exponent range 1 < 0 < ∞. By “unimodal” we mean that the density initiates at the origin, increases monotonically to a global maximum, and then decreases to zero as v → ∞. The shape of a a unimodal density is that of a skewed bell curve (drawn on the positive half line). Hence, in the exponent range 1 < 0 < ∞ the amalgamated macroscopic size distribution admits the UPL structure noted in the Introduction, a statistical structure that is prevalently encountered across the sciences [10–12,33–35]. The theoretical limit laws established here in the context of rank distributions are thus in full agreement with empirically observed UPL size distributions and yield the following conclusions: (i) The rigid inverse-Pareto asymptotics manifests the macroscopic university of the bottom-up model, which turns out to govern small sizes. (ii) The flexible unimodal bulk manifests the macroscopic versatility of the global model, which turns out to govern intermediate sizes. (iii) The rigid Pareto asymptotics manifests the macroscopic university of the top-down model, which turns out to govern large sizes. Combined and intertwined, the three rank-distribution models establish an amalgamated model from which UPL size distributions emerge macroscopically. In the case of the distribution of wealth the poor are governed by the bottom-up model, the rich are governed by the top-down model, and the middle class is governed by the global model with sigmoidal cumulative distribution functions. C. Energy landscapes

As noted above, in the socioeconomic state of socialismcapitalism only the global model yields macroscopic versatility. Moreover, in the context of this state and this model, Eq. (22) establishes an explicit connection between the society’s wealth values Ws (r) (r = 1,2, . . . ,s) and the quintile function Q (u) (0 < u < 1), which quantifies the society’s macroscopic size distribution. An elemental paradigm in statistical physics is entropy maximization, asserting that a system, within its restrictive constraints, attains a configuration of maximum entropy; the system’s constraints are often quantified by a potential function manifesting the system’s underlying energy landscape. Thus, having observed the society’s macroscopic size distribution, statistical physics naturally leads us to the following question: What is the society’s underlying energy landscape? In this section we address this reverse engineering question. Consider a system whose configuration is represented by a random variable taking values in the real range (a,b) (where a and b are, respectively, the range’s lower and upper bounds), and governed by the probability density function (x) (a < x < b). Also, let the real-valued function U (x) (a < x < b) denote the system’s potential function. Then, the

entropy maximizing configuration is given by   1 (x) = c exp − U (x) τ

(31)

(a < x < b), where c is a positive normalizing constant, and where τ is a positive parameter often interpreted as the system’s effective temperature. The readers are referred to [40] for a detailed derivation and discussion of Eq. (31). The macroscopic size distribution is supported on the positive half line, and is quantified by the quintile function Q (u). Hence, the corresponding real range is (a,b) = (0,∞), and the corresponding density function is (x) = −F> (x) (x > 0), where the tail distribution function F> (x) is the inverse of the quintile function Q (u). A straightforward inversion of Eq. (31) yields the reverse engineered potential function U (x) = u + τ ln{−Q [Q−1 (x)]}

(32)

(x > 0), where u = τ ln (c). Size distributions, especially in economics, are often the outcome of multiplicative processes. In such cases it is natural to consider the log-size distributions, rather than the size distributions themselves. Transforming to a logarithmic scale yields the real range (a,b) = (−∞,∞), the probability density  ˜ (x) = exp (x) exp (x) (x real), and the reverse function engineered potential function U˜ (x) = U [exp (x)] − τ x

(33)

(x real). A general methodology for the reverse engineering of potential functions underlying the equilibria of multiplicative processes was recently introduced in [42]. Equations (32) and (33) represent the society’s underlying energy landscape in terms of the quintile function Q (u) quantifying the society’s macroscopic size distribution; Eq. (32) does so on a standard scale, while Eq. (33) does so on a logarithmic scale. In the case of UPL size distributions the standard-scale potential function U (x) is asymptotically logarithmic (at the limits x → 0 and x → ∞), and the logarithmic-scale potential function U˜ (x) is asymptotically linear (at the limits x → −∞ and x → ∞). Hence, as noted in the Introduction, in the context of convex U-shaped logarithmic-scale potential functions U˜ (x), the energy landscapes underlying UPL size distributions are indeed characterized as being “on the edge of convexity” [42]. VI. CONCLUSION

In this paper we have established a general framework that classifies rank distributions into five macroscopic socioeconomic states: monarchy, oligarchy-feudalism, criticality, socialism-capitalism, and communism. Monarchy and communism—the extremes of the socioeconomic spectrum— are diametric deterministic states. Oligarchy-feudalism and socialism-capitalism are antithetical stochastic states governed, respectively, by discrete macroscopic rank distributions and by continuous macroscopic size distributions. Criticality is a phase-transition state separating oligarchy-feudalism and socialism-capitalism. Illustratively, oligarchy-feudalism can be visualized by a round pie, with unit area, that is cut into slices representing

