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Interacting Individuals Leading to Zipf’s Law Matteo Marsili and Yi-Cheng Zhang Institut de Physique Théorique, Université de Fribourg, CH-1700 (February 1, 2008) 8 We present a general approach to explain the Zipf’s law of city distribution. If the simplest 9 interaction (pairwise) is assumed, individualstendto form cities in agreement with thewell-known 9 statistics 1 n PACS numbers: 89.50.+r, 05.20.-y, 05.40.+j a J Zipf[1],halfacenturyago,hasfoundthatthecitysizes US/10^5 8 obey an astonishingly simple distribution law, which is 1000 India/101^/4x Model 2 attributedtothemoregenericleasteffortprincipleofhu- man behavior. Let us denote by q the number of cities m ] ∞ ′ h having the population size m, then R(m)= m qm′ dm c defines the rank, i.e. the largest city has RR = 1, the 100 e second largest R = 2, etc. Zipf found that empirically, m R(m) 1/mγ, with γ 1 (see figure 1). Remarkably ∼ ≈ - Zipfshowedthatthescalingexponentγ =1isveryclose t a to reality for many different societies and during various 10 t s time periods. More recent data [3,2] shows some varia- t. tions from the pure γ = 1 result as also shown in figure a 1. Countrieswhichhavea peculiarsocialstructure,such m as the former USSR or China, do not follow Zipf law. 1 d- For other developed countries Zipf law remains a rather FIG.11. Zipf plot for cit1i0es of population l1a0r0ger than 105 n goodapproximation. Such a generic law calls for generic in the US and in India, and for a simulation of the master o explanation, since different countries (e.g. Germany and equation with p=10−3 and W =106, α=2. c USA) have different cultural, economic structures, and [ theirpeoplehaveinnumerablereasonstochoosewhether m citizensdepartssothatm m 1. Wealsoassume 1 to live in a big or small city. Yet collectively the society i i → i− that there is a small probability pdt that a new city is v self-organises, without express wishes of authorities, to 9 created, with a single citizen. Assigning the transition obey the Zipf’s law. 8 rates w (m) and w (m) specifies the model. Note that Human settlements on the earth’s surface appear to a d 2 birth and death are not explicitly considered, these can 1 be clustered, hence cities. This is because individuals beincludedinthetransitionprobabilities. Notealsothat 0 interact with each other through social, economic and once an individual leaves a city, not necessarily he/she 8 cultural ties. In very primitive times an individual (or 9 willsettleinanothercityrightaway,thusthe totalnum- a family) should perform all basic activities to survive; / ber of city dwellers is not conserved. Thus the whole t there was no need to form a large cluster beyond the a system is composed of the city dwellers and a reservoir m size of a tribe. In modern times, ever refined mutual of unsettled travellers whose number is unregistered. cooperation/competition brings people to live together. - One can study this problem introducing the average d Yetthistendencydoesnotseemtoleadtoasinglemega- number q of cities of size m at time t, which satisfies n city in the world: many of us prefer to live in a big city, m,t o the master equation: equally many may hate and escape it for all its negative c : impacts. Somehow the ensuing compromise results in a ∂ q =w (m+1)q w (m)q +pδ v robust statistical distribution, the Zipf’s law. t m,t d m+1,t− d m,t m,1 Xi Alltheseverygeneralfeaturescallforaequallygeneral +wa(m−1)qm−1,t−wa(m)qm,t. (1) r approach to model city distribution. Below we propose The parameters of the equation are the transition rates a a general framework using master equations. Let there w (m), p, w (m). be Q cities and m citizens in the ith city. The model is d a i Thetotalnumbern(t)= mq ofpersonsandthe defined in terms of a master equation, assigning transi- m