Limiting Value of the Kolkata Index for Social Inequality and a Possible Social Constant
aa r X i v : . [ phy s i c s . s o c - ph ] F e b Limiting Value of the Kolkata Index for Social Inequality and a PossibleSocial Constant
Asim Ghosh ∗ and Bikas K Chakrabarti
2, 3, 4, † Raghunathpur College, Raghunathpur, Purulia 723133, India. Saha Institute of Nuclear Physics, Kolkata 700064, India. Economic Research Unit, Indian Statistical Institute, Kolkata 700108, India. S. N. Bose National Centre for Basic Sciences, Kolkata 700106, India
Based on some analytic structural properties of the Gini and Kolkata indices for social inequality,as obtained from a generic form of the Lorenz function, and some more new observations on thecitation statistics of individual authors (including Nobel laureates), we make a conjecture that about14% of people or papers or social conflicts tend to earn or attract or cause about 86% of wealth orcitations or deaths in very competitive situations in markets, universities or wars. This is a modifiedform of the (more than a) century old 80 −
20 law of Pareto in economy (not visible today becauseof various welfare and other strategies) and gives an universal value (0 .
86) of social (inequality)constant or number.
I. INTRODUCTION
Unlike the universal constants in physical sciences, like the Gravitational Constant of Newton’s Gravitylaw, Boltzmann Constant of thermodynamics or Planck’s Constant of Quantum Mechanics, there is noestablished universal constant yet in social sciences. There have of course been suggestion of severalpossible candidates.Stanley Milgram’s experiment [1] to determine the social ‘contact-distance’ between any two personsof the society, by trying to deliver letters from and to random people through personal chains of friendsor acquaintances, suggested ‘Six Degrees of Separation’. Studying similar distance through coauthormeasurements, between any two scientists (e.g., the Erdos number [2], describing the collaborativedistance between mathematician Paul Erdos and another mathematician, as measured by coauthorshipof papers) indicated similar but not identical numbers. Later, the (internet) network structure studies[3, 4] linked the (separation) number to be related to the network size (typically going as log of thenetwork size) and not any really as universal as six. The Dunbar number [5], suggesting that we canonly maintain one hundred and fifty distinct social relationships (as may be seen in the sizes of the oldvillage groups), has also been questioned. It is observed to vary from much smaller numbers, for closershell relationships, to order of magnitude larger number, for social weblinks and can be extracted,say, from the sizes of individual’s mobile call list (see e.g., [6, 7]).We find that the limiting magnitude of a particular social inequality measure shows a robust anduniversal value across different social contexts. In a series of papers [8–12] (see also [13] for a recentreview), we introduced the Kolkata index ( k ) for measuring social inequality ( k = 1 / k = 1 corresponds to extreme inequality). In the economic context [8], it says(1 − k ) fraction of people posses k fraction of wealth, while in the context of an university [8, 10], itsays (1 − k ) fraction of papers published by the faculty of the university attracts k fraction of citations,or even in the context of wars or major social conflicts, it says [11] (1 − k ) fraction of social conflicts orwars cause k fraction of deaths. We observed [8, 10, 11] that for such a wide range of social contexts,the limiting values of Kolkata index k concentrate around 0 .