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the society’s discrete macroscopic rank distribution; picking at random a point of the pie’s area manifests picking at random one dollar of the society’s overall wealth, and the pie slice into which the random point falls is the probability that the “random dollar” belongs to the respective oligarch-lord that “owns” this slice. On the other hand, socialism-capitalism can be visualized by the area under the probability density function governing the society’s continuous macroscopic size distribution; picking at random a point of the area under the curve manifests picking at random one member of the society’s overall population, and the x coordinate of the random point is the size of the randomly picked population member. Applying the general framework to three different rankdistribution models—top-down, bottom-up, and global— yields a panoramic macroscopic outlook of rank distributions. Interestingly, these models display both macroscopic universality and macroscopic versatility. In the top-down model the macroscopic rank distributions are versatile, whereas the macroscopic size distributions are universally governed by Pareto statistics. In the bottom-up model the macroscopic rank distributions are universally governed by exponential statistics, and the macroscopic size distributions are universally governed by inverse-Pareto statistics. In the global model the macroscopic rank distributions are universally governed by power-law statistics, whereas the macroscopic size distributions are versatile. Moreover, at criticality, both the top-down model and the global model display allometric scaling governed by power-law multifractal spectra. The application of the general framework to the global model yields a macroscopic socioeconomic classification of the generalized Zipf law, an omnipresent form of rank distributions observed across the sciences; the classification is based on the law’s pair of exponents—one governing the behavior of the high ranks, and the other governing the behavior of the low ranks. An amalgamation of the three models establishes a universal rank-distribution explanation for the macroscopic emergence of UPL size distributions: continuous size distributions governed by unimodal densities with inverse-Pareto power-law tails at zero and with Pareto power-law tails at infinity. The wide-scope aim of this paper is to serve scientists as a post from which the macroscopic structures of rank distributions, as well as their connections to size distributions, can be sharply observed and classified. Since rank distributions and size distributions are ubiquitously encountered across the sciences, we hope that this service will be used by researchers from diverse scientific fields.

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described in Sec. IV A. To that end we replace the limit of Eq. (7) with the limit ω (θ ) = lim

x→∞

(A1)

(0 < θ < 1), where ρ (x) is an arbitrary perturbation function with positive limit ρ (∞) = limx→∞ ρ (x). Namely, the limit ω (θ ) should be independent with respect to perturbation function ρ (x). All the statements of Sec. IV A holds valid for the generalized limit of Eq. (A1). Recall that Ns (p) denotes the number of the society’s richest members that collectively hold a nontrivial proportion p of the society’s overall wealth (0 < p < 1). In terms of the aggregate wealth the number Ns (p) is implicitly given by As [Ns (p)] = pAs (s) .

(A2)

Hence, substituting the allometric scaling of Eq. (5) into Eq. (A2) we arrive at the limit As (cs η(p) ) = p. s→∞ As (s) lim

(A3)

Equation (13). In the top-down model the aggregate wealth is given by As (k) = A (k) (k = 1,2, . . . ,s), and hence, As (cs η(p) ) A(cs η(p) ) = lim . s→∞ s→∞ As (s) A (s) lim

(A4)

Setting B (x) = A[exp (x)] and x = ln (s) we have A(cs η(p) ) B [η (p) x + ρ (x)] = lim , s→∞ x→∞ A (s) B (x) lim

(A5)

with perturbation function ρ (x) ≡ ln (c) . Equations (A4) and (A5) imply that the limit of Eq. (A3) holds if and only if the function B (x) is regularly varying, in which case p = [η (p)]β ,

(A6)

where β is the regular-variation exponent of B (x). Equation (A6) yields the multifractal spectrum of Eq. (13). Equation (23). In the global model the aggregate wealth As (k) is given by Eq. (19), and hence,

η(p)  A cs s+1+1 As (cs η(p) )

1  . lim (A7) = 1 − lim s→∞ s→∞ A As (s) s+1  Setting B (x) = A exp (−x) and x = ln (s + 1) we have

η(p)  A cs s+1+1 B {[1 − η (p)] x + ρ (x)}

1  = lim lim , s→∞ A x→∞ B (x) s+1

ACKNOWLEDGMENT

This work was supported by a DIP (German-Israeli Project Cooperation Program) Grant.

φ [θ x + ρ (x)] φ (x)

(A8)

with perturbation function satisfying ρ (∞) = ln (c). Equations (A7) and (A8) imply that the limit of Eq. (A3) holds if and only if the function B (x) is regularly varying, in which case,

APPENDIX

p = 1 − [1 − η (p)]1/β ,

In this appendix we prove Eqs. (13) and (23). The proofs require a generalization of the notion of regular variation

where β is the regular-variation exponent of B (x). Equation (A9) yields the multifractal spectrum of Eq. (23).

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(A9)

IDDO I. ELIAZAR AND MORREL H. COHEN [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14] [15] [16] [17] [18]

[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

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