m,t total number Q of cities arePgenerally not constant: tion rates for the growth w (m ) or decrease w (m ) of a i d i the population of a city of size mi. In other words we ∂tQ=p wd(1)q1,t (2) assume that with a probability wa(mi)dt a new citizen − ∞ arrives in city i in the time interval (t,t+dt), so that ∂ n=p [w (m) w (m)]q . (3) t d a m,t m m +1. With a probability w (m )dt one of the − X − i i d i m=1 → 1 Let us focus onthe stationarystate solutions∂ q = more familar form of Zipf m(R) 1/R. Therefore we t m,t ∼ 0, for which q is simply denoted q , independent of may draw the conclusion that it is the pairwise interac- m,t m time, and n and Q are constant on average. This leads tion that is behind the Zipf law for city distribution, a us to the equation general explanation indeed. Recently Zanette and Man- rubia have proposed a city formation model [3]. They w (m+1)q w (m)q +pδ d m+1 d m m,1 use a multiplication and diffusion process and find that − +wa(m 1)qm−1 wa(m)qm =0, (4) their results also reproduce the Zipf’s law. A possible − − motivation for such a multiplicative process is that citi- wheretheratewarealsoindependentoftime. Theprob- zens of the same city are subject to the same aggregate lemcanbereadilyhandledwiththeaidofthegenerating shocks,whichtendtoincreaseordecreasethepopulation function g(s)= smq . m m size. Theimplicitassumptionisthatthestrengthofsuch Let us first coPnsider the linear case w (m)=Dm and d random shocks does not depend on the size m of city i. i w (m) = Am. This corresponds to independent deci- a Our present approach is complementary to theirs: since sions by the individuals. It is convenient to choose the therearemoreorlessm2 departuresandarrivals,thenet constants to be A=(1 p)/n,D=1/n, where n counts increase (or decrease) δm of the population of a city in − the average total number of citizens (constant). In gen- a unit time interval, is linearly proportional to m. This eralthereisadeficitforeachexistingcityw >w ,since d a leads to a system with multiplicative noise. This argu- a departingindividualhavefinite chanceto create anew ment shows, on one side that multiplicative noise results city. Theaboveequationisreadilysolvedwiththeresult: from a pairwise interaction, on the other that aggregate np (1 p)m 1 shocks, i.e. random events which affect equally each cit- qm = − e−pm. (5) izen of a given city, also would lead to the m2 transition 1 p m ∼ m − rates in our model. One can verify that the average number of citizens is On the other hand, the linear case w m is char- i ∼ indeed mqm =nwhilethe averagenumberofcitiesis: acterized by fluctuations δm which are proportional to P √m. As it has been discussed in another reproduction- np lnp Q= | |. (6) diffusion system [4], this is typical of a system of non- 1 p interacting individuals. − Severalremarksareinorder: duetotheabovedeficit,the These results are confirmed by numerical simulations. odds are always against the existing cities. If the deficit We show in fig. 1 the Zipf plot of a population of cities is small (or p 1), fluctuations can still make a large obtained for wa(m), wd(m) m2. This compares well cityarise. The≪parameterpsets the cutoffsizem∗ 1/p with actual data [1]. ∼ of the power law behavior q 1/m. The con≃dition One may argue that, in reality, both linear and square m p 1 is equivalent to the statem∼ent that the the above terms should be present, since an individual’s decision de≪ficitissmall,orthatthecreationofanewcityisarare must have both his independent as well as interactive event, which seems realistic. The distribution q 1/m parts. Thereforeitis naturalto considerthe mixedcase. m is verybroad. Invertingthe rankfunction R(m), w∼e find For the ease of presentation we assume the transition m(R)=m∗exp( R+1),verydifferentfromtheobserved probabilities to be: − Zipf’s