86, except in the case of world economiestoday (where k is much smaller and observed to be in the range 0 . − . k indexis a quantitative generalization of the century old 80 −
20 rule of Vilfredo Pareto [14], who observedtowards the end of eighteenth century that in most of the European countries (Italy, in particular) ∗ Email: [email protected] † Email: [email protected] almost 80% of the land are owned by 20% of the people (i.e., k ∼ . c u m u l a t i v e f r a c t i on o f w ea l t h he l d b y t he m cumulative fraction of people from poorest to richest(k,1-k)k - S e q u a li t y li n e Lo r en z c u r v e S FIG. 1. Lorenz curve or function L ( x ) (in red here) represents the accumulated fraction of wealth (or citationsor deaths) against the fraction ( x ) of people (or papers or social conflicts) possessing (or attracting or causing)that, when arranged from poorest (or lowest) to richest (or highest). The diagonal from the origin representsthe equality line. The Gini index ( g ) can be measured [15] by the area ( S ) of the shaded region in-betweenthe Lorenz curve and the equality line, when normalized by the area ( S + ¯ S = 1 /
2) of the triangle below theequality line): g = 2 S . The Kolata index k can be measured by the ordinate value of the intersecting point ofthe Lorenz curve and the diagonal perpendicular to the equality line. By construction, it says that k fractionof wealth (or citations or deaths) is being held by (1 − k ) fraction of top people ((or papers or social conflicts). We will first discuss here analytically some indications of a limiting behavior of the Kolkata index,suggesting its value k near 0.86. Next we will provide some detailed analysis of data from differentsocial sectors like citations of papers published by different universities and in different journals,human deaths in different wars or social conflicts, and citations of papers published by individualauthors (including Nobel Laureates) showing that the limiting value of the inequality index k suggeststhat typically 86% of citations (or deaths) come from 14% papers (or conflicts). II. LORENZ CURVE: GINI & KOLKATA INDICES
Our study here is based on the Lorenz curve (see Fig. 1) or function [16] L ( x ), which gives thecumulative fraction of (total accumulated) wealth (or citations or human deaths) possessed (attractedor caused) by the fraction ( x ) of the people (or papers or social conflicts respectively) when countedfrom the poorest (or least or mildest) to the richest (or highest or deadliest). If the income/wealth(or citations or deaths) of every person (or paper or war) would be identical, then L ( x ) would be astraight line (diagonal) passing through the origin. This diagonal is called the equality line. The Ginicoefficient or index ( g ) is given by the area between the Lorenz curve and the equality line (normalizedby the area under the equality line: g = 0 corresponds to equality and g = 1 corresponds to extremeinequality.We proposed [8] the Kolkata index ( k ) given by the ordinate value (see Fig. 1) of the intersectingpoint of the Lorenz curve and the diagonal perpendicular to the equality line (see also [9–13]). Byconstruction, 1 − L ( k ) = k , saying that k fraction of wealth (or citations or deaths) is being possessed(owned or caused) by (1 − k ) fraction of the richest population (impactful papers or deadliest wars). As g , k k c k c FIG. 2. Gini ( g ) and Kolkata ( k ) indices obtained numerically for the generic form of the Lorenz function L ( x ) = x n (eqn. (1); n is a positive integer) for different values of 1 /n . For n = 1, g = 0 and k = 0 . n → ∞ g = 1 = k . However, at g ≃ . ≃ k , g crosses k and then turns again and become equal atextreme inequality. This multi-valued equality property of k as function of g seems to restrict the inequalitymeasure at the limiting value of k (= g ) at 0.86 below its extreme value at unity. The inset shows the k c values obtained by fitting the different k and g values (for different n ) to the linear equation (2) and thensolving for k = k c = g . k g FIG. 3. Shows the plot of the different k values vs. corresponding g values (for different n in eqn. (1); see Fig.2). Also the k = g line is shown. A linear extrapolation (2) of the Initial part of k vs. g suggests k = g = 0 . k = g ≃ . such, it gives a quantitative generalization of more than a century old phenomenologically established80-20 law of Pareto [14], saying that in any economy, typically about 80% wealth is possessed by only20% of the richest population. Defining the Complementary Lorenz function L ( c ) ( x ) ≡ [1 − L ( x )],one gets k as its (nontrivial) fixed point [12, 13]: L ( c ) ( k ) = k (while Lorenz function L ( x ) itselfhas trivial fixed points at x = 0 and 1). Kolkata index ( k ) can also be viewed as a normalizedHirsch index ( h )[16] for social inequality as h -index is given by the fixed point value of the nonlinearcitation function against the number of publications of individual researchers. We have studied themathematical structure of k -index in [12] (see [13] for a recent review) and its suitability, comparedwith the Gini and other inequality indices or measures, in the context of different social statistics, in[8–13]. III. NUMERICAL STUDY OF g AND k FOR A GENERIC FROM OF LORENZFUNCTION
For various distributions of wealth, citations or deaths, the generic form of the Lorenz function L ( x )is such that L (0) = 0 and L (1) = 1 and it grows monotonically with x . As a generic form, we assume L ( x ) = x n , (1)where n is a positive integer. For n = 1, the Lorenz curve falls on the equality line and one gets g = 0, k = 0 .
