Econophysics Through Computation
EEconophysics Through Computation
Antika Sinha, ∗ Sudip Mukherjee, † and Bikas K Chakrabarti
3, 4, 5, ‡ Asutosh College, Kolkata, 700026, India Barasat Government College, Barasat, Kolkata 700124, India. Condensed Matter Physics Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India. S. N. Bose National Centre for Basic Sciences, Kolkata 700106, India. Economic Research Unit, Indian Statistical Institute, Kolkata-700108, India.
We introduce here very briefly, through some selective choices of problems and through the samplecomputer simulation programs (following the request of the editor for this invited review in the Jour-nal of Physics Through Computation), the newly developed field of econophysics. Though relatedattempts could be traced much earlier (see the Appendix), the formal researches in econophysicsstarted in 1995. We hope, the readers (students & researchers) can start themselves to enjoy theexcitement, through the sample computer programs given, and eventually can undertake researchesin the frontier problems, through the indicated survey literature provided.
I. INTRODUCTION
The research field of econophysics has emerged recently as economics-inspired statistical physics. Though the attemptsare not new and in fact almost a century old (see the Appendix), the institutionalization of this research field, where(statistical) physicists can and do regular researches in their own departments and publish their relevant results intraditional and contemporary physics journals, is new (see e.g., [1–6]). Indeed the term econophysics had been formallycoined in 1995 (see the Appendix and Fig. 1) and we are now in the silver jubilee celebration year! This review isintended for interested students for self-studies and self-learning through computational modelings of a few selectedproblems in econophysics. Some elementary computer programs or codes (in Fortran or Python) are added for readysupport. We first introduce a few popular research problems in econophysics and continue discussion on them in thefollowing. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ phy s i c s . s o c - ph ] J a n F r e qu e n c y Year
Econophysics in Google Scholar
FIG. 1: Histogram plot of numbers of entries per year containing the term econophysics versus the corresponding year. Thedata are taken from Google Scholar (dated: 31 December, 2019). It may also be noted from Google Scholar that while this24-year old econophysics has today typical yearly citation frequency of order 1 . × , more than 100-year old subjects likeastrophysics (Meghnad Saha published his ionization equation for solar chromosphere in 1920), biophysics (Karl Pearson coinedthe term in his 1892 book ‘Grammar of Science’) and geophysics (Issac Newton explained planetary motion, origin of tides, etcin ‘Principia Mathematica’, 1687) today have typical yearly citation frequencies of order 36 . × , 40 . × and 45 . × respectively. Inequality in income or acquired wealth has been ubiquitous: not only today, even in the earliest days of thehuman civilization. It is hard to find any society with fare amount of equality in income or wealth for everyone. Ithas continued to exist and sometimes have grown enormously in societies for centuries, and even today threateningour existence and wellbeing. Studies on inequality in wealth has a long history and has fascinated generations ofphilosophers, economists, social scientists and thinkers alike. Analysis of real data on wealth distribution is not new:with the advent of digital era, researchers from social science, even from interdisciplinary branches have been studyingthe recorded bulk amount of human social interaction data to explore the hidden structure of these data and alsoinvestigate the reason behind such inequality and so on. There are many ways to quantify inequality present in somesocial context or opportunity, e.g. income, wealth, as we know income is taken as a measure of economic growth ofany country. The popular measures, summarized in one value, are Gini, Pietra indices etc. We will also discuss therecently introduced Kolkata index for measuring social inequality. All these make use of the Lorenz curve plot. On theother hand, one can also study the probability distribution and find the trend present in empirical data. Interestingly,the above mentioned indices computed from the Lorenz curve which is also plotted using the cumulative probabilitydistribution of some context data against number of occurences. The study can also be extended to uncover thetemporal pattern of social inequality from those data spread over years.Along with income or wealth inequality, we also put together inequality indices values measured over several othersocial context, e.g., academic citation count, city size or population, voting data, human death counts from socialconflict which can be man-made like war, battle or natural disasters like earthquake, flood etc. These studies,sometimes done over couple of century-wide data (which are publicly available), have helped to uncover interestingpatterns (of the presence of higher inequality but not highest inequality).Next we study about the Kolkata Paise Restaurant (KPR) problem. KPR is a repeated many-player many-choicegame played by a large number of players ( N ). Every day each player will choose a restaurant and visit there forlunch. Each restaurant prepares a single dish that costs the same at any restaurant. Player can only make singlechoice per day, and lunch at chosen restaurant is guaranteed if visited alone there for that day. Any day if morethan one player choose the same restaurant then only one of them gets the dish and others arriving there would misstheir lunch. Information about the restaurant occupations for few finite previous days (depending on the memorycapacity of the players) will be publicly available. But players do not interact with others while making decision i.e.choosing a restaurant for lunch that day. In such set up, how the players should set individual choice towards sociallyoptimal solution i.e. no food waste as well as no player staying hungry that day. A simple solution could be to hire anon playing captain (dictator) who would assign some restaurant to the players, thus all of them get their food fromfirst day and following this setting till end. But in a democratic setup, players would like to make their own choicefollowing some collectively learning strategy. Here the objective is to achieve maximum social utilization of the scarceresources in absence of some external coordinator. This makes the KPR game interesting. Some results on statisticsof KPR dynamics over variety of strategies developed, as well as an interesting phase transition phenomena have beendiscussed in respective section.Another important and popular research-work deals with the economic/social networks evolved due to social inter-actions through market dynamics and game. In this context we will discuss about the Indian Railway network (IRN)as a complex (transport) network where each railway station is vertex and track between any two connecting stationsis the edge between them. Study reveals IRN as a small world network, a popular network model where mean distancebetween any two nodes becomes constant as network size grows. Thus the graph like study has successfully answeredseveral interesting questions like how many trains one passenger would require to switch to reach any destinationstation within country while traveling by train etc.We discuss the attempts made in the developments of the microscopic dynamical models (mainly based on kinetictheory), which can explore the underline dynamics behind the making of the real-world income or wealth distribution.We also highlight few models which show the possible ways of minimizing the socioeconomic inequality. cumulative fraction of people from poorest to richest c u m u l a t i v e f r a c t i o n o f w e a l t h h e l d b y t h e m AE FBD C
S S (k,1-k)k e q u a li t y li n e L o r e n z c u r v e FIG. 2: Lorenz curve (in red here) plot represents the accumulated wealth against the fraction of people possessing that whenarranged from poorest to richest. The diagonal from the origin represents the equality line. The Gini index ( g ) can be measuredfrom the (shaded) area ( S ) normalized by the area ( S + ¯ S = 1 /
2) of total area of the triangle below the equality line): g = 2 S .The k index can be measured by the ordinate value of the intersecting point of the Lorenz curve and the diagonal perpendicularto the equality line. It says that k fraction of wealth is being held by 1 − k fraction of top richest population. II. MEASURING SOCIAL INEQUALITY AND THE KOLKATA INDEX
Inequality present in socio-economic data can be quantified in several ways. Mostly used is the Gini index ( g )introduced by Italian statistician and sociologist Corrado Gini in 1912 [7]. There are others as well like Kolkata index( k ), Pietra index ( p ) etc. Usually we plot histogram or frequency distribution to get the initial idea of any data inhand. And those indices mentioned above are easily measurable scalars from the cumulative probability distributionof the respective quantity. This distribution when plotted is known as Lorenz curve. Study of real data on the socio-economic context has been reported to follow fat tail, lognormal or gamma like distribution, see. Below we discusssome standard measures and techniques to get a general idea on how to measure inequality present in some quantity. Lorenz curve:
Lorenz curve was proposed by Max O. Lorenz in 1905. The curve always begins from point (0 , ,
1) as seen in Fig. 2. If one plots the fraction of population or household in increasing order of X -axis against fraction of wealth held by them gives the cumulative probability distribution function F . Thus, thefraction of total wealth held by bottom x% population is represented by y = F ( x ). More the fractions grow close toeach other, i.e. if x % ∼ y % then the Lorenz curve becomes a straight line representing perfect equality in wealth orincome distribution. Far the curve deviates from the diagonal or 45 ° line mean presence of greater inequality in thedistribution. Intuitively, the curve never rise above the equality line ( X = Y ). We also discuss about some inequalitymeasuring indices, to be obtained using the Lorenz curve. Gini index ( g ): Gini index ( g ) is a standard measure of inequality, used not only by economists but also researchersacross other disciplines e.g. physicist, social scientists etc. For computing the Gini index ( g ), one can fit Lorenz curvein a unit square, where the ratio of area between the Lorenz curve and equality line to the area below the equalityline gives the Gini index ( g ) value. In Fig. 2, S represents the area between the Lorenz curve and equality line and import matplotlib.pyplot as pltimport random as rddata=[9,14990,78940,21480,27490,63337,184396,10410,92650,0,174,64000,71,10280,46050,3116,62340,21754,5620,11409,12840,78730,47000,40468,1170,2650,275300,5000,4110,14162,18,16543,39700,0,24780,35689,365099,1440,82360,304100,885208,14653,60,10463,970,51770,115319,0,0,2580,3200,15490,65810,83722,11920,10316,25340,436800,1600,3200,1250,1930,58310,25850,176799,44203,430,770,23940,590,61810,2]X=[]Y=[]j=0data=[x for x in data if x>=0]data.sort()for i in range(len(data)+1):X.append(j/float(len(data)))Y.append(sum(data[:j])/float(sum(data)))j=j+1x1=0.0x2=1.0rdm=1000lenx=len(X)step=1/float(lenx) ax=[]for i in range(int((x2-x1)/step)):ax.append(x1+(i*step)) csum=0for i in range(len(ax)):for j in range(rdm):u=ax[i]d=Y[i]f=rd.random()if f<=u and f>=d:csum+=1total=csum/float(rdm*lenx)g=2*totalprint('g',g)dY=[]for i in range(len(X)):dY.append(1-X[i])plt.axis([0,1,0,1])plt.plot(X,X,'g-')plt.plot(X,Y,'b-' )plt.plot(X,dY,'m:' )plt.ylabel('cumulative share of Y')plt.xlabel('cumulative share of X in increasing order of Y')plt.show() FIG. 3: Python (version 2 .
7) program to plot Lorenz curve based on hypothetical data, and also to compute gini index value.If run this program, one obtains g value around 0 . ( S + ¯ S ) represents the area below the equality line, then (cid:16) g = SS + ¯ S (cid:17) . It ranges from 0 to 1 where g = 0 denotesperfect equality, say every individual has same income and plotting this should give the equality line (see Fig. 2). Andwhen g = 1, it represents maximum possible inequality where in a society only one person has every wealth and restleft with none. But then, Gini index ( g ) is a summary measure. Below we discuss Kolkata index ( k ), Pietra index ( p )etc. to be estimated using the same Lorenz curve plot. Kolkata index ( k ): Ghosh et al. has recently proposed ‘Kolkata index’( k ) [8], another inequality measuring indexto be obtained from the same Lorenz curve plot. The k value can be estimated from the X -axis point where theLorenz curve intersects the diagonal line perpendicular to the line of equality, see Fig. 2. So, the k -index ranges from0 . F ( n ) = 1 − F ( n ), denotes k -index as anon-trivial fixed point on x -coordinate such that ˜ F ( k ) = k where ˜ F ( n ) intersects the diagonal line spanning between(0 ,
0) and (1 , g ) is frequently used by the economist. Note that the g -index represents theoverall summary of inequality, whereas k -index is the indicator of the fraction of x quantity held by 1 − x fraction ofpopulation. Already mentioned that, g varies between 0 to 1, whereas k ranges from 0 . A. g and k index values for several types of real data: Here we discuss results from various kinds of social events where inequality is measured (for e.g., citations, income,expenditure, vote, city size, human-death counts from social conflicts etc.) and some interesting results are observed.Chatterjee and Chakrabarti, 2016 [10] studied probability distribution of many publicly available (by several univer-sities and peace research institutes) data on human death counts occured from wars, armed-conflicts as man-madedisasters as well as natural disasters like earthquake, forest fires etc. The distribution plots showed up to follow powerlaw, in the fat-tail region, with exponent ζ around -1 .