law. The linear model implies no interaction among citi- w (m)= m2 +m, w (m)=e1/m∗(m2 +am). (8) a d m m zens. Fromanindividual’sviewpoint,thechancetoleave 0 0 (or arrive at) a city is independent of the city’s size m, The simplest way to derive the distribution of city sizes, everybody is free to move around. The simplest interac- in this case, is to use detailed balance: The number of tion we may consider is pair-wise type, this leads us to cities of size m becoming of size m 1 is q w (m). In m,t d w m2. In this case we choose wa(m) = (1 p)m2/W the steady state,this hasto balance−the number ofcities ∼ − andwd(m)=m2/W. Thecalculationis,again,standard of size m 1 becoming of size m, i.e. qm−1,twa(m 1). and the result is that This readi−ly gives − ∞ n= pW|lnp|, Q= n (1−p)m π2n , e−m/m∗ Γ(m+m0) 1−p |lnp|mX=1 m2 ≃ 6|lnp| qm =C m Γ(m+1+am0), and wheretheconstantC dependsontheratepatwhichnew Wp (1 p)m cities are created. qm = 1 p −m2 . (7) For sizes m ≪ m0 the distribution is practically the − sameasthatforthe linearcase,q 1/m. Ifthecut-off m Note that q 1/m2, for m m∗ 1/p. Again us- size m∗ m , we conclude that th∼e quadratic term is m 0 ∼ ≪ ≃ ≪ ∗ ing the rank relation, we find R(m) 1/m, or in the not relevant. If, on the other hand m m , the linear 0 ∼ ≫ 2 behaviorq 1/mholdsform m andthenitcrosses H = lnm was considered. The equilibrium distri- m ∼ ≪ 0 i i over to a power law behavior butionPat inverse temperature β = α is clearly given by a power law distribution. However for α > 2, there are qm ∼m−2−(a−1)m0for m0 ≪m≪m∗. two phases: a fluid phase for ρ<ρc(α) which is well de- scribed by a power law distribution with an exponential The Zipf exponent is therefore γ = 1/[1+(a 1)m ]. 0 cutoff,andacondensedphasewherea“mega-city”nucle- − Zipf’s law γ 1, obtains only if the pairwise interac- ates, containing a finite fraction of the total population. ≃ tion is dominant, i.e. more precisely if m (a 1) 1. 0 It is easy to understand this transition from equilibrium | − | ≪ In the original Zipf’s work [1], the scaling law is valid considerations: Let us consider the state in which we as- only for large cities. The above results with a 1 1, | − |≪ signtoeachcityiarandomnumberofcitizensmi drawn allow for a scenario where the distribution of city sizes fromapowerlawdistributionwithexponentα>2. This crosses over from Zipf’s law with γ = 1 for large cities ∼ stateminimizestheentropyandithasadensitywhichis m m , to a non-interacting situation q 1/m for 0 m given by ρ = n/Q = m = ρ (α). If we want to find a ≫ ∼ i c smalltownsm m . Thepopulationdynamicsinsmall h i 0 state with a lower density, one can introduce a chemical ≪ towns is dominated by the linear term in the transition potential,whichisequivalenttoanexponentialcutoffon rates, which describes non-interacting individuals. In thedistributionofm . Ontheotherhandifonewantsto i largecities,ontheotherhand,interactiondominatesand build a state of higher density ρ>ρ one gets into trou- c itleadstoZipf’slaw. NotethatZipf’slaw,withageneral bles. There are two ways out: The first is to put some valueofγ canalsooccurifthecondition m (a 1) 1 0 δm = ρ ρ extra citizens in each city. This results in | − |≪ i c is not met. − a state which has a free energy cost δF Qln(ρ/ρ ). c It is also interesting to consider a general model with ≃ The second way out is to put all the Q(ρ ρ ) excess w (m) = (1 p)mα/W and w (m) = mα/W. This in- − c a d citizens in only one city. This leads to a free energy cost − deed allows us to investigate the effects of multi-person δF ln[Q(ρ/ρ 1)]. This only grows logarithmically c interactions. For example α=3 would represent a three ≃ − withthe systemsize whereasthe firstvariantleadstoan person interaction. Having found that for α = 1,2 the extensiveincreaseofthefreeenergy. Itisthenclearthat solution is: in the equilibrium state