5. For n = 2, g = 1 / k = (( √ − /
2) becomes inverse of the Golden ratio [9, 12, 13]. Forincreasing values of n , both g and k approach unity ( g = ( n − / ( n + 1) and k n + k − g and k (see also [9] for similar features in the case of special distributions), and shown in Figs. 2 and 3. k g k c k c FIG. 4. Plot of estimated values of k against g from the the web of science data for citations against paperspublished by authors from different universities or institutes and also of the publications in different journals(from refs. [8, 10]). Similar data for the death distributions in various social conflicts or wars [11] are alsoshown. The inset shows the linear extrapolation (2) for k c = k = g plotted against k . IV. DATA ANALYSIS FOR CITATIONS OF PAPERS BY INSTITUTIONS ANDINDIVIDUALS
First we reanalyze the Web of Science data [8, 10] for the citations received by papers published byscientists from a few selected Universities and Institutions of the world and citations received by paperspublished in some selected Journals. We also added the the analysis of the data from various WorldPeace Organizations and Institutions [11] for human deaths in different wars and social conflicts. In K o l k a t a i nde x ( k ) Hirsch index (h) mean ( - k) = 0.83standard deviation ( σ ) = 0.04 within standard deviationexcluding Nobel laureatesonly Nobel laureates 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 k h - k = 0.82 σ = 0.03excluding Nobel laureates 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 k h - k = 0.86 σ = 0.04only Nobel laureates 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 FIG. 5. Plot of the estimated values of Kolkata index k against Hirsch index h [17] of 100 individual scientists,including 20 Nobel Laureates, from the citation data of Google Scholar (Table I). The separate insets clearlyshow that the average value of k index for Nobel Laureates ( k = 0 .
86) is distinctly higher than that ( k = 0 . K o l k a t a i nde x ( k ) Gini index (g) k c k c = k line FIG. 6. Plot of k against g for the citation statistics of individual scientists from Table I. It gives k =0 . ± .
06. This plot may be compared with similar plot in Fig. 4 for paper citation statistics of theuniversities, institutions or journals.
Fig. 4, we plot the estimated values of k against g for citations received by papers published byauthors from different universities or institutes and also of the publications in different journals, aswell as from data for deaths distributions in various social conflicts. Noting (see Fig. 2) that k hasapproximately a piece-wise linear relationship with g as k = 0 . Cg, (2)with a constant C , we estimate the C values from the data points in Fig. 2, and using that we makea linear extrapolation for k c = k = g (see the inset).We have estimated here the values of Kolkata index k against the respective Hirsch index h [17] for100 individual scientists, including 20 Nobel Laureates (each having more than 100 papers/entries andminimum h index value 20, in their, ‘e-mail-site-verified’, Google Scholar page) from the respectivepaper citations (Table I). In Fig. 5, we plot the estimated values of k against h of all these 100individual scientists. The statistics suggests the k index value (0 . ± .
04) to be independent of the h index value (in the range 20 ≤ h ≤ k values ( k = 0 . ± .
03) plottedagainst respective h values for 80 scientists who are not Nobel Laureates and another exclusively forthe 20 Nobel Laureates. This clearly shows that the limiting values of k index for the Nobel Laureateson average are higher ( k = 0 . ± . k against the g values of theirpublication statistics and the inset shows the estimated k c values obtained using eqn. (2) and solvingfor k c = k = g .An interesting observation from Table I has been that the h index value of an author seems to growwith number N of publications, statistically speaking, following a power law h ∼ √ N (see Fig. 7,where the inset for the Nobel Laureates suggests a better fit). h i nde x ( h ) papers/documents (N) excluding Nobel laureatesonly Nobel laureatesh ~ N h N only Nobel laureatesh ~ N FIG. 7. The data for Hirsch index h values in Table I suggest the relation h ∼ N γ , with N denoting totalnumber of papers and γ ∼ . V. SUMMARY AND DISCUSSIONS
Social inequalities in every aspects, resulting from competitiveness are described by various distribu-tions (like Log-normal, Gamma, Pareto, etc., see e.g., [18, 19]). Economic inequality has long beencharacterized [15] by the Gini index ( g ) and a few other (much) less popular geometric characteri-zations (see e.g., [20]) of the Lorenz curve or function L ( x ) [16] (see Fig. 1). We introduced [8] theKolkata index ( k ) as a fixed point of the Complementary Lorenz function L ( c ) ( x )( L ( x ) has trivialfixed points at x = 0 and 1). Unlike the Gini index, which measures some average properties of theLorenz function, Kolkata index gives a tangible interpretation: (1 − k ) fraction of rich people or papersor social conflicts possess or attract or cause k fraction of wealth or citations or deaths respectively.Assuming a generic form L ( x ) ∼ x n (as in eqn. (1), giving L (0) = 0 and L (1) = 1 and monotonicincrease parametrized by n , as desired), we see (in Figs. 2 and 3) that as inequality increases (withincreasing n ) from equality k = 0 . g = 0 for n = 1 to extreme inequality k = g = 1 as n → ∞ , k has a non-monotonic variation with respect to g such that k and g crosses at k = g ≃ .