5, see TABLE I. Extending their work, Sinha and Chakrabartiin [11] measured the corresponding g and k index values by plotting Lorenz curves using many of the data discussedin [10] and the g and k values were found to be very high, see TABLE II and TABLE III. Such high values indicatessevere inequalities which is rarely seen in economic states of different countries (presumably because of deliberatesupports to economically weaker groups, etc.). Those indices are also measured for the physical quantities (in caseof natural disasters only, e.g., Richter magnitudes for earthquake, areas affected in sq. km. for floods, maximumwater height in m for tsunami) causing such unequal occurences of human deaths TABLE IV. Surprisingly not muchinequality is observed in this case in comparison to those social effects caused by them. To establish the ‘similarityclasses’ of social inequality, these results were compared against those inequality measures found from man-madecompetitive societies like academic institution (paper citation counts), see TABLE V. And the study indicated thegrowing recent trend of economic inequality across the world, may be because of encouraging competitiveness insocieties as well as due to fast disappearance of social welfare measures in recent years. TABLE I: Estimated value of power law exponent ζ for man-made, natural conflicts from [10]. Type of conflicts/disasters Time period ζ conflict 1946-2008 1 . ± . . ± . . ± . . ± . . ± . . ± . Type of conflicts Time period g -index k -index war 1816-2007 0 . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± B. Relationship between Gini ( g ) and Kolkata ( k ) index In Fig 2, the Lorenz curve is represented by the red color line obtained using a probability distribution function F ( x ). Assume, X denotes the cumulative fraction of x taking lowest to highest order and Y as the cumulative fractionof y . The Lorenz curve cuts the anti-diagonal Y = 1 − X at point C ( k, − k ). This gives the k -index defined as: k TABLE III: Estimated inequality index values for social damages by natural disasters from [11]
Type of disasters Social Damage measuresTime period g -index k -index earthquake 1000-2018(July) 0 . ± . ± . ± . ± . ± . ± Type of disasters Physical Damage measuresTime period g -index k -index earthquake 2013-2018(July) 0.35 0.64flood 1900-2018(July) 0.76 0.79tsunami 1000-2018(July) 0.53 0.69 fraction of wealth is possessed by 1 − k fraction of people. Here, the shaded area S is enclosed by the Lorenz curve( ACB ) and the equality line (
ADB ), and the The Gini index g is represented by: g = area of shaded regionarea of the triangle ABE = 2 S (1)Here we would study some approximate ways to measure S . In Fig. 2, we can assume the Lorenz curve to be givenby two broken straight lines AC and CB . Here, AB = √ DF = √ , CF = (1 − k ) √
2, thus CD = ( DF − CF ) = √ (2 k − CAB is A = 12 · AB · CD = 12 (2 k −
1) (2)Note that this is an exact result where equality in the relation holds for g = k at g = k = 1. Thinking other way,consider that the Lorenz curve is represented by the arc ACB of a circle with radius r (= AE = BE ), see Fig. 4(a).Find that DE is perpendicular to base AB with ∠ BED = θ . And the area of arc ACB is the difference between thesector
BEAC and the triangle
ABE . The area of sector
BEAC is θr . And, the area of triangle ABE is given by · DE · AB = · rcosθ · rsinθ = r cosθsinθ . Thus the required area ACBD representing the Lorenz curve is given by: A (cid:48) = r ( θ − sinθcosθ ) (3) A C BDE θ (a) π /6 π /4 π /3 π /2 γ = / α θ (b) FIG. 4: (a) Approximating Lorenz curve as an arc of a unit circle (here ACB) and equality line as the chord AB. (b) Plot ofapproximated slope ( γ = α ) of k - g linear relation for different values of θ . TABLE V: Values of the inequality indices ( g and k ) for some of the academic institutions (from [8], see also [12]; source-datataken from the Web of Science). [*Cambridge: University of Cambridge, Harvard: Harvard University, MIT: Massachusetts Institute of Technology,Oxford: University of Oxford, Stanford: Stanford University, Stockholm: Stockholm University, Tokyo: The University of Tokyo, BHU: Banaras HinduUniversity, Calcutta: University of Calcutta, Delhi: University of Delhi, IISC: Indian Institute of Science, Madras: University of Madras, SINP: SahaInstitute of Nuclear Physics, TIFR: Tata Institute of Fundamental Research.] Inst./Univ.* Year Index values for Inst./Univ.* Year Index values for
Gini( g ) Kolkata( k ) Gini( g ) Kolkata( k )Cambridge 1980 0.74 0.78 BHU 1980 0.68 0.761990 0.74 0.78 1990 0.71 0.772000 0.71 0.77 2000 0.64 0.742010 0.70 0.76 2010 0.63 0.73Harvard 1980 0.73 0.78 Calcutta 1980 0.74 0.781990 0.73 0.78 1990 0.64 0.742000 0.71 0.77 2000 0.68 0.742010 0.69 0.76 2010 0.61 0.73MIT 1980 0.76 0.79 Delhi 1980 0.67 0.751990 0.73 0.78 1990 0.68 0.762000 0.74 0.78 2000 0.68 0.762010 0.69 0.76 2010 0.66 0.74Oxford 1980 0.70 0.77 IISC 1980 0.73 0.781990 0.73 0.78 1990 0.70 0.762000 0.72 0.77 2000 0.67 0.752010 0.71 0.76 2010 0.62 0.73Stanford 1980 0.74 0.78 Madras 1980 0.69 0.761990 0.70 0.76 1990 0.68 0.762000 0.73 0.80 2000 0.64 0.732010 0.70 0.76 2010 0.78 0.79Stockholm 1980 0.70 0.76 SINP 1980 0.72 0.741990 0.66 0.75 1990 0.66 0.732000 0.69 0.76 2000 0.77 0.792010 0.70 0.76 2010 0.71 0.76Tokyo 1980 0.69 0.76 TIFR 1980 0.70 0.761990 0.68 0.76 1990 0.73 0.772000 0.70 0.76 2000 0.74 0.772010 0.70 0.76 2010 0.78 0.79 Now if we write A = A (cid:48) = α · AB · CD , then α = θ − sinθcosθsinθ (1 − cosθ ) (4)where α is a fraction incorporated to get the approximate result of A (cid:48) as α A . Thus one gets g (cid:39) A (cid:48) = 2 α A = α (2 k −
1) using Eq. 2, and this gives: k = 12 + 12 α g (5)This is the general result of g and k relationship with slope γ = α . Fig. 4(b) also shows variation of α against θ .From Fig. 5, one can find the observed approximate value of γ = 0.365 which corresponds to θ = π/ E such that2 θ = π/ k valueusing Eq. 5 that should come as 0 . k = 0 . . g . For more details see [12, 13]. As the k -index ranges from 0 . k -1 is also considered which represents the maximum vertical distance between the equalityline and Lorenz curve. This is how done as alike g -index, the 2 k − , g and 2 k − k g k = . + . Inst CitationJour CitationIN consu 2009-10 MPCEIN consu 2009-10 MPECEIN consu 2004-05IN consu 2011-12BR consu 2002-03BR consu 2008-09IT consuUS Incomeincome data of Ref. [23]IN voteWB vote UP voteMP voteAP voteBihar voteUK voteCA voteBD voteTZ voteOPE voteCity Populationk=0.5+0.365gk=g
FIG. 5: Plot of the relationship between Kolkata index( k ) and Gini index ( g ) found from various types of data, e.g., citation,income, city population, vote share, etc. The data fits Eq. 5 quite well. Above figure is taken from ref. [12]. calamities etc. Two slope is observed when 2 k − g : about 0 .
73 (for lower range of g ) and 1 . g ). Though for both the cases, g values are observed to be higher than corresponding 2 k − g k Man-made losses (social)Natural damages (social)Natural damages (physical)slope ~ 1.50slope ~ 0.73
FIG. 6: Plot of 2 k − g measured for human death count data from social conflict, also see TABLE II & TABLE III.Both the estimates range from 0 to 1. For both of the slopes, 2 k − g values(see Ref. [11]). III. ECONOPHYSICS OF INCOME AND WEALTH DISTRIBUTION
As discussed already, irrespective of history, culture and economic policies followed, socioeconomic inequalities arefound to be omnipresent across the globe and throughout the ages. Indeed some robust features of such unequaldistributions of income or wealth are already established (see e.g., [5, 14]): While the overall the income or wealthdistribution fits generally a Gamma-like curve (see some typical income distribution in Fig. 7; whereas economistsstill like to fit it to a log-normal curve). The tail end of the distribution (for large income or wealth) decays followinga robust power law, called the Pareto law (see Fig. 7). Physicists today have been trying to capture such genericfeatures of the income or wealth distribution, using models based on the kinetic theory of ideal gases, where theinteractions among the ‘social-atoms’ or agents (traders) due to a trade (involving money exchanges), are consideredas a two-body scattering problem where the total money (like energy) before and after the trade, remains conserved.To best of our knowledge, the first text book (‘A Treatise on Heat’) on the statistical thermodynamics, which discussedthe application of the kinetic theory (of ideal gas od ‘social-atoms’) to the derive the income or wealth distribution,was written by Saha and Srivastava [15]. U nno r m a li s ed p r obab ili t y , P ( m ) Individual weekly income in GBP, m (a)UK U nno r m a li s ed p r obab ili t y , P ( m ) Individual weekly income in $, m
USA 1992 (b) manufacturingservices 0 0.02 0.04 0.06 0.08 0.1 0.12 0 20 40 60 80 100 120 P r obab ili t y den s i t y Individual annual income, k$Census 1996 -5 -4 -3 -2 -1 -1 C u m u l a t i v e p r obab ili t y , Q ( m ) Income in thousand USD, m(c) USA 2001 -7 -6 -5 -4 -3 -2 -1 C u m u l a t i v e p r obab ili t y , Q ( m ) Income in thousand JPY, m(d) Japan 2000
FIG. 7: (a) Distribution P ( m ) of individual weekly income in UK for 1992, 1997 and 2002. (b) Distribution P ( m ) of individualweekly income for manufacturing and service sectors in USA for 1992; data for US Statistical survey. The inset shows theprobability distribution of individual annual income, from US census data of 1996. (c) Cumulative probability Q ( m ) = (cid:82) ∞ m P ( m ) dm of rescaled adjusted gross personal annual income in US for IRS data from 2001, with Pareto exponent α p ≈ . ν = 1 .