the system prefers to create a pW (1 p)m “mega-city” to accommodate the excess population. qm = 1 p −mα , (9) This discussion clearly refers to an equilibrium model. − Metropolis or Montecarlo dynamics of this equilibrium we can try this solution in the equation. Treating sep- system [6] is different from the dynamics of our master arately the cases m = 1 and m > 1 we see indeed that equation. Furthermore, and more importantly, we deal this is the solution. Such a solution is also readily found with a system in which nor the number of cities Q nor using detailed balance. The numbers n and Q are given the population size n are fixed. In other words the den- by sity ρ is not fixed. For p 1 the averagedensity is very ≪ close to ρ (α). The density, no matter how it fluctuates, pW ∞ tα−2dt will sweepc across ρ in time. This suggests that, in our n= . (10) c Γ(α 1)Z et 1+p model,dynamicnucleationofa“mega-city”shouldoccur − 0 − for α>2. This agrees indeed with the results of numer- Ifα>2theintegralisfiniteasp 0andonehassimply → ical simulations. We show in fig.2 the distribution of the n pW. For 1 < α < 2 the integral diverges as p 0 an∼d the leading term is n pα−1W. In the same→way fractionmmax/nofcitizenswhichliveinthebiggestcity. ∼ Forα=3,the distributionis peakedatvaluesveryclose one can calculate the number of cities to1(forα=2thecriticaldensitydivergesρ = . Some c ∞ ∞ pW ∞ tα−1dt precursor effects of the transition are however visible). Q= qm = Γ(α)Z et 1+p. (11) This leads to our second observation, that processes mX=1 0 − with α > 2 must result in the nucleation of a “mega- city”, which is not very realistic. We can conclude that Forα>1theintegralisfiniteforp=0,sothatQ pW. ∼ interactions of higher orders (than pairwise) are not rel- The interesting point here is that the average size of evant in the dynamics of city formation. citiesisfiniteforα>2,n/Q ζ(α 1)/ζ(α),whereζ(x) ≃ − WeseethattheinteractionleadingtotheZipf’slawis, is Riemann’s Zeta function. This observation raises the ononehand,thesimplestpossible(pairwiseinteraction). question of what happens when the population density On the other it is a rather special one, since it is the ρ=n/Q grows beyond the value ρ =ζ(α 1)/ζ(α). c − “lowest order” of interaction which does not lead to the The answer is that the excess population concentrates formation of a mega-city, which draws a good portion injustonecity,whichthereforehasafinite,largefraction of the whole population. The absence of a mega-city of the total population. This effect has been studied re- suggests that in an expansion of the transition rates in cently [5]inanequilibriummodelwhereanHamiltonian powers of m, we should neglect terms of order higher 3 4.5 4 3.5 3 max)/n] 2.5 P[m( 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 m(max)/n FIG.2. Distributionofthefractionofcitizensinthelargest city. than the second. This leads us to the mixed interaction case, which gives very realistic results. In many disparate societies, it is not unnatural to assume that individuals make their city-dwelling decision based on their own opinions as well as on its interaction with other citizens. If this indeed is the case, we show that the larger cities obey approximately the Zipf’s law. WethankRicardSoleforhisearlierparticipation. This work is supported in part by Swiss Science Foundation through the grant 20-46918.96. [1] G.K. Zipf, Human Behavior and the Principle of Least Effort (Addison-Wesley Press, Inc.,Cambridge 1949). [2] Data are drawn from thesame sources as ref. [3]. [3] D. Zanette and S. Manrubia, Phys. Rev. Lett. 79, 523 (1997). [4] Y.-C. Zhang, M. Serva and M. Polikarpov, J. Stat. Phys.58,849(1990);Y.-C.Zhang,Phys.Rev.E55,R3817 (1997) [5] P. Bialas, Z. Burda, D. Johnston, preprint cond- mat/9609264. [6] C. Godreche, J. M. Luck,preprint cond-mat/9707052. 4