86 and theyfinally meet at k = g = 1. As the Gini index ( g ) is identified (see [21]) as the information entropy TABLE I. Statistical analysis of the papers and their citations for 100 ‘randomly chosen’ scientists (in-cluding 20 Nobel Laureates; denoted by * before their names) in physics (Phys), chemistry (Chem), biol-ogy/physiology/medicine (Bio), mathematics (Maths), economics (Econ) and sociology (Soc), having individ-ual Google Scholar page (with ‘verifiable email site’) and having at least 100 entries (papers or documents,latest not before 2018), with Hirsch index ( h ) [17] value 20 or above. These authors (including the NobelLaureates) have Hirsch index in the range 20-222 and number of papers ( N ) in the range 111-3000. The datawere collected from Google Scholar during 1st week of January 2021 and names of the scientists appear herein the same form as in their respective Google Scholar pages.name total total index valuespaper citations h g k k c *Joseph E. Stiglitz(Econ) 3000 323473 222 0.90 0.88 0.86H. Eugene Stanley(Phys) 2458 200168 192 0.86 0.84 0.83C. N. R. Rao(Chem) 2400 121756 157 0.77 0.80 0.81*Hiroshi AMANO(Phys) 1300 44329 97 0.80 0.81 0.83didier sornette(Phys) 1211 46294 103 0.80 0.81 0.82Hans J. Herrmann-Phys) 1208 36633 100 0.75 0.79 0.81Giorgio Parisi(Phys) 1043 88647 123 0.83 0.83 0.82George Em Karniadakis(Math) 1030 53823 105 0.84 0.83 0.83Richard G M Morris(Bio) 950 70976 110 0.89 0.87 0.85debashis mukherjee(Chem) 920 15169 59 0.83 0.83 0.83*Joachim Frank(Chem) 853 48077 113 0.80 0.81 0.82R.I.M. Dunbar(Soc) 828 65917 124 0.81 0.82 0.82C. Tsallis(Phys) 810 36056 78 0.88 0.86 0.84Biman Bagchi(Chem) 803 23956 75 0.77 0.79 0.81Srinivasan Ramakrishnan(Phys) 794 6377 38 0.78 0.80 0.82*William Nordhaus(Econ) 783 74369 117 0.87 0.86 0.85Ronald Rousseau(Soc) 727 15962 57 0.83 0.83 0.82*David Wineland(Phys) 720 63922 112 0.88 0.87 0.86*Jean Pierre Sauvage(Chem) 713 57439 111 0.73 0.77 0.80*Gregg L. Semenza(Bio) 712 156236 178 0.81 0.82 0.82*G´erard Mourou(Phys) 700 49759 98 0.82 0.83 0.83Jean Philippe Bouchaud(Phys) 688 44153 101 0.82 0.82 0.82*Frances Arnold(Chem) 682 56101 127 0.75 0.79 0.81Dirk Helbing(Phys) 670 60923 104 0.86 0.85 0.84T. Padmanabhan(Phys) 662 26145 74 0.86 0.84 0.84Gautam R. Desiraju(Chem) 661 59333 95 0.84 0.83 0.83Brian Walker(Bio) 656 136565 96 0.93 0.91 0.89A. K. Sood(Phys) 626 24076 62 0.82 0.81 0.81Masahira Hattori(Bio) 618 80069 98 0.90 0.87 0.85Joshua Winn(Phys) 611 45701 85 0.88 0.85 0.84Kaushik Basu(Econ) 584 21506 66 0.86 0.85 0.84*Abhijit Banerjee(Econ) 578 59704 91 0.89 0.88 0.86Kimmo Kaski(Phys) 567 19647 67 0.80 0.81 0.82*Esther Duflo(Econ) 565 69843 92 0.91 0.89 0.87*Serge Haroche(Phys) 533 40034 90 0.87 0.86 0.85Peter Scambler(Bio) 518 31174 92 0.81 0.81 0.82Spencer J. Sherwin(Maths) 496 15383 63 0.83 0.83 0.83*Michael Houghton(Bio) 493 49368 96 0.83 0.83 0.83*A. B. McDonald(Phys) 492 20346 50 0.91 0.88 0.86Mauro Gallegati(Econ) 491 10360 50 0.80 0.82 0.83A. K. Raychaudhuri(Phys) 470 12501 56 0.78 0.81 0.82Sidney Redner(Phys) 409 26287 74 0.78 0.80 0.81Janos Kertesz(Phys) 407 20115 69 0.80 0.81 