96. The data and analysis adapted from [14].
A. Saha and Srivastava’s Kinetic Theory for ideal gases with ‘atoms and ‘social agents
In thermodynamic systems, the number atoms or molecules is typically of the order of Avogadro number ( ∼ )whereas the number of social agents even in a global market is about 10 . Still one can imply the statistical physicsprinciples in such economic systems. In their famous book, Saha and Srivastava had put a discussion in the sectionMaxwell-Boltzmaan velocity distribution of ideal gas, which highlights the idea of applying kinetic theory in marketto evaluate the income distribution of a society (see Fig. 8).In case of an ideal gas, at temperature T , the number density of particles P ( (cid:15) ) (atoms or molecules) having energybetween (cid:15) to (cid:15) + d(cid:15) is given by P ( (cid:15) ) d(cid:15) = g ( (cid:15) ) f ( (cid:15) ) d(cid:15) . Here the density of states g ( (cid:15) ) d(cid:15) gives the number of possibledynamical states between the energy (cid:15) to (cid:15) + d(cid:15) . The energy distribution function is denoted by f ( (cid:15) ). We shouldmention that, in case of an ideal gas, the particle can only have the kinetic energy (cid:15) = | (cid:126)p | / w , where (cid:126)p and w are the momentum and mass of the particle respectively. Since momentum is a vector, one can clearly understandthat the particle can have same energy for different momentum vector (cid:126)p . The density of states can be written as g ( (cid:15) ) d(cid:15) = 4 π | (cid:126)p | d | (cid:126)p | = 2 π (2 m ) / √ (cid:15)d(cid:15) . As the energy is conserved during any energy exchanging process, then theenergy distribution should satisfy f ( (cid:15) ) f ( (cid:15) ) = f ( (cid:15) + (cid:15) ) (for any arbitrary values of energy (cid:15) , (cid:15) ). Therefore one canexpect the exponential form of the energy distribution function, which is f ( (cid:15) ) ∼ exp( − (cid:15)/ ∆), where we will identifylater ∆ = K B T . This essentially gives P ( (cid:15) ) d(cid:15) ∼ √ (cid:15) exp( − (cid:15)/ ∆) and from such expression one can evaluate the pressure P of an ideal gas having volume V and temperature T . Comparing such result with the equation of state of ideal gas P V = K B T , one can identify ∆ = K B T .In their chosen example in the section velocity distribution, Saha and Shrivastava indicated that in a closed economicsystem (where no migration of labour, no growth etc are considered) the money m distribution of the agents of the0 S ca nn e d by C a m S ca nn e r FIG. 8: Image of the pages 104 and 105 of the book “A Treatise on Heat” (1931) by Saha and Srivastava [15]. system should have the similar form of the energy distribution function of ideal gas atoms or molecules. This is evidentbecause in a closed economic system the money M is conserved. Hence the money distribution should also satisfythe condition f ( m ) f ( m ) = f ( m + m ). Like the collision (the energy exchange process) between the particles, themoney exchange (due to any kind of trading) between the agents are also random. Therefore, one can expect themoney distribution should be f ( m ) ∼ exp( − m/σ ), where σ is constant. As in the economic system there is no quantitywhich is equivalent to the momentum vector of the gas particle, in this case the density of states is constant. Hencethe number density of agents P ( m ) having money m should be P ( m ) = C exp( − m/σ ). Here C is another constant.Both σ and C can be evaluated using two conditions. One of the condition is N = (cid:82) ∞ P ( m ) dm , where N is the totalnumber of agents. There is another condition M = (cid:82) ∞ mP ( m ) dm . Calculating these two integrations one would get C = 1 /σ where σ = M/N , which is actually the average money per agent. Therefore instead of Maxwell-Bolzmann orGamma distribution of energy in the ideal gas, the money distribution becomes an exponentially decaying function(like Gibbs distribution). Here most number of agents have zero money. For the students, Saha and Shrivastva leftthe task of making this exponential distribution more like Gamma distribution.
B. Data analysis of Income and Wealth distributions
Italian scientist Pareto, from his socioeconomic study in Europe [16] showed that the income distribution containsa power law tail ( f ( x ) ∼ x − (1+ α p ) ). Such an observation is often called Pareto’s law ( α p the Pareto exponent). Lateranother economist Gibrat [17] showed that the part of the income distribution which corresponds to the higher incomerange actually fits well with power law type function whereas the rest of the distribution can be characterized by thelog-normal function f ( x ) ∼ x √ πσ exp − log ( x/x )2 σ . From the analysis of Japanese personal income distribution dataranging from 1887 to 1908, Souma [18] reported about the two type nature of income distribution. also indicated1both Pareto index ( α p ) and Gibrat index ( β g = 1 / √ σ ) varies with time.Several physicists have been rigorously studied the financial data of many countries to capture the form of theincome distribution. From the investigation of the Japanese income tax data of the year 1998, Aoyama et al. [19]observed the power law decay of the income distribution. Dr˘agulescu and Yakovenko analyzed the USA [20] andUK [21] income tax data. From the examinations of such data, they conjectured that the major part of the incomedistribution fits with the exponential function whereas in the higher income range the distribution follows powerlaw decay. Dr˘agulescu and Yakovenko explained the exponential nature of the income distribution in the basis ofstatistical mechanics [22]. They treated the closed economic system as a closed thermodynamic ideal gas systemwhere the money in the economic system is equivalent to the energy in the idea gas system. They considered thefinancial transaction between any two agents is equivalent to the scattering between the two gas molecules or atomsin which they exchange energy. Like total energy of ideal gas system, the total money is also conserved in the closedeconomic system. Therefore the money distribution should follow the Gibbs distribution (exponential) if the densityof state is constant. There are many suggestions on the shape of income distribution. Ferrero [23] proposed that theincome distribution follows Gamma distribution f ( x ) ∼ x n − exp( − x/a ), where n and a are the fitting parameters.Clementi [24] proposed κ generalized function f ( x ) = α β x α − exp κ ( − β x α ) √ κ β x α for the income distribution. Here κ isthe deformation parameter and α , β are fitting parameters. Using κ generalized function, they found good data fitwith USA income data. In their analysis the low income part of the distribution is exponential, which is retrieved bytaking the limit κ →
0. The low income part is actually corresponds to the low energy regime of the physical system,where one can treat such system non-relativistically. As a result of that the nature of the income distribution in thelow income part is exponential. On the other hand the high income part is associated with the physical system in highenergy scale. Therefore the physical system should be treated relativistically and one could not expect exponentialdistribution in the high income regime. The Pareto’s law can be obtained by taking the limit x → ∞ with κ (cid:54) = 0.Analyzing the 1996 Forbes magazine data Levy and Solomon [25] found the existence of Pareto’s law in the wealthdistribution. The Forbes [1] magazine data from year 1988 to 2003 was investigated by Klass et al. [26]. Theyordered rich Americans according to their wealth where the wealth of the r -th person is w r . They found a power law w r ∼ r − (1 /α p ) , where the Pareto exponent α p (cid:39) .
43. The exponent γ is called Zipf exponent. The existence of powerlaw in the wealth distribution of India was reported by Sinha [27].From the analysis of Internal Revenue Service data of USA, Silva and Yakovenko [28] studied the time evolution ofthe income distribution. They found that the form of the income distribution qualitatively remains similar throughoutthe entire period of the observation. Most interestingly the nature of the time evolutions of the lower and the upperparts of the income distribution are different. The lower income part of the distribution for all the years can befitted with a single exponential curve, which actually indicates the thermal equilibrium in this part of the incomedistribution. On the other hand Silva and Yakovenko noticed that the power law tail in high income regime evolvessignificantly with time. They found the Pareto’s exponent α p changes from 2 . . C. Models of the Income and Wealth distributions
There are several attempts have been made to model the observed universal income or wealth distributions ofvarious countries. Physicists try to make models which can highlight the basic mechanism behind the formation ofsuch universal income or wealth distribution in the society. Their another motivation is to explain the global economicinequality using the elementary ideas of physics.There are few works on the modeling of wealth distribution based on generalized Lotka-Volterra model (e.g., [33, 34]).Following the model, the time evolution equation of m i,t can be written as m i,t +1 = (1 + ξ t ) m i,t + aN (cid:88) m j,t − c (cid:88) m i,t m j,t , (6)where m i,t is the money of the i -th agent at time t . Here N is the total number of agents and a , c are two parameters.The variance of the distribution of random number ξ t (always positive) is V . Due to the presence of the second termin the right hand side of the equation (6), the money of any agent should not go to zero at any instant of time. Suchterm may be considered as the effect of some kind of social security policy. The overall growth of the total money is c . Since the total money of the system can change with time, the equation 6 doesnot have any stationary solution. The relative money of an agent can be defined as x i,t = m i,t / (cid:104) m t (cid:105) , here (cid:104) m t (cid:105) is theaverage money per agent at any instant of time t . The x i,t becomes independent of time if the ratio a/V is constant.As a result of that, even in a non-stationary system, after some amount of time, one can eventually get a time invariantrelative money distribution. In mean field approximation the distribution function f ( x ) has the following form f ( x ) = exp [ − ( ν − /x ] x ν . (7)Here ν (positive exponent) is the ratio of a and V . For large value of x , the form of equation (7) eventually becomespower-law-like.Bouchaud and M´ezard [35] proposed a generalized model for the growth and redistribution of wealth. Their model(BM) able to reproduce the Pareto law. They used the physics of directed polymers in economical framework. In theBM model, the dynamics of the wealth w i of the i -th is governed by the set of stochastic equation, dw i ( t ) dt = η i ( t ) w i ( t ) + (cid:88) j (cid:54) = i J ij w j ( t ) − (cid:88) j (cid:54) = i J ji w i ( t )Here η i ( t ) follows a Gaussian distribution which has mean µ and variance 2 σ . The Gaussian multiplicative processsimulates the investment dynamics. The J ij is the linear exchange rate of between i -th and j -th agents. EmployingFokker-Planck equation under mean field approximation, one can obtain a stationary solution of the distributionfunction P ( w ), where w = (cid:80) i w i /N is the mean wealth. Here N is the total number of agents. The form of the P ( w )is given by P ( w ) = A exp[(1 − ν ) /w ] w ν , where A = (1 − ν ) ν / Γ( ν ) and ν = 1 + J/σ . For large value of w , the P ( w ) decays in a power law with exponent ν . The BM model indicates about the two phases, in one phase only a few number of agents hold the entire amountof wealth. Such phase appears when ν <
1. In the another phase the wealth is distributed among the finite numberof agents. Under mean field approximation the agents in BM model exchange the same percentage of wealth theyhave. That means a relatively poor agent receives an unrealistic amount of wealth from the rich agent. In the fieldapproximation, the wealth of the individual agents asymptotically converges to the mean wealth w . That means afterlong time, all the agents have same amount of wealth, which is again an unrealistic situation. To introduce economicinequality, Scafetta et al. [36] modulate the investment term of BM model. Garlaschelli and Loffredo [37] accountedBM model in different types of networks.Employing kinetic theory, physicists built models which can reproduce the income or wealth distribution of thesociety. Chakraborti and Chakrabarti [38] proposed a generalized model (CC model) of income distribution. In theirproposed money exchange dynamics, during the economic transaction the participating agents (two people) alwayskeep some fraction of their money and the sum of their remaining money is distributed randomly among them. Thedynamical equations of CC model are given by m i ( t + 1) = λm i ( t ) + (cid:15) ij [(1 − λ )( m i ( t ) + m j ( t ))] m j ( t + 1) = λm j ( t ) + (1 − (cid:15) ij )[(1 − λ )( m i ( t ) + m j ( t ))] . (8)Here λ is the saving propensity and (cid:15) ij is random fraction. Both λ and (cid:15) ij are ranging from zero to unity. One cansimulate the dynamics of CC model using the Fortran code given in the Fig. 9. The money distribution P ( m ) fordifferent values of λ are shown in the Fig. 10. For any non-zero value of λ the most probable position of the P ( m )will be located at the non-zero value of the money. For zero value of λ the distribution is essentially exponentialwhereas for non-zero value of λ the P ( m ) fits approximately to a Gamma function. Patriarca and Chakraborti [39]numerically evaluated the mathematical form of the P ( m ) which is given by P ( m ) = 1Γ( n ) (cid:16) n (cid:104) m (cid:105) (cid:17) n m n − exp (cid:16) − mn (cid:104) m (cid:105) (cid:17) ,n ( λ ) = 1 + 3 λ − λ . We can see in the limit λ →
1, the distribution function becomes sharply peaked about some non-zero value ofmoney, which indicates the money is uniformly distributed among the agents. Almost simultaneous to the CC model,3
File: /home/sudip/ccm/code_for_muktish_rev/cc.f90 Page 1 of 2
MODULE
CSEED
INTEGER :: SEED
END
MODULE
CSEED program income_distribution
USE
CSEED implicit noneinteger :: i,j,k,l,i7,eqitr,step,i1,ii,m integer , parameter ::n=100,eitr=100000,itr=eitr+100000 integer , parameter ::no_total_seed=10,bin=1000,p_dim=n*bin integer , dimension (:), allocatable ::p real ::RANF,u1,u2,u3,exchange,value_bin,value_y,value_z real , parameter :: money=1.0d0,lamda=0.0d0 real , dimension (:), allocatable ::a, seed_array eqitr= eitr+1step=(itr-eitr) allocate (a(n),seed_array(no_total_seed),p(0:p_dim)) a(:)=moneyp(:)=0.0d0 SEED=313890299 do m=1, no_total_seedseed_array(m)= SEED + m enddo open (33333, file ='cc_n=100_seed=10_lamba=0.0') ! seed loop do m=1, no_total_seedSEED = seed_array(m)a(:)=money ! iteration loop do i=1, itr ! one monte carlo step do j=1, n! selection of first agentu1=RANF()k=int(1 + (u1*n)) ! selection of second agent71 u2=RANF()l=int(1 + (u2*n)) if (k==l) then goto endif !money exchangeexchange=(1 - lamda)*a(k)+(1 - lamda)*a(l) u3=RANF()a(k)= (lamda)*a(k) + u3*exchangea(l)= (lamda)*a(l) + (1 - u3)*exchange enddo ! end of one monte carlo step File: /home/sudip/ccm/code_for_muktish_rev/cc.f90 Page 2 of 2 if (i>=eitr) thendo ii=1, np(int(a(ii)*bin))=p(int(a(ii)*bin))+1 enddoendif enddo ! end of iteration loop enddo ! end of seed loop ! evaluation of distribution do l=0, p_dimvalue_z=(float(p(l))/float(step*no_total_seed*n))value_y=value_z*binvalue_bin=float(l)/float(bin) if (value_y .ne. thenwrite (33333,*) value_bin, value_y endifenddo end program income_distribution FUNCTION
RANF()
RESULT (CR)!! Function to generate a uniform random number in [0,1]! following x(i+1)=a*x(i) mod c with a=7** 5 and! c=2**31-1. Here the seed is a global variable.!
USE
CSEED
IMPLICIT NONE
INTEGER :: H, L, T, A, C, Q, R
DATA
A/16807/, C/2147483647/, Q/127773/, R/2836/
REAL :: CR H = SEED/Q L = MOD(SEED, Q) T = A*L - R*H IF (T .GT. THEN
SEED = T
ELSE
SEED = C + T
END IF CR = SEED/FLOAT(C)
END
FUNCTION
RANF
FIG. 9: The Fortran code for simulating the dynamics of CC model.
Dr˘agulescu and Yakovenko [22] proposed another model (DY model), in which they mapped the two body collisionprocess (exchange energy in the collision) into the economic system, where in each financial transaction (equivalentto collision) between two agents, they exchange money (equivalent to energy). The stochastic equations of moneyexchange DY model are given by, m i ( t + 1) = m i ( t ) − ∆ mm j ( t + 1) = m j ( t ) + ∆ m, (9)where at time t the i -th agent contains m i ( t ) amount of money. The financial transaction is only allowed if both m i ( t ) and m j ( t ) are greater than zero. Here ∆ m is the random fraction of the average money of the two participatingagents. There is no provision of saving propensity in the DY model, which is the fundamental difference with the CCmodel. Using the dynamical money exchange equation of DY model, one will get a exponential money distribution,which appears as a special case λ = 0 in the CC model. Therefore one would get an equivalent dynamics of DY modelby putting λ = 0 in the CC model.4 P ( m ) m λ = 0.0 λ = 0.1 λ = 0.5 λ = 0.9 FIG. 10: The money distribution P ( m ) for different values of saving propensity factor λ = 0 , . , . , .
9. Here the number ofagents N = 100. -8 -7 -6 -5 -4 -3 -2 -1 -3 -2 -1 P ( m ) m N=1000
FIG. 11: For uniformly distributed saving propensity factor λ , the money distribution P ( m ) contains a power law tail whichgoes as m − . Here the number of agents N = 1000. Although we observe power law tail in the income distributions of various countries, both CC and DY models failto generate such power law tail by using their proposed dynamical rules. We find in the CC model, the value of λ is same for every agent but in realistic situation the saving propensity should vary from agent to agent. Chatterjeeet al. [40] discussed the CC model in more general way. Instead of constant value of saving propensity, they consider λ is distributed among the agents. Therefore the modified dynamical equations of their model (CCM model) can bewritten as m i ( t + 1) = λ i m i ( t ) + (cid:15) ij [(1 − λ i ) m i ( t ) + (1 − λ j ) m j ( t ))] m j ( t + 1) = λ j m j ( t ) + (1 − (cid:15) ij )[(1 − λ i ) m i ( t ) + (1 − λ j ) m j ( t ))] . (10)The values of the saving propensities of i -th and j -th agents are λ i and λ j respectively and they are in general different.Employing the dynamical equations (10), Chatterjee et al. [40] numerically found a money distribution which containsa power tail. Such power law tail actually reveals the Pareto law. The existence of the power law tail is robust to thetype of the distribution function ρ ( λ ) of saving propensities but the power law exponent depends on the nature of the5 ρ ( λ ). In fact for the distribution function ρ ( λ ) ∼ | λ − λ | α λ , the Pareto exponent value becomes unity for all valuesfor α λ ( (cid:54) = 0). For uniformly distributed λ , the money distribution decays as P ( m ) ∼ m − . The money distribution forsystem size N = 1000 is shown in Fig. 11, here the relaxation time is of order of 10 . Chatterjee et al. [40] reportedanother important result regarding the fluctuation in money of individual agent. They showed that in case of CCmodel the fluctuation in the money of individual agent increases with the decrease in the value of λ whereas in case ofCCM model exactly opposite trend is observed. Patriarca et al. [41] investigated the relaxation behavior of income orwealth distribution. They found the equilibrium time is proportional to the number of agents. They also noticed thatfor a given value of λ , the equilibrium time τ λ ∼ / (1 − λ ). Chakraborty and Manna [42] found a distribution withpower law tail by using the dynamics of CC model and their power law exponent is similar with the result obtainedin the CCM model. In contrast to the CC model, Chakraborty and Manna [42] considered the probability of anagent in participating in a financial transaction is proportional to the positive power of his/her money. That meansthey evoked a dynamics where the richer class of people essentially get more opportunity in trading rather than thelow-income people. Reduction of the Cobb-Douglas utilization maximization principle to the CC model of exchangedynamics form was shown by Chakrabarti and Chakrabarti [43] (see also [44] for a recent discussion).Heinsalu and Patriarca [45] introduced another gas like model of income or wealth distribution. Their model isoften called immediate exchange (IE) model. Their proposed dynamical equations of exchanging money between j -thand k -th agents are given by m (cid:48) j = (1 − (cid:15) j ) m j + (cid:15) k m k m (cid:48) k = (1 − (cid:15) k ) m k + (cid:15) j m j , (11)where m and m (cid:48) are the money of any agent before and after the exchange respectively. Here (cid:15) j ) and (cid:15) k ) are tworandom numbers, uniformly distributed in (0 , f α s ( m ) has the shape of Γ-function. f α s ( m ) = α sα s m α s − Γ( α s ) exp( − α s m ) . (12)Here the shape parameter α s = 2. Heinsalu and Patriarca [45] assert that for small values of wealth, the distributionfunction obtained from IE model matches better with the real data than the earlier models. Along with new dynamics,they introduced an acceptance criterion of trading for the agents. The acceptance probability of j -th agent for makingtransaction with the k -th agent is a function of ∆ m jk (= (cid:15) k m k − (cid:15) j m j ). Moreover, the equilibrium money distributiondoes not affected by the introduction of acceptance criterion. There are few analytical works on the IE model.Katriel [46] performed analytical investigations on the IE model. He analytically showed the equilibrium moneydistribution of IE model converges to Γ-function in infinite population limit. Lanchier and Reed [47] realized the IEmodel on connected graph, where agents are located at the vertex set of the graph and they can interact only withtheir neighbors. Lanchier and Reed [47] analytically proved the conjectures made by Heinsalu and Patriarca [45]. O θ -3 -2 -1 -3 -2 -1 O | θ - θ c | bx -3 -2 -1 -3 -2 -1 O a θ = 0.59 (a) O θ -2 -1 -3 -2 -1 O | θ - θ c | x (b) O θ O θ - θ c x (c) FIG. 12: (a) The numerical data of the variation of O with the threshold money θ (in the mean field case), where O is thenumber of agents having money below the threshold money θ in the equilibrium state. The top inset shows that for θ = 0 . O → N → ∞ . The bottom inset shows the power law fit of the numerical data of O against ( θ − θ c ), which gives β (cid:39) .
91, where O ∼ ( θ − θ c ) β . Here the number of agents N = 10 . (b) The variation of O with the threshold money θ in onedimension. Inset shows the power law fit of the numerical data of O against ( θ − θ c ). Such scaling fit gives β (cid:39) .
41. Here thenumber of agent N = 10 (c) The variation of O with the threshold money θ in two dimension. Inset shows the power law fitof the numerical data of O against ( θ − θ c ). Such scaling fit gives β (cid:39) .
67. The dimension of the lattice is 1000 × D. A Kinetic Exchange Model with self-organized poverty-line
Along with the modeling of the observed income or wealth data, there are few attempts made on illustratingintriguing models which indicate the potential way of reducing economic inequality in the society. Pianegonda etal. [48] proposed conservative exchange market (CEM) model where a group of agents is realized in one dimensionallattice. According to their proposed dynamics, in each transaction one of the participating agent is necessarily thepoorest agent of the group. Due to the transaction the poorest agent may gain (or lose) some amount of wealth. Suchamount of wealth is equally deducted (or distributed) from the two nearest neighbors of the poorest agent. In themean field version of the model (globally coupled), such deduction (or addition) of wealth is done from two randomlychosen agents. Considering the nearest neighbors interactions, they found a wealth distribution in which almost allagents are beyond a certain threshold η T . Their numerical results indicated the value of η T ≈ . η T ≈ .
2. They noticed the probability of an agent becoming wealthier decreases with time andfinally it converges to a value ≈ .