0.82Jayanta K Bhattacharjee(Phys) 394 3674 30 0.74 0.78 0.81Alex Hansen(Phys) 393 9678 50 0.76 0.80 0.82Prabhat Mandal(Phys) 386 4780 35 0.75 0.79 0.81Bikas K Chakrabarti(Phys) 384 10589 44 0.81 0.82 0.83Ashoke Sen(Phys) 379 33342 97 0.69 0.76 0.80*Paul Milgrom(Econ) 365 102043 82 0.90 0.89 0.87Ramasesha S(Chem) 362 7188 44 0.78 0.80 0.82 name total total index valuespaper citations h g k k c Noboru Mizushima(Bio) 347 117866 122 0.82 0.83 0.83William S. Lane(Bio) 334 72622 123 0.74 0.78 0.80Debraj Ray(Econ) 322 23558 65 0.85 0.85 0.85Beth Levine(Bio) 321 103480 116 0.81 0.82 0.83Debashish Chowdhury(Phys) 320 8442 36 0.88 0.86 0.84Toscani Giuseppe(Math) 299 10129 54 0.75 0.79 0.82Matteo Marsili(Phys) 294 8976 48 0.77 0.80 0.82Rosario Nunzio Mantegna(Phys) 289 29437 63 0.88 0.86 0.85Diptiman Sen(Phys) 286 6054 41 0.74 0.78 0.80J. Barkley Rosser(Econ) 281 5595 38 0.81 0.81 0.82*David-Thouless(Phys) 273 47452 67 0.89 0.87 0.86Sanjay Puri(Phys) 271 6053 39 0.79 0.81 0.82Maitreesh Ghatak(Econ) 263 11942 43 0.89 0.87 0.86Serge GALAM(Phys) 258 7774 41 0.82 0.83 0.84Sriram Ramaswamy(Phys) 257 13122 46 0.87 0.85 0.84*Paul Romer(Econ) 255 95402 54 0.96 0.93 0.90Krishnendu Sengupta(Phys) 251 7077 36 0.86 0.85 0.84Chandan Dasgupta(Phys) 248 6685 42 0.76 0.79 0.81Scott Kirkpatrick(CompSc) 245 80300 64 0.95 0.91 0.88*richard henderson(Chem) 245 27558 62 0.84 0.84 0.84*F.D.M. Haldane(Phys) 244 41591 68 0.87 0.86 0.86Kalobaran Maiti(Phys) 235 3811 32 0.86 0.84 0.83Amitava Raychaudhuri(Phys) 235 3522 34 0.74 0.78 0.81Bhaskar Dutta(Econ) 232 6945 43 0.82 0.83 0.84Ganapathy Baskaran(Phys) 232 6863 29 0.91 0.89 0.87Hulikal Krishnamurthy(Phys) 231 14542 46 0.86 0.85 0.84Rahul PANDIT(Phys) 226 6067 35 0.82 0.82 0.82W. Brian Arthur(Econ) 225 47014 52 0.92 0.90 0.88Pratap Raychaudhuri(Phys) 224 4231 34 0.80 0.82 0.83Jose Roberto Iglesias(Phys) 217 1819 22 0.77 0.80 0.82Hongkui Zeng(Bio) 208 18914 60 0.82 0.82 0.83Deepak Dhar(Phys) 200 7401 43 0.77 0.80 0.82Sitabhra Sinha(Phys) 193 2855 32 0.76 0.80 0.83Amol Dighe(Phys) 189 8209 49 0.76 0.80 0.82Arup Bose(Maths) 186 1965 20 0.73 0.77 0.79Abhishek Dhar(Phys) 177 5004 38 0.73 0.78 0.80S. M. Bhattacharjee(Phys) 171 2268 27 0.72 0.78 0.81Martin R. Maxey(Maths) 168 10124 43 0.86 0.84 0.83Arnab Rai Choudhuri(Phys) 164 6115 39 0.81 0.82 0.83Victor M. Yakovenko(Phys) 158 7699 43 0.72 0.78 0.81Md Kamrul Hasan(Phys) 147 1844 23 0.66 0.74 0.79Shankar Prasad Das(Phys) 145 2476 24 0.81 0.81 0.81Amit Dutta(Phys) 137 2845 28 0.79 0.81 0.82Anirban Chakraborti(Phys) 135 4809 28 0.83 0.84 0.85Parongama Sen(Phys) 129 3062 21 0.82 0.83 0.83Roop Mallik(Bio) 122 3363 26 0.83 0.83 0.84Wataru Souma(Phys) 117 2607 24 0.82 0.82 0.83Subhrangshu S Manna(Phys) 117 4287 28 0.75 0.80 0.83Damien Challet(Math) 112 5521 27 0.86 0.85 0.85*Donna Strickland(Phys) 111 10370 20 0.95 0.92 0.90 of social systems and the Kolkata index ( k ) as the inverse of effective temperature of such systems(increasing k means decreasing average money in circulation and hence decreasing temperature [18]),this multivaluedness of (free energy) g/k as function of (temperature) k − at g = k ≃ .