76. Pianegonda et al. [48] observed the fraction of rich agents is independent ofthe size of the market. The CEM model reveals an exponential distribution (beyond η T ) in case of nearest neighborsinteractions whereas such distribution has almost linear form in the mean field limit. Iglesias [49] simulated thedynamics of the CEM model through inclusion of several types of taxes. He considered a situation where the amountof wealth gain (or lose) by the poorest agent is equally collected from the all agents. Such kind of deduction of wealthcan be treated as an implementation of tax (uniform for every agent) in the society for the development of the poorclass of the people. The distribution obtained by imposing such global-uniform tax is similar with the distributionacquired in case of the mean field version of the CEM model but in this case the value of η T ≈ .
25. It should benoted that in case of mean field CEM model the value of Gini coefficient g ≈ . g ≈ . η T ≈ . .
32. The value of the Gini coefficient isnearly 0 . g in the original local version of the CEM model. In case of global-proportional tax, thenature of the wealth distribution is power law and the value of g ≈ .
16. Ghosh et al. [50] proposed gas-like modelwhich shows an effective way of improving the financial condition of poor people. In their model they initially setan threshold θ value of money or wealth. Like the CEM model, in each interaction one of the agent must have themoney below the θ and the other j -th agent is randomly selected (mean field case) from the rest of the agents. Thedynamical equation of money exchange are given by m (cid:48) i = (cid:15) ( m i< + m j ) m (cid:48) j = (1 − (cid:15) )( m i< + m j ) . (13)Here m i< < θ , the money of the i -th agent before the interaction whereas m (cid:48) i is the money after the interaction. The (cid:15) is a random number between 0 to 1. Each financial transaction is considered as unit time t . The dynamical exchangeof money continues until the money of the all agents cross the threshold or poverty line θ . After sufficiently longtime t > τ (relaxation time) one will get a steady state distribution, in which a small perturbation cannot affect thedistribution. Such perturbation is implemented by forcibly bringing down a agent below the level θ and to ensure themoney conservation his/her money is given to anyone else. Addition of such perturbation cannot alter the relevantresults obtained in the steady state.The equilibrium money distribution is independent of the initial states of agents which essentially reflects theergodicity of the system. Ghosh et al. [50] computed O , the average number of agents below the threshold θ , whichis certainly zero in steady state up to a critical value of the threshold θ c . That means for θ > θ c one will never get adistribution where O is zero (see Fig. 12). Interestingly the relaxation time τ divergences at θ = θ c . These outcomesessentially indicate a phase transition in the system, where O is the order parameter. In the mean field case theof θ c ∼ . β = 0 . ± . z = 0 . ± .
01 and δ = 0 . ± .
01. Theseexponents β , z and δ are obtained from the power fit of O ∼ ( θ − θ c ) β , τ ∼ ( θ − θ c ) z and O ( t ) ∼ t − δ respectively.Ghosh et al. [50] studied their model in one dimension. In this case during a financial transaction one of the agentcontains money m i< < θ . The other agent is randomly selected from the two nearest neighbors (which can containany money whatsoever) of the i -th agent. The variation of O in the one dimension is shown in the Fig. 12. The valueof θ c = 0 . ± . β = 0 . ± . z = 0 . ± . δ = 0 . ± .
01. Ghosh et al. [50] also simulated the model in two dimension where again in a financial transaction oneof the agent has money m i< < θ and the other agent is one of the four nearest neighbors of the i -th agent. The nearest7neighbors can have any money whatsoever. The values of the critical point and exponents are θ c = 0 . ± . β = 0 . ± . z = 1 . ± .
01 and δ = 0 . ± .
02. In should be mentioned that except the value of z , the values of β and δ in mean field, one dimension and two dimensional cases are very close to the values of these exponents in theManna model [51–53] in the respective dimensions. Ghosh et al. [50] commented that the mismatch in the values of z may arise due to the limitation in the system size and they conjectured that their model might belong to the Mannauniversality class. E. The Yard-Sale Model and effects of taxes
Another kinetic exchange model considered had been (see Chakraborti [54]) that m i ( t + 1) = m i ( t ) − m j ( t ) + (cid:15) (2 m j ( t )) m (cid:48) j ( t + 1) = (1 − (cid:15) )(2 m j ( t ))when m i ( t ) ≥ m j ( t ). In many economic transactions this may be a natural feature, particularly for Yard-Sale modeland hence the name. Here, the wealthier agent saves exactly the excess amount and trading takes place with doublethe poorer agent’s money or wealth. The attractive and stable fixed point for the dynamic corresponds to wealthcondensation in the land of one agent. This is obvious, as m j ( t ) at any t becomes zero, the agent gets isolated fromany further trade and this continues for all others until one agent grabs all! This absolute level of condensation orinequality in the model made it unrealistic [1] . However, as discussed in the footnote, several strategies could savethe model from such condensation with extreme inequality ( g = 1), and one suggested recently [56] to impose thenatural tax collection by government, eventually to redistribute it among all the agents in the form of public goodsand services, e.g., road construction etc, is extremely successful and gives very good fit to the data. IV. KOLKATA PAISE RESTAURANT PROBLEM
The city Kolkata was once the capital of India and till date continues to be one of the oldest trading centre of thiscountry. This old city has attracted large number of labors migrating from all parts of India. A century has passedsince this city lost its preeminent position, but the migrant inflow has made Kolkata highly populated till date. Mostof the migrants belongs to unorganized labor class, generally lacking secure wages even without fixed working hours.Sometime in Kolkata, there used to be an array of cheap restaurants at road side, namely ‘Paise Hotels’: Paise is thesmallest Indian currency. Everyday, each of those restaurant would prepare limited number of dishes, that costs verylow (at rate of basic cost of cooking). And this cost matches well with the affordability of those labors. Budget is alsonot a constraint while choosing any restaurant. Thus the Paise hotels become much popular among the poor laborsduring their lunch hour. Everyday without discussing with others, they themselves would choose some restaurant forlunch that day. During lunch hour, they would walk down the street and visit his chosen restaurant for lunch. And,only one choice can be afforded per agent per day due to strict lunch hour.For simplicity, we assume that only one dish would get prepared by each of those restaurants. If some day, only onelabor arrives at some restaurant during lunch hour, he will be served the only dish prepared there. And he would goback to work happily. Problem arises if more than one people visit any one restaurant for lunch. Then the restaurantwould choose one of them randomly and he gets the lunch. Others arrived there would miss their lunch for thatday and report back to work staying hungry for rest of the working hour. Nobody would like to starve. Choosingrestaurants intelligently could guarantee lunch for every labor that day. But how to choose in that way makes thisproblem interesting.So every day, these labors face a decision making problem. They do not discuss with peers while making his choice.Only information they have is the crowd distribution of every restaurants for some finite number of past days. End ofthe day, no one would like to continue work skipping lunch. Ideal case would be: if every labor arrives at a restaurantwhere nobody else visit for that day to assure his lunch. The maximum possible social utilization fraction is 1 . [1]The model was studied by the group members in Saha Institute of Nuclear Physics during 2000-2005 and income/wealth condensationphenomenon was taken initially as a signature of the absurdity of the model and the study was temporarily abandoned. Later, however,some interesting slow dynamical behavior of the model was observed [54]. It was also noted that a ‘mixed kinetic exchange strategy’ (seee.g., Pradhan [55]) can destabilize the condensation (with Gini coefficient g = 1) and can lead to some extreme but realistic distributions,having g (cid:46) νN agents choosing among N restaurants (however we will consider ν =1) for lunch every day. Agents do not interact with others while making his decision any day. Information regardinglast day’s restaurant fill up statistics is available publicly. With this, agents choose and visit the chosen restaurantduring lunch hour. Dish is guaranteed only if a restaurant is visited by some agent alone that day. Any day if somerestaurant is visited by more than one agents then only one of them gets the food and others return remaining hungryfor that day. But, lunch hour is strict. Visiting another restaurant would delay to return back to work and henceonly one choice can be made per day per agent. End of the day, the social utilization fraction (number of restaurantsvisited by at least one agent by total number of restaurant) will be calculated. Maximum possible social utilizationfraction is unity. This is when every agent is able get lunch at some restaurant, and no dish gets wasted that day. Thedictatorial solution, as discussed in previous para, though works well though we will encourage the readers to evolvesome strategy following which those agents will learn to make decision their own and also social utilization fractioncan be maximized as much as possible. Below we discuss few interesting strategies along with their results developedby several econophysics researchers. A. Learning Strategies
Here we will study the dynamics of Kolkata Paise Restaurant game problem following several strategies proposedin [58–60]. They are: No Learning (NL), Limited Learning (LL), One Period Repetition (OPR), Crowd Avoidingstrategy (CA), Stochastic Crowd Avoiding strategy (SCA). Among them No Learning strategy is considered to bethe base strategy throughout, often compared with other mentioned strategy. No Learning (NL)
In this strategy, νN agents randomly chooses among N restaurants. We consider no past history i.e. memory.For simpilicity, restaurant occupying density ν is considered to be 1 throughout study. The probability of choosing arestaurant by n < N agents is: ˜ P ( n ) = (cid:18) νNn (cid:19) p n (1 − p ) νN − n (14)The restaurants being equi-probable, the probability of choosing one restaurant among N is p = N and for N → ∞ one gets (using Poisson Limit theorem): ˜ P ( n ) = ν n n ! exp ( − ν ) (15)So, fraction of restaurants not chosen by any agent is ˜ P ( n = 0) = exp ( − ν ), and this gives the average fraction ofrestaurants chosen by at least one agent on that day ¯ f is 1 − exp ( − ν ) (cid:39) .
63. This we will consider as the basestrategy and will compare with remaining cases for improvement in f . Results Following No Learning strategy givenin Fig. 14 can be obtained using program given in Fig. 13.9 import randomfrom random import gaussimport numpy as npfrom numpy import arrayimport matplotlib.pyplot as pltn=500t=100csum=0arr=np.array([])f=np.array([])for i in range(t):arr = np.array([0]*n)for j in range(0,n):arr[random.randint(0,n-1)]+=1arr=array(arr)csum=sum(x>0 for x in arr)f=np.append(f,csum) f=f/nplt.plot(np.arange(t),f,'m:')plt.plot(np.arange(t),array([np.mean(f)]*t ),'k-')plt.xlabel('number of days (t)')plt.ylabel('utilization fraction (f)')plt.show() FIG. 13: Python (version 2 .
7) program for No Learning strategy. Results (see Fig. 14) can be obtained if one runs this program. number of days ( t ) u t ili z a t i o n f r a c t i o n ( f ) FIG. 14: Plot of social utilization fraction ( f ) in magenta color vs. days ( t ) following No Learning (NL) strategy. Averageutilization fraction ¯ f ( (cid:39) .
63) over 100 days is shown in black color straight line. Limited Learning (LL)
On first day, agents will randomly choose some restaurant similar to No Learning strategy with f = 0 .
63. Nextday onwards, they will make individual choice depending upon their last day lunch availability: below we discuss theLL(1) inspired strategy proposed in [58]. If an agent gets lunch from some restaurant on t -th day, then he opts forthe best restaurant on day t + 1. If he did not get lunch on any t -th day, then next day t + 1 he randomly choose oneamong the other ( N −
1) restaurant with equal probability. Say x t fraction of agents or rather f t N number of agents( f t is utilization fraction at t -th day) on getting their lunch on some day t will visit the best restaurant (restaurant1), and only one of them will get lunch there and others will not get their lunch for that day. Remaining ( N − f t N )agents will try from the remaining ( N −
1) restaurant following no learning case. And the recursion relation will be: f t +1 = 1 x t + (1 − exp ( − ν t )); ν t = 1 − f t (16)0The first term of the summand will contribute 0 as N → ∞ , and one gets the steady state utilization fraction f (cid:39) . One Period Repetition (OPR)
On first day say t −
1, agents will randomly choose a restaurant following NL strategy. If an agent gets his lunch onday t − t . This is one period repeat. If some agent got his lunchfrom same restaurant for two consecutive days t − t , then he will compete for the best restaurant (ranking ofrestaurant is agreed upon by all agents) on day t + 1. For any day t if an agent fails to get lunch, next day t + 1 hewill randomly choose one among restaurants which remained vacant yesterday.Here the fraction f t (representing social utilization at day t ) is made of two parts: x t − fraction of agents who willcontinue their lunch at last day chosen restaurant, and rest of the fraction x t who have chosen today: f t = x t − + [1 − x t − ](1 − exp ( − x t of agents who have chosen today is given by NL strategy where N (1 − x t − ) left out agents finds oneout of N (1 − x t − ) yesterday’s vacant restaurants, so ν t = 1.On next day, the fraction f t +1 will be: f t +1 = 1 x t − + x t + (1 − x t − − x t )(1 − exp ( − x t − would be very small and gets ignored, and replacing x t with (1 − x t − )[1 − exp ( − f t +1 = [1 − x t − ](1 − exp ( − − x t − − (1 − x t − )(1 − exp ( − − exp ( − f t − = f t = f t +1 = f and x t − = x t = x t +1 , so dropping the subscript t and equating f t = f t +1 one calculates x to be 0.19 and f = 0.71. So this strategy is an improvement over NL case, though an agentafter getting lunch at some restaurant k revisits there the very next day with probability 1. The fluctuations in thesocial utilization fraction is found to be Gaussian in the simulation results reported by [58]. Crowd Avoiding strategy (CA)
As the name suggests, agents following this strategy will randomly choose some restaurant on day t where nobodyhad visited last day i.e. day( t − f following Crowd Avoiding strategy. Computer simulation results the distribution of social utilizationfraction f to be Gaussian with peak around 0.46. It can be understood following way: as the fraction of restaurantsfilled last day will strictly get avoided by agents today, so the number of available restaurants today is N (1 − f ) whichwill be chosen randomly by all agents. With ν = − f , the recursion relation becomes: f = (1 − f )[1 − exp ( − − f )] (20)Solving above equation, we get f = 0.46, which fits well along the simulation result (Fig. 16). Stochastic Crowd Avoiding strategy (SCA)
We consider the strategy be following: if an agent arrives at restaurant j for getting his lunch on day ( t -1), thennext day (i.e. day t ) probability of visiting back to the same restaurant for that agent to have his lunch will dependon how crowded was the last day’s restaurant. So, probability of visiting restaurant j on day t is p j ( t )= n j ( t − . Andthe probability of visiting any other restaurant j (cid:48) ( (cid:54) = j ) goes as p j (cid:48) ( t )= (1 − p j ( t ))( N − . Both the numerical and analyticalresults of average utilization fraction ¯ f following this strategy is found to be 0 .
8. The distribution is Gaussian withpeak around f (cid:39) . f following the above strategy can be made as: Suppose a i denotes the fraction of restaurants having exactly i number of agents ( i = 0 , , ...N ) arrived on day t . And also1 import randomfrom random import gaussimport numpy as npfrom numpy import arrayimport matplotlib.pyplot as pltn=500t=100csum=0f=np.array([])arr_0=np.array([])arr_1=np.array([])arr_0=np.array([0]*n)j=0it=0for i in range(0,n):loc=random.randint(0,n-1)arr_0[loc]+=1arr_0=array(arr_0)for it in range(t):arr_1=array([0]*n)for j in range(0,n):loc=int(random.randint(0,n-1))i=arr_0[loc] if i>0: while(i!=0):loc=int(random.randint(0,n-1))i=arr_0[loc] arr_1[loc]+=1elif i==0:arr_1[loc]+=1else: continuecsum=sum(x>0 for x in arr_1)f=np.append(f,csum) arr_0=arr_1f=f/nplt.plot(np.arange(t),f,'m:')plt.plot(np.arange(t),array([np.mean(f)]*t),'k-')plt.xlabel('number of days (t)')plt.ylabel('utilization fraction (f)')plt.show() FIG. 15: Python (version 2 .
7) program for Crowd Avoiding strategy. Results (see Fig. 16) can be plotted if one runs thisprogram. assume that a i = 0 for i ≥ t as the dynamics gets stabilized in steady state. Thus, a + a + a = 1, a + 2 a = 1 for large t gives a = a . Following the strategy, a fraction of agents will try to leave their last day’svisited (each with a probability of 1/2) restaurant on day t + 1 and of course, no such activity will take place in somerestaurant where nobody (fraction a ) or only one agent (fraction a ) visited last day. Fraction of a agents withsuccessful leave attempt will now get equally distributed into N − a , fractionvisiting any vacant restaurant ( a ) of last day is a a , at present fraction of vacant restaurant is a = a − a a .In this process, the vacancy which will also pop up in those restaurants where exactly two agents visited last day is( a / − a ( a / a = a − a a + a − a a a = a , we get a = a = 0 .
2, and also a = 0 .
6. Thus, social utilization fraction f becomes a + a = 0.8. Thisis an approximate result considering nil contribution from a i with i ≥ t , as seen numerically. Simulationresults also confirm the approximated result of steady state utilization fraction, see Fig. 17. B. Phase Transition in KPR game
Recently, a novel phase transition phenomena is reported in Kolkata Paise restaurant game problem [61] if n i ( t − i with weight [ n i ( t − α and 1 / ( N −
1) for any other2 number of days ( t ) u t ili z a t i o n f r a c t i o n ( f ) FIG. 16: Plot of social utilization fraction ( f ) in magenta color vs. days ( t ) following Crowd Avoiding strategy. Averageutilization fraction ¯ f ( (cid:39) .
46) over 100 days is shown in black color straight line. number of days ( t ) u t ili z a t i o n f r a c t i o n ( f ) FIG. 17: Plot of social utilization fraction ( f ) in magenta color vs. days ( t ) following Stochastic Crowd Avoiding strategy.Average utilization fraction ¯ f ( (cid:39) .
79) over 100 days is shown in black color straight line. restaurant, then the steady state ( t independent) utilization fraction becomes f ( t ) = [1 − (cid:80) Ni =1 [ δ ( n i ( t )) /N ]]. Here,the critical point α c is found near point 0 + where (1 − f ) is found to vary as ( α − α c ) β where β (cid:39) .
8. See Fig. 18,where (1 − f ) is plotted against α , fitting is done using maximum likelihood estimation. Inset plot shows the directfunctional relationship between (1 − f ) and α . C. Application of KPR: Vehicle for Hire problem in mobility market
KPR game model has been applied in competitive resource allocation systems where scarce resources need to beallocated effectively for repeated times. Areas like dynamic matching in mobility markets [60], mean field equilibriumstudy for resource competitive platform reported in [62] are such examples. In mobility market, agents generallyhave their individual preference/ranking towards resources. Vehicle for hire markets is one of such platform wheredrivers individually decide some pick up location to increase their own utilization and accordingly offer individualtransportation to customers by car. Martin & Karaenke [63] has extended the KPR game model to generalize theVehicle for Hire Problem (VFHP) and applied playing strategies discussed in previous section assuming drivers asagents and customers as resources in hire market. Below we discuss some of the results where drivers will have their3 [ f ] . f FIG. 18: Figure 18 Plot of [1 − f ] against α fitted to ( α − α c ) β where β (cid:39) . f denoting the utilization fraction and α denotingthe weight factor power as defined earlier). The critical point α c is fount to be at 0 + [61]. individual ranking over customers unlike original KPR game model having commonly agreed resource ranking. No Learning (NL) νN drivers,independent of their individual preference ranking and any last trip history, choosing randomly among N customers for their next ride. For simplicity ν is considered as 1. The probability of choosing any particularcustomer by n < N drivers is: ˜ P ( n ) = (cid:18) νNn (cid:19) p n (1 − p ) νN − n (22)Assuming each of the customers to be equi-probable with probability p = N and for N → ∞ one gets (using PoissonLimit theorem): ˜ P ( n ) = ν n n ! exp ( − ν ) (23)Thus fraction of customers not chosen by any driver is ˜ P ( n = 0) = exp ( − ν ), and one obtains the average fraction ofcustomers chosen by at least one driver for ride ¯ f is 1 − exp ( − ν ) (cid:39) .
63. This result is exactly similar to NL or basestrategy of KPR and again we will compare this result with other strategies. Limited Learning (LL)
Following LL strategy of KPR, drivers choose a customer randomly for ride at time t and go to their most preferredcustomer at time t + 1 if they got a tour at time t . otherwise they choose randomly again. One obtains the utilizationfraction by following formula: f t +1 = f t · (1 − exp ( − − exp ( − ν t )); ν t = 1 − f t (24)The left summand of above equation represents all those drivers who are successful in choosing their most preferredcustomer at time t + 1 after choosing randomly at time t or earlier. The right summand models those who chooserandomly or successful in choosing their top priority customer at time t and return there. On solving the aboveequation, one gets the steady state utilization f = 0 . One Period Repetition (OPR)
At time t −
1, drivers will randomly choose customers for trip. For the next ride he will repeat trip with the samecustomer of time t −
1, if successful in previous ride choosing any customer. For drivers with two consecutive successfultrip with same customer, would go for individual best ranked customer at time t + 1. Any driver follows NL strategyfor next trip if fails in getting a successful ride at any time slot. So, overall utilization fraction at time t , say f t ,comprises of two parts: x t − fraction of drivers who continues riding with last time chosen customer and rest of thefraction who choose any customer at current time slot. So f t becomes: f t = x t − + [1 − x t − ](1 − exp ( − x t fraction of drivers who have chosen in current trip will be given by NL strategy where N (1 − x t − ) left out drivers finds one out of the N (1 − x t − ) customers waiting for a ride.On next trip, utilization fraction f t +1 will be: f t +1 = x t − + x t + (1 − x t − − x t )(1 − exp ( − x t with (1 − x t − )[1 − exp ( − f t +1 = x t − + [1 − x t − ](1 − exp ( − − x t − − (1 − x t − )(1 − exp ( − − exp ( − f t − = f t = f t +1 = f and x t − = x t = x t +1 , so dropping the subscript t and equating f t = f t +1 one calculates x to be 0 .