86 and g = k = 1 (Figs. 2 and 3) indicates a first order like (thermodynamic) phase transition [22] at g = k ≃ . k index value, in extreme limits of social competitiveness, convergetowards a high value around 0 . ± .
03 (see Fig. 4), though not near the highest possible value k = 1(maximum possible value for extreme inequality). Indeed, k index gives a quantitative generalizationof the century old 80 −
20 rule ( k = 0 .
80) of Pareto [14] for economic inequality (though, as mentionedearlier, the economic inequality statistics today for various countries of the world shows [8] much lower k values in the range 0 . − .
73, because of various economic welfare measures).In summary, using a generic form (valid for all kinds of inequality distributions) of the Lorenzfunction L ( x ) (= 0 for x = 0 and = 1 for x = 1 and monotonically increasing in-between), we showed(see Fig. 2) that as inequality increases (with increasing values of n in eqn. (1)), the differencein values between k (initially higher in magnitude) and g , both increasing, the difference vanishes at k = g ≃ .
86 (after which g becomes higher than k in magnitude) until the point of extreme inequality( n → ∞ ) where k = 1 = g , where they touch each other in magnitude. We consider this crossingpoint of k = g ≃ .
86, which is higher than the Pareto value 0 .
80 [14]), as an instability point andinduces a limiting and universal value for the inequality measure k for the various distributions indifferent social sectors.Our earlier citation analysis [8, 10] from the web of science data for citations against papers publishedby authors from different established universities or institutes and also of the publications in differentcompetitive journals indicate the limiting value of k to be 0 . ± .
03. Similar analysis for humandeaths in different deadly wars or social conflicts [11] also suggests similar limiting value of k (see Fig.4). These are a little higher than the Pareto value [14] of k (= 0 . N in theirrespective Google Scholar page, with ‘verifiable email site’, and having the Hirsch index h value 20or more) suggests k = 0 . ± .
04 (see Fig. 5) and independent of the h index value in the range20 − k = 0 . ± .
4, again independent h index value) saying that for any of them, typically about 14%of their successful papers earn about 86% of their total citations. It may be interesting to note thatGoogle Scholar has developed a page [23] on Karl Marx, father of socialism, and it is maintained bythe British National Library. The page contains 3000 entries (books, papers, documents, published byMarx himself, or later collected, translated, and edited by other individuals and different institutions.According to this page, his total citation counts 387995 and his Hirsch index value ( h ) is 213. Ourcitation analysis says, his Kolkata index value ( k ) is 0.88, suggesting again that 88% of his citationscomes from 12% of his writings! From Table I, we also note that individual’s h index value grows, onaverage, with the total number N of his/her publications as h ∼ √ N (more clearly so for the NobelLaureates; see the inset of Fig. 7), and as such Hirsch index has no limiting value (as a universallimiting social number).To conclude, our analytic study of the properties of the Gini ( g ) and Kolkata ( k ) indices for socialinequality, based on a generic form (eqn. (1)) of the Lorenz function L ( x ) (in section II), and someobservations on the citation statistics of individual authors (including Nobel laureates), institutionsand journals (also on the statistics of deaths in wars etc), we make a conjecture that about 14% ofpeople or papers or social conflicts earn or attract or cause about 86% of wealth or citations or deathsin very competitive situations in the markets, universities or wars. This is a modified form of the(more than a) century old 80 −
20 law of Pareto in economy which is not visible in today’s economiesbecause of various welfare strategies to mitigate poverty). This limiting value of the Kolkata indexfor inequality k ( ≃ .
86) gives perhaps an universal social constant or number.
ACKNOWLEDGMENTS