28 and f = 0 . D. Application of KPR: Resource allocation in wireless IoT system
Recently, Park et al. has proposed and analyzed a KPR inspired learning framework [64] for resource allocationin the IoT environment. Here, the IoT devices has been modeled as non-cooperative agents choosing their preferredresource block with limited past information made available by their neighbors. However, socially optimal solution isreported for denser as well as lesser dense IoT environment, discussed below.Internet of Things (IoT) technology is behind many smart city application. In an IoT environment, huge numberof devices gets deployed for task and given limited source of energy (battery) for each device makes it challenging toensure efficient resource allocation per time slot. The transmissions are often random and infrequent in nature. Now,up link of such an IoT system in time division multi-access way is considered with one base station (BS) to serve N IoT devices transmitting short packets whenever they want to. For each time slot t , the channel will be divided into b resource blocks (RB), each of which to be used by exactly one transmitting device: multiple devices choosing sameRB end up transmission failure. At time slot t , the probability of N t = n t devices transmitting with probability p is:˜ P ( N t = n t ) = (cid:18) Nn t (cid:19) p n t (1 − p ) N − n t (28)Generally in an IoT system, N t is much larger than b . In order to obtain maximum socially optimal solution, thetrivial solution could be a centralized solution where the BS allocates RBs to transmitting devices. But for severalreasons this solution is impractical. The simple solution is to let the transmitting device randomly choose a RB withequal probability. Let S n be a random variable representing number of successful transmission with support functionsupp( S n ) as [0, N t ] for N t ≤ b and [0, b -1] when N t > b . Taken N t = n t , the probability of having minimum s successfultransmission is: P r ( S n ≥ s | N t = n t ) = s − (cid:89) i =0 b − ib [ b − sb ] ( n t − s ) (29)From equation and , one gets probability of having s successful transmission as: P r ( S n = s ) = N (cid:88) i = s P r ( S n = s | N t = i ) P r ( N t = i ) (30)Following [64], Park et al. has modeled the one to one association between RBs and IoT devices as a KPR gamewhere each player i ∈ ρ has a set of actions A i of selecting set of b RBs. Also the RBs have different channel gain5˜ g i and a device would prefer to transmit using a channel block with higher gain. Let the utility function for an IoTdevice i choosing an action a i , t of selecting RB j with channel gain ˜ g i , j at slot t is: u ( a i , t ) = (cid:40) ˜ g i , a i , t if a i , t (cid:54) = a k , t ∀ k ∈ ρ, k (cid:54) = i, .0 otherwise. (31)Unlike original KPR game model, multiple transmitting devices choosing single RB results into zero utility. Withincertain communication range ( r c ), during time slot t each transmitting device learns from their neighbor’s RB usage;on ( t −
1) slot, if neighbor’s transmission is successful then choose some better ranked RB else transmit with lesspreferable RB than neighbor’s last preference, otherwise transmit randomly. Nash equilibrium is reported for certainvalues of r c for even denser IoT network with service rate of 27 . V. SOCIAL NETWORKS
Society evolves in to many complex network structures in the hyperspace of inter-agent interactions (viewed aslinks) among the agents (viewed as nodes). Examples may be transport networks, bank networks, etc. Geometricallydefined, for example, if one puts N random dots (nodes) on an unit area the minimum distance of separation betweenany two dots or nodes will decays as 1 / √ N . The same problem in an unit three dimensional volume will give theinter-node separation decaying as 1 /N / , and decays as 1 /N /d for a d dimensional system. For complex socialnetworks such inter-node distance (for a fixed embedding volume) will decay as 1 / (log N ), as in an effective infinite-dimensional space. This indicates the minimum contact number required for the spread of disease or information in asociety. In other words, the number of transits for transport between any two nodes can be much lower than naivelyexpected for two or three dimensional world! A. Indian Railway network analysis
Here we discuss one of the highly cited research work from Kolkata: the structural properties of the 160 years’old Indian Railway network (IRN) as complex network. Indian Railway is the largest medium of transport in thecountry. Question is, how many trains one passenger would need to change while reaching any destination withincountry while traveling by train. Passengers would not like to change many trains to avoid the hassle and latency.How would if a train runs between every stations (junction, regular as well as remote) to make journey hassle-freei.e. switching trains during journey become less; considering 579 trains running between 587 stations. But it will endup incurring too much latency. In [65], for the first time Indian Railway network has been studied by Sen et al. as agraph where each railway-station is considered a node and the physical track joining any two stations as the edge sothat there exist minimum one train running between them using the track. For simplicity, edges are considered to beone unit distant, ignoring actual geographical distance between the two stations it connect as an unweighted graph.Rigorous investigation of the structural properties of IRN as a complex network are discussed below.With a motive of being fast and economic, railways run several trains covering short as well as long route. IRN isquite a large network consisting more than 8000 stations over which almost 10000 trains run over the country. Forthe purpose of coarse-grained study of IRN, [65] has considered only 587 stations ( N ) with 579 tracks ( L ) representedas a grant rectangular matrix G ( N, L ) such that G [ i, j ] = 1 if train j has a stop at station i . In 1998, Watts andStrogatz proposed a model of network in [66] with properties like: diameter of the network changes very slowly as thesize of the network grows over time and the network will possess dense connectivity among the neighbors of a node,termed as clustering coefficient C ( N ). They named this network type as Small World Network (SWN) arguing it’sdiameter growth is similar to random network i.e. ln ( N ) /N → N → ∞ with large value of clustering coefficients C ( N ) ∼
1. Sen et al. has measured both of the metric over 25 different subsets of IRN and concluded it to behavesimilar to a small world network (see Figs. 19, 20). Practically this implied that over the years IRN has grown up tobe economic, fast i.e. very few trains need to be changed to reach any arbitrary station over the whole network. Asper Graph theory, how well connected a node is primarily agreed upon by the number of neighbor it has i.e. degreeof the node. Hence it is important to study the degree distribution P ( k ) of IRN. The cumulative degree distribution F ( k ) = P ( k ) dk of IRN when plotted on a semi-logarithmic scale, has been found to fit an exponentially decayingdistribution F ( k ) ∼ exp ( − βk ) with β = 0 . F ( k ) does not tell much about how well connected are theneighbors of a high degree station i.e. the correlation (whether positive or negative) of the average degree < k nn ( k ) > of neighbors to that of the respective node. Fig. 21 shows the plot of < k nn ( k ) > against respective vertex’s degree6 N D ( N ) l P r o b ( l ) FIG. 19: Plot of the mean distance D ( N ) considering 25 different subsets of IRN consisting different number of nodes ( N ).The fitted function over the whole range of nodes is D ( N ) = A + B log( N ) where A ≈ .
33 and B ≈ .
13. The probabilitydistribution Prob( (cid:96) ) of the shortest path lengths (cid:96) on IRN is shown in the inset. The mean distance D ( N ) of the IRN networkis ≈ .
16, varying maximum up to 5 links. N C ( N ) FIG. 20: Plot of the computed clustering coefficients C ( N ) over 25 different subsets of IRN with different number of nodes N .Initially very high followed by fluctuations, C ( N ) has finally seen to stabilize at 0 .
69 as N → ∞ , over the whole IRN. to find the general assortative behavior of the network. Using rigorous measure discussed in [65], the value came forIRN is very small ( ∼ − . VI. SUMMARY AND DISCUSSION
Unequal distribution of income, wealth and other social features are not only common, they have been the persistentfeature throughout the world and through the history of human civilization. Apart from the thinkers and philosophers,the economists and other social scientists have studied extensively about it. Recently, physicists are trying to measureand explore the cause of such inequalities.We first discuss (in Section II) about measures of social inequalities, including the Kolkata index k , giving thefraction of wealth k possessed by (1 − k ) fraction of the reach population: As such it generalizes the Pareto’s ‘80/20law’. We have discussed in this section about the Gini and Kolkata index values measured in various social contexts,7 k < k nn ( k ) > FIG. 21: Plot of the fluctuations of average degree < k nn ( k ) > of neighbors over the degree of nodes k over the whole range ofIRN. e.g., of the income and wealth, deaths in social conflicts and natural disasters, citations of papers across the institutionsand journals etc.Next, in section III we investigate the nature of income and wealth distribution in various societies, and find adominant feature in that typically more the ninety percent of the population in any society has a distribution whichfits a Gamma distribution, while the upper tail part (for the super-rich fraction of the population) fits a robust powerlaw or Pareto law. We show, the kinetic theory of ideal gas, where the trading agents are like ‘social atoms’ of thegas as in two person trading of scattering process with conserved money (like the conserved energy for the gas atoms)and saving a fraction of respective money in each trading indeed gives a Gamma-like distribution which crosses overto Pareto-like power law when the saving fractions are inhomogeneous.Econophysics is interdisciplinary by nature, contributed by physicists, economists, statisticians, social scientists,computer scientists etc. Here, an effort has been put to give a glimpse of recent studies from Econophysics bycomputation. Inequality measuring techniques starting from simple histogram to standard gini index as well asKolkata index and others (see e.g., [8, 12, 13, 67]). One of the major goal of econophysics has been to search for asuccessful theory or model which can capture the behavior of the real economic data of income or wealth distribution.In this review we briefly discuss some of the models inspired by the kinetic theory of ideal gases which are able to givesome insights into the mechanisms of the income distributions. For application of kinetic exchange models to socialopinion formation, see [68–70]. There are a few extensive reviews and books [4, 5, 71] highlighting these developmentsof the econophysics (of income and wealth distribution) as well as in sociophysics. Beside this field of study there aremany applications of physical laws in financial and stock markets (see e.g., [2, 3, 72, 73]). Social resource allocationmodels (see for e.g., [74],[75]) along with several intelligent collective learning strategies are discussed along withtheir programs. We also discussed the connectivity structures of social networks (see e.g., [6, 76]). In particular wediscussed here for the Indian Railway network, how the minimum number of train connections (links) one need tohop for going from one destination to another within India (or, for that matter, any other country), grows with thetotal number of service stations (nodes) in India (network of the respective country). Such shortest number of linksin any network gives the idea for the time required to spread rumor or contact diseases, etc. in a networked society.In order to give a birds’-eye-view of the developments of econophysics, we give the chronological entries in ‘Timelineof econophysics’ (Appendix) for the major developments in the initial phase. For further studies and search of researchproblems in these and related fields, see Refs [2–6, 71, 74, 76–79].8 VII. ACKNOWLEDGEMENT
We are grateful to Muktish Acharyya for the invitation to write this review. BKC is grateful to J. C. Bose fellowship(DST, Govt. India) for financial support. [1] K C Dash.
The Story of Econophysics . Cambridge Scholars Publishing, Newcastle upon Tyne, (2019).[2] R N Mantegna and H E Stanley.
An Introduction to Econophysics . Cambridge University Press, Cambridge, (2000).[3] S Sinha, A Chatterjee, A Chakraborti, and B K Chakrabarti.
Econophysics: An Introduction . Wiley, Berlin, (2010).[4] V M Yakovenko and J Barkley Rosser Jr. Colloquium: Statistical mechanics of money, wealth, and income.
Reviews ofModern Physics , 81:1703, (2009).[5] B K Chakrabarti, A Chakraborti, S R Chakravarty, and A Chatterjee.
Econophysics of Income and Wealth Distributions .Cambridge University Press, Cambridge, (2013).[6] P Sen and B K Chakrabarti.
Sociophysics: An Introduction . Oxford University Press, Oxford, (2014).[7] C Gini. Measurement of inequality of incomes.
The Economic Journal , 31:124–126, (1921).[8] A Ghosh, N Chattopadhyay, and B K Chakrabarti. Inequality in societies, academic institutions and science journals:Gini and k -indices. Physica A: Statistical Mechanics and its Applications , 410:30–34, (2014).[9] I I Eliazar and I M Sokolov. Measuring statistical heterogeneity: The pietra index.
Physica A: Statistical Mechanics andits Applications , 389:117–125, (2010).[10] A Chatterjee and B K Chakrabarti. Fat tailed distributions for deaths in conflicts and disasters.
Reports in Advances ofPhysical Sciences , 1:1740007, (2017).[11] A Sinha and B K Chakrabarti. Inequality in death from social conflicts: A gini & kolkata indices-based study.
Physica A:Statistical Mechanics and its Applications , page 121185, (2019).[12] A Chatterjee, A Ghosh, and B K Chakrabarti. Socio-economic inequality: Relationship between gini and kolkata indices.
Physica A: Statistical Mechanics and its Applications , 466:583, (2017).[13] J Inoue, A Ghosh, A Chatterjee, and B K Chakrabarti. Measuring social inequality with quantitative methodology:analytical estimates and empirical data analysis by gini and k indices. Physica A: Statistical Mechanics and its Applications ,429:184–204, (2015).[14] A Chatterjee and B K Chakrabarti. Kinetic exchange models for income and wealth distributions.
The European PhysicalJournal B , 60:135–149, (2007).[15] M Saha and B Srivastava.
A Treatise on Heat . Indian Press, Allahabad, (1931).[16] V Pareto.
Cours d’ ´Economie Politique . Rouge, Lausanne, (1897).[17] R Gibrat.
Les In´egalit´es ´Economiques . Libraire du Recueil Sirey, Paris, (1931).[18] W Souma. Universal structure of the personal income distribution.
Fractals , 9:463–470, (2001).[19] H Aoyama, W Souma, Y Nagahara, M P Okazaki, H Takayasu, and M Takayasu. Pareto’s law for income of individualsand debt of bankrupt companies.
Fractals , 8:293–300, (2000).[20] A Dr˘agulescu and V M Yakovenko. Eevidence for the exponential distribution of income in the usa.
The European PhysicalJournal B-Condensed Matter and Complex Systems , 20:585–589, (2001).[21] A Dr˘agulescu and V M Yakovenko. Exponential and power-law probability distributions of wealth and income in theunited kingdom and the united states.
Physica A: Statistical Mechanics and its Applications , 299:213–221, (2001).[22] A Dr˘agulescu and V M Yakovenko. Statistical mechanics of money.
The European Physical Journal B-Condensed Matterand Complex Systems , 17:723–729, (2000).[23] J C Ferrero. The statistical distribution of money and the rate of money transference.
Physica A: Statistical Mechanicsand its Applications , 341:575–585, (2004).[24] F Clementi, T Di M, M Gallegati, and G Kaniadakis. The κ -generalized distribution: A new descriptive model for the sizedistribution of incomes. Physica A: Statistical Mechanics and its Applications , 387:3201–3208, (2008).[25] M Levy and S Solomon. New evidence for the power-law distribution of wealth.
Physica A: Statistical Mechanics and itsApplications , 242:90–94, (1997).[26] O S Klass, O Biham, M Levy, O Malcai, and S Solomon. The forbes 400, the pareto power-law and efficient markets.
TheEuropean Physical Journal B , 55:143–147, (2007).[27] S Sinha. Evidence for power-law tail of the wealth distribution in india.
Physica A: Statistical Mechanics and its Applica-tions , 359:555–562, (2006).[28] A C Silva and V M Yakovenko. Temporal evolution of the thermal and superthermal income classes in the usa during1983–2001.
Europhysics Letters , 69:304, (2004).[29] A K Gupta. Money exchange model and a general outlook.
Physica A: Statistical Mechanics and its Applications , 359:634–640, (2006).[30] M A Saif and P M Gade. Emergence of power law in a market with mixed models.
Physica A: Statistical Mechanics andits Applications , 384:448–456, (2007).[31] M A Saif and P M Gade. Effects of introduction of new resources and fragmentation of existing resources on limitingwealth distribution in asset exchange models.
Physica A: Statistical Mechanics and its Applications , 388:697–704, (2009). [32] Z Neda, I Gere, T S Biro, G Toth, and N Derzsy. Scaling in income inequalities and its dynamical origin. arXiv preprintarXiv:1911.02449 , (2019).[33] P Richmond and S Solomon. Power laws are boltzmann laws in disguise. International Journal of Modern Physics C , 12.[34] S Solomon and P Richmond. Stable power laws in variable economies; lotka-volterra implies pareto-zipf.
The EuropeanPhysical Journal B , 27:257–261, (2002).[35] Jean-Philippe Bouchaud and Marc M´ezard. Wealth condensation in a simple model of economy.
Physica A: StatisticalMechanics and its Applications , 282:536–545, (2000).[36] N Scafetta, S Picozzi, and B J West. A trade-investment model for distribution of wealth.
Physica D: Nonlinear Phenomena ,193:338–352, (2004).[37] D Garlaschelli and M I Loffredo. Effects of network topology on wealth distributions.
Journal of Physics A: Mathematicaland Theoretical , 41:224018, (2008).[38] A Chakraborti and B K Chakrabarti. Statistical mechanics of money: how saving propensity affects its distribution.
TheEuropean Physical Journal B , 17:167–170, (2000).[39] M Patriarca, A Chakraborti, and K Kaski. Statistical model with a standard γ distribution. Physical Review E , 70:016104,(2004).[40] A Chatterjee, B K Chakrabarti, and SS Manna. Pareto law in a kinetic model of market with random saving propensity.
Physica A: Statistical Mechanics and its Applications , 335:155–163, (2004).[41] M Patriarca, A Chakraborti, Els Heinsalu, and G Germano. Relaxation in statistical many-agent economy models.
TheEuropean Physical Journal B , 57:219–224, (2007).[42] A Chakraborty and SS Manna. Weighted trade network in a model of preferential bipartite transactions.
Physical ReviewE , 81:016111, (2010).[43] A S Chakrabarti and B K Chakrabarti. Microeconomics of the ideal gas like market models.
Physica A: Statisticalmechanics and its applications , 388:4151–4158, (2009).[44] H Chunhua, L Shaoyong, and L Chong. Investigations to the price evolutions of goods exchange with ces utility functions;https://doi.org/10.1016/j.physa.2019.123938.
Physica A: Statistical Mechanics and its Applications , (2019).[45] Els Heinsalu and M Patriarca. Kinetic models of immediate exchange.
The European Physical Journal B , 87:170, (2014).[46] G Katriel. The immediate exchange model: an analytical investigation.
The European Physical Journal B , 88:19, (2015).[47] N Lanchier and S Reed. Rigorous results for the distribution of money on connected graphs.
Journal of Statistical Physics ,171:727–743, (2018).[48] S Pianegonda, J R Iglesias, G Abramson, and J L Vega. Wealth redistribution with conservative exchanges.
Physica A:Statistical Mechanics and its Applications , 322:667–675, (2003).[49] J R Iglesias. How simple regulations can greatly reduce inequality.
Science and Culture , 76:437–443, (2010).[50] A Ghosh, U Basu, A Chakraborti, and B K Chakrabarti. A threshold induced phase transition in the kinetic exchangemodels.
Physical Review E , 83:061130, (2011).[51] S L¨ubeck. Universal scaling behavior of non-equilibrium phase transitions.
International Journal of Modern Physics B ,18:3977–4118, (2004).[52] SS Manna. Critical exponents of the sand pile models in two dimensions.
Physica A: Statistical Mechanics and itsApplications , 179:249–268, (1991).[53] SS Manna. Two-state model of self-organized criticality.
Journal of Physics A: Mathematical and General , 24:L363, (1991).[54] A Chakraborti. Distributions of money in model markets of economy.
International Journal of Modern Physics C , 13:1315–1321, (2002).[55] S Pradhan. Random trading market: Drawbacks and a realistic modification. arXiv preprint physics/0503105 , (2005).[56] B Boghosian. Is inequality inevitable?
Scientific American , 321:70–77, (2019).[57] B K Chakrabarti. Kolkata restaurant problem as a generalised el farol bar problem. In
Econophysics of Markets andBusiness Networks , pages 239–246. Springer, (2007).[58] A S Chakrabarti, B K Chakrabarti, A Chatterjee, and M Mitra. The kolkata paise restaurant problem and resourceutilization.
Physica A: Statistical Mechanics and its Applications , 388:2420–2426, (2009).[59] A Ghosh, A Chatterjee, M Mitra, and B K Chakrabarti. Statistics of the kolkata paise restaurant problem.
New Journalof Physics , 12:075033, (2010).[60] L Martin. Extending kolkata paise restaurant problem to dynamic matching in mobility markets.
Juniour ManagementScience , 4:1–34, (2019).[61] A Sinha and B K Chakrabarti. Phase transition in the kolkata paise restaurant problem. arXiv preprint arXiv:1905.13206 ,(2019).[62] P Yang, K Iyer, and P Frazier. Mean field equilibria for resource competition in spatial settings.
Stochastic Systems ,8:307–334, (2018).[63] L Martin and P Karaenke. The vehicle for hire problem: A generalized kolkata paise restaurant problem. https://mediatum.ub.tum.de/doc/1437330/1437330.pdf , (2017).[64] T Park and W Saad. Kolkata paise restaurant game for resource allocation in the internet of things. In , pages 1774–1778. IEEE, (2017).[65] P Sen, S Dasgupta, A Chatterjee, PA Sreeram, G Mukherjee, and SS Manna. Small-world properties of the indian railwaynetwork.
Physical Review E , 67:036106, (2003).[66] D J Watts and S H Strogatz. Collective dynamics of small-world networks.
Nature , 393:440, (1998).[67] S Banerjee, B K Chakrabarti, M Mitra, and S Mutuswami. On the kolkata index as a measure of income inequality. arXiv:1905.03615 , 2019. [68] M Lallouache, A S Chakrabarti, A Chakraborti, and B K Chakrabarti. Opinion formation in kinetic exchange models:Spontaneous symmetry-breaking transition. Physical Review E , 82:056112, (2010).[69] S Biswas, A Chatterjee, and P Sen. Disorder induced phase transition in kinetic models of opinion dynamics.
Physica A:Statistical Mechanics and its Applications , 391:3257–3265, (2012).[70] S Mukherjee and A Chatterjee. Disorder-induced phase transition in an opinion dynamics model: Results in two and threedimensions.
Physical Review E , 94:062317, (2016).[71] L Pareschi and G Toscani.
Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods . Oxford UniversityPress, Oxford, (2013).[72] P Bhadola and N Deo. Extreme eigenvector analysis of global financial correlation matrices. In
Eds. F. Abergele et al.,Econophysics and Sociophysics: Recent Progress and Future Directions , pages 59–69. Springer, (2017).[73] N Bertschinger, I Mozzhorin, and S Sinha. Reality-check for econophysics: Likelihood-based fitting of physics-inspiredmarket models to empirical data. arXiv:1803.03861 , 2018.[74] B K Chakrabarti, A Chatterjee, A Ghosh, S Mukherjee, and B Tamir.
Econophysics of the Kolkata Restaurant Problem andRelated Games: Classical and Quantum Strategies for Multi-agent, Multi-choice Repetitive Games . Springer, Switzerland,(2017).[75] D Dhar, V Sasidevan, and B K Chakrabarti. Emergent cooperation amongst competing agents in minority games.
PhysicaA: Statistical Mechanics and its Applications , 390:3477–3485, (2011).[76] B K Chakrabarti, A Chakraborti, and A Chatterjee.
Econophysics and Sociophysics: Trends and Perspectives . WileyVCH, Berlin, (2007).[77] F Slanina.
Essentials of Econophysics Modelling . Oxford University Press, Oxford, (2014).[78] H Aoyama, Y Fujiwara, and W Souma.
Macro-Econophysics: New Studies on Economic Networks and Synchronization .Cambridge University Press, Cambridge, (2017).[79] F Jovanovic and C Schinckus.
Econophysics and Financial Economics: An Emerging Dialogue . Oxford University Press,Oxford, (2017).
Appendix
Econophysics TimelineI. Developments up to 2000: