Avalanches in an extended Schelling model: an explanation of urban gentrification
RRoughness and avalanches in an extended Schelling model: an explanation of urbangentrification
Diego Ortega, Javier Rodr´ıguez-Laguna, and Elka Korutcheva
1, 2 Dto. F´ısica Fundamental, Universidad Nacional de Educaci´on a Distancia (UNED), Spain G. Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria. (Dated: July 17, 2020)Residential segregation is analyzed via the Schelling model, in which two types of agents attemptto optimize their situation according to certain preferences and tolerance levels. Several variantsof this work are focused on urban or social aspects. Whereas these models consider fixed valuesfor wealth or tolerance, here we consider how sudden changes in the economic environment orthe tolerance level affect the urban structure both in the closed city and open city frameworks,i.e. depending on whether migration processes are relevant or not. In the closed city framework,agents tend to group into clusters, whose boundary can be characterized using tools from kineticroughening. On the other hand, in the open city approximation agents of a certain type may enteror leave the city in series of avalanches, whose statistical properties are discussed.
I. INTRODUCTION
People with similar features (culture, income, etc.)tend to group together in the same neighborhood, giv-ing rise to segregation on a social scale. More than 40years ago, Schelling put forward a seminal model that de-scribes this reality, linking individual preferences to themacroscopic behaviour of the system [1]. Two differentsocial groups, which we may call red and blue , are dis-tributed over a square lattice with some vacancies on it.Agents are characterized by a tolerance T : the fraction ofdifferent agents in their neighborhood that he or she cantolerate. The model proceeds through the following dy-namical rules: a random agent i is selected and his/herfraction of diverse neighbors is evaluated. If this frac-tion value is lower or equal than T , the agent remainsat his/her location. Otherwise, he/she relocates to thenearest vacancy that meets his/her demands. For inter-mediate values of T we observe segregation, and clustersare formed with different types of agents.This model has attracted a great deal of attention, dueto its simplicity and insight, giving rise to a wealth ofvariants. System behaviour when one kind of agents aretolerant and the vacancies are differently priced was char-acterized in [2]. Differences between constrained models,where only unhappy individuals are allowed to move, andunconstrained ones, in which all agents can relocate tovacancies as long as they keep or increase their happiness ,were also studied [3]. In [4], attempted relocations suc-ceed with a probability which is modulated by a power-law linking their current happiness and the attractivenessof the offered place. The effect of the city shape, size andform is investigated in [5], finding that the properties ofthe system in equilibrium are weakly affected by these pa-rameters. The authors of [8] proposed a thermodynamicapproach to segregation based on their cluster geometry,and considered quantities analogous to the specific heatand susceptibility, along with a connection with spin-1models. Moreover, an open city model in which agentscan leave or enter the system was described in [9]. In addition to showing different kinds of interfaces betweenclusters, economic aspects of the system were introducedby means of a chemical potential. Recently, the use of dif-ferent tolerance levels for the agents was proposed in [10],in a system with no vacancies, where agents could onlyexchange locations with agents of a different type. Onthe other hand, in [11] each cell of the system is consid-ered a building containing many agents, and segregationwas considered both at a microscopic and a macroscopiclevel, giving rise to a complex phase diagram. Some ofthe mentioned works take into account the importance ofthe initial conditions [2, 10], and some others also con-sider migratory movements [9, 10].In this article we consider how a closed city, in whichno agents can enter or leave, adapts to drops in the tol-erance level. Under some circunstances, a vacancy inter-face is observed separating the main clusters. We estab-lish its statistical properties, specially its roughness. Onthe other hand, we also consider an open city model inwhich the system becomes more hostile towards one typeof agent (economic handicap) and more friendly towardsthe other (economic advantage). This phenomenon cangive rise to a partial or total overtake of the favored typeof agent, which may proceed through avalanches. Theseavalanches are shown to present a power-law behaviorwhich is usual of similar processes [6, 7]. Our generalframework is established in similarity with the Blume-Emery-Griffiths (BEG) model in presence of an externalmagnetic field [8, 9].The paper is organized as follows. In Section II wedefine our BEG model and discuss the dynamics of bothregimes (open and closed city) and the evolution process.In Section III we describe our results for the closed citymodel III A and the open city model III B, linking themto well-known social mechanisms: confrontation betweenequal forces within the closed city or the overtaking ofone kind of agents over the other due to economic supe-riority, which resembles a social process known as gen-trification [12]. Our main conclusions and proposals forfurther work are discussed in section IV. a r X i v : . [ phy s i c s . s o c - ph ] J u l II. MODEL
The Blume-Emery-Griffith model [13] was introducedto study the behaviour of He -He mixtures. In thismodel the spin values considered are s i = 0 , ±
1. In thepresence of a magnetic external field, the Hamiltoniancan be written as: H = − (cid:88) (cid:104) i,j (cid:105) (cid:0) J s i s j + K s i s j (cid:1) + (cid:88) i (cid:0) D B s i + H B s i (cid:1) , (1)where (cid:104) i, j (cid:105) stands for the eight nearest neighbors ofa Moore neighborhood. This Hamiltonian represents aspin-1 Ising model with coupling constant J , biquadraticexchange constant K , crystal field D B and an externalmagnetic field of intensity H B . The crystal field D B acts as a chemical potential that controls the entry ofcells with non-zero spin value. Dissimilar entry fluxesfor s = +1 and s = − H B .In our interpretation of the BEG model spin values willbe associated with blue agents ( s i = +1), red agents ( s i = −
1) and vacancies ( s i = 0). A positive value of J yields anegative energy for each pair of neighboring agents of thesame type, while a positive value of K assigns a negativeenergy to every pair of neighboring agents, disregardingtheir type. If D B >
0, the system reduces its energy byexpelling agents, and if H B > blue agents or attracting red ones.Under certain conditions, the Hamiltonian provided byEq. (1) can only decrease along the actual dynamicaltrajectories of the system, thus serving as a Lyapunovfunction [14].In order to make an explicit connection between ourphysical model and social realities, let us now define ameasure of the level of unhappiness of an agent in relationwith the parameters of Eq. (1). The lack of happiness ofagent i is measured by the dissatisfaction index I dis ( i ) I dis ( i ) = N d ( i ) − T [ N s ( i ) + N d ( i )] + D + H ( i ) , (2)where N s and N d are, respectively, the number of neigh-boring agents of the same (s) and different (d) type, D is a measure of the global economic level and is the samefor all agents. Meanwhile H ( i ) = + H for blue agentsand − H for red ones. It can be understood as half theincome gap between both types of agents. We will dropthe dependency on i when it is clear from context. Thecondition for satisfaction will be, therefore, I dis ( i ) ≤ . (3)Notice that when D <
D >
0, the economic opportu-nities have disappeared and the system becomes hostile.Even when N s > N d , an agent might be forced to leavethe system under such circumstances. Moreover, the pos-sibility that the wealth levels of both communities are notsimilar is taken into account by the parameter H . Whenits absolute value is large, agents of one type might beforced to leave while the other type is attracted.The number of similar and different neighbors can beeasily obtained from the spin variables of sites neighbor-ing i , N s ( i ) − N d ( i ) = s i (cid:88) (cid:104) i,j (cid:105) s j , (4) N s ( i ) + N d ( i ) = s i (cid:88) (cid:104) i,j (cid:105) s j , (5)where the sum over (cid:104) i, j (cid:105) should be understood as a sumover all j which are neighbors of i . Substituing Eq. (4)and (5) into Eq. (1), we can rewrite our condition for thesatisfaction of agent i , Eq. (3), as − s i (cid:88) (cid:104) j (cid:105) s j − (2 T − s i (cid:88) (cid:104) j (cid:105) s j + 2 Ds i + 2 Hs i ≤ , (6)where j runs over his/her eight closest neighbors in theMoore neighborhood. We will only allow moves that ei-ther preserve or reduce the dissatisfaction index for eachagent, and their types are fixed. For constant values of T , D and H , the energy of the BEG model becomes aLyapunov function: H = − (cid:88) (cid:104) i,j (cid:105) (cid:0) s i s j + (2 T − s i s j (cid:1) + 2 (cid:88) i (cid:0) Ds i + Hs i (cid:1) , (7)with an external magnetic field. This Hamiltonian repre-sents a spin-1 Ising model with coupling constant J = 1,biquadratic exchange of strength 2 T −
1, crystal field ofstrength 2 D and a magnetic field of intensity 2 H , as itcan be seen by comparing with Eq. (1). A. Closed city dynamics
Let us consider our agents to be living in an N × N square lattice with open boundaries and a fixed vacancydensity ρ , in similarity to [8]. Agents will not be allowedto enter or leave the system and, therefore, the globaleconomic level D and income inequality H need not betaken into account. In this case, Eq. (2) reduces to I dis ( i ) = N d ( i ) − T [ N s ( i ) + N d ( i )] . (8)At each iteration, a random occupied site i and a ran-dom vacancy j are selected. The proposed relocation isaccepted if I dis ( j ) ≤
0, where I dis ( j ) is calculated usingEq. (8). We must note that a relocation of an agentto another place where his/her happiness level is infe-rior is possible, if the destination environment verifies T ≥ N d ( j ) / ( N s ( j ) + N d ( j )). In this situation the en-ergy, E S = (cid:88) (cid:104) i,j (cid:105) s i s j − (2 T − (cid:88) (cid:104) i,j (cid:105) s i s j , (9)is not strictly a Lyapunov function, because relocationsthat increase the dissatisfaction level are allowed.The parameters of the model in this situation are thesystem size N , the tolerance level T and the vacancydensity ρ . Typical values for ρ in urban environmentsare under 0 .
1. All our simulations start with a randomconfiguration.
B. Open city dynamics
Agents can enter or leave the N × N lattice dependingon their dissatisfaction level, Eq. (2), and the economicenvironment H and D must be explicitly considered.The system dynamics is similar to the one describedin [9], and can be explained as follows: at each iterationwe choose a random site i . If it corresponds to a vacancywe attempt to occupy it with an agent of a random type.The movement is accepted if the selected agent presents I dis ( i ) ≤
0, see Eq. (2). On the other hand, if the selectedsite is occupied we attempt an internal or an externalchange with equal probabilities. For the internal change,a vacancy is randomly chosen at site j (implying an in-finte range interaction), and the dissatisfaction index forthe agent in the offered place, I dis ( j ), is calculated. Theinternal change is accepted only if the dissatisfaction inthe new place is preserved or reduced: I dis ( j ) ≤ I dis ( i ). Ifthe external change is chosen, agent i attempts to leavethe system, which will take place if I dis ( i ) >
0. Opencity systems are controlled by a variety of parameters:the economic environment variables, D and H , and thetolerance level T . III. RESULTS AND DISCUSSIONA. Closed city
As we have discussed in Sec II, two equal populationsof agents inhabit an N × N square lattice. We will use N = 50 and a fixed vacancy ratio ρ = 6%, unless other-wise specified. Agents can not enter or leave the system,i.e. no external changes are allowed. However, an agent i is able to move into an empty cell j , randomly offered, if I dis ( j ) ≤
0. Note that the relocation process may in-crease the dissatisfaction index of some of the old or newneighbors of the chosen agent.A phase diagram for different values of T and ρ waspresented in [8], which we will take as our starting point.Once the density ρ is fixed, T remains as the only controlparameter of the system. Thus, depending on its value,the system can be found in three different states, whichwe will call frozen , segregated and mixed , characterizedby the stationary morphologies and the acceptance ratefor relocations.1. Low T , or frozen . Few changes are accepted andthe system remains close to the random initial con-figuration: a random mixture of red agents, blueagents and vacancies.2. Medium T or segregated . Two big clusters are cre-ated and the accepted change rate is close to 50%in equilibrium.3. High T or mixed . Almost all changes are accepted,so no clusters are formed and the configuration re-mains close to random.We will put special emphasis on the sizes of the differ-ent clusters, measured through the segregation coefficient[15], s.c. , given by s.c. = 2 N (1 − ρ ) (cid:88) { c } n c , (10)where c indexes all the clusters in the system and n c isthe number of agents in each cluster. This coefficientranges from values close to 0, where clusterization hasnot taken place, to 1, where only two clusters remainand n c = N (1 − ρ ) /
1. Continuous evolution of T
As it was discussed previously, we will focus on how thesystem adapts to changes in tolerance. So, first, we char-acterize the system behavior when the tolerance value T decreases according to the following law T ( t ) = 1 − tanh (cid:18) tt (cid:19) , (11)where t is the time measured in Monte Carlo steps and t is a factor controlling the overall speed of the process.Eq. (11) describes the evolution of the tolerance in a citythat changes from being extremely tolerant ( T ∼
1) tototally intolerant ( T ∼ T as an analogue of the systemtemperature [8], so the process can also be understoodas a cooling process. . . . t se g r e g a t i on c o e ff i c i e n t t =10000t =1000t =100t =10t =1 Figure 1. Evolution of the segregation coefficient as a functionof time (in Monte Carlo steps) for different t values, shownin the figure key. The central lines correspond to the averageover 50 runs, while the coloured zone represents the regionfor a confidence interval of 70%. Time ranges from 0 to 30 t . The evolution of the segregation coefficient for differentvalues of t ranging from 1 (long dash) to 10 (solid) isshown in Fig. 1. As t increases, the system spends moretime on intermediate values of T , rising the clusterizationeffect. As a consequence, the segregation coefficient valuebecomes larger. For t (cid:29) t , T becomes very low, thelattice freezes and the structure created in the previousstage remains. Time is measured in Monte Carlo (MC)steps, corresponding to N × N iterations of the systemdynamics.
2. Sudden change in T
Let us consider the possibility of a sudden drop in thetolerance towards a final value of T = 1 /
4. Two types ofsocieties are considered, depending on the initial value ofthe tolerance denoted as T i : one is highly tolerant, with T i = 7 /
8, and the other one is segregated, with T i = 1 / T , the vacancies of the system are groupedat the interfaces between red and blue agents, see Fig. 2(a) and (b).Let us consider a vacancy at a flat segment of the bor-der between red and blue agents. We can calculate thecritical tolerance value for its location as T ∗ = 4 / (4+4) =1 /
2. Before the drop in tolerance, T = 1 /
2, so agentscan accept to move into these vacancies. After the drop,when T = 1 /
4, no relocation can take place anymore to-wards this place, because
T < T ∗ , so the vacancy mustremain empty. After a short time, most vacancies insideclusters have been transferred into the interface and itbecomes flat. From a social perspective, nobody wantsto be placed between two intolerant social groups: redagents find this place too close to the blue ones and viceversa. In order to characterize the interface we define the grouped vacancy ratio , which is defined as the proportionof vacancies that have either one or two more vacanciesin their neighborhood.We can provide a theoretical estimate for this magni-tude as a function of the population density ratio. Noticethat the grouped ratio should not depend on the toler-ance before the cooling procedure, since vacancies followa random distribution. We can assume a binomial dis-tribution for the presence of one or two vacancies amongthe eight cells that comprise the Moore neighborhood, (cid:18) (cid:19) ρ (1 − ρ ) + (cid:18) (cid:19) ρ (1 − ρ ) . (12)The grouped ratio is evaluated in four different situa-tions in Fig. 2 (c). Measurements before the drop in T are shown as red circles, which can be compared with thetheoretical estimates, shown as blue squares. Measuresafter the drop are shown for T = 1 / T = 7 / T , the grouped ratio presents major differences betweenthe segregated society (black circles, T = 1 /
2) and themixed society (purple circles, T = 7 / N vacancies, i.e., thesystem length. The corresponding value of ρ is, there-fore, ρ = 1 /N . In our case, N = 50 and the maximumgrouped ratio value reached corresponds to ρ = 0 .
02, aswe can see in Fig. 2 (c). If there are less vacancies, theywill be randomly placed along the boundary between theclusters. However, when ρ > /N , some vacancies maydiffuse into the bulk of the clusters, while some other va-cancies may allow the boundary to become rough. For ρ (cid:29) /N most vacancies are located in the bulk, andthe grouped ratio approach the predictions following thebinomial distribution.For T i = 7 /
3. Characterization of the boundary
Even when the two clusters are well formed the inter-face between them is not static. It tends to be flat inaverage, but it always presents fluctuations. Thus, it isrelevant to estimate its roughness, W , defined as W = (cid:115) N b (cid:88) i h i , (13)where N b is the total number of border vacancies, h itsheight, defined as its distance to the average flat line, andthe summation index i runs over all the vacancies in the (a)(b) . . . . . . r g r oup e d r a t i o TheoreticalBeforeT=1/2 AfterT=7/8 After (c)Figure 2. Long term configurations for (a) T = 0 . T = 0 .
25 for N = 50 and ρ = 6%. (c) Grouped vacancy ratio as afunction of ρ . 50 runs and 20000 iterations are considered for each run. interface. This roughness typically scales with systemsize, W ∼ N α , where α is usually called the roughnessexponent [16], as we can see in Fig. 3. . . . log N l og W Figure 3. Roughness of the cluster interface as a function ofthe system size N for flat boundary configurations, in log-log scale. Each value represents the average of 50 runs over150000 MC steps. The interface is subject both to a smoothing effectand a random noise. In other words, the system willtend to flatten the boundary in order to minimize thedissatisfaction of the agents, yet agent relocations arerandom events that will disturb the shape of the inter-face. The balance between both effects is reminiscentof the Edward-Wilkinson (EW) and Kardar-Parisi-Zhang(KPZ) universality classes [16]. Both of them present aroughness exponent α = 1 /
2. Our estimate is close tothis value: α = 0 . ± .
4. Avalanche processes
The arrival of an agent into a new neighborhood candecrease the satisfaction of one or more agents who maywish to be relocated. When these agents occupy newplaces, other agents close to them may feel frustratedand desire a new location in the system, giving rise to achain reaction. Such avalanche phenomena in this setupwere already anticipated by Schelling [1].In our closed city model, agents are relocated whentheir new level of satisfaction is not acceptable, indepen-dently of the effects on the neighborhood, as discussed inSec. II A. Under these circumstances, the energy is nota Lyapunov function. This fact is illustrated in Fig. 4,where we can see the time evolution of the energy of a sin-gle run with T = 1 / − . − . . t N o r m a li z e d E n e r g y ClosedOpen
Figure 4. Normalized energy evolution in MC steps for closedand open city dynamics. Two single runs are considered for asystem with N = 50, ρ = 6% and constant T = 1 /
2. Energyis normalized per agent and link by the factor 4 N (1 − ρ ). B. Open city
Let us discuss the case in which the number of agents ofeither type is not conserved. Again, as discussed in Sec.II, we will consider agents of two types in a 100 × T , the economical level of the system D , and also the gapbetween the financial incomes for each type of agent, H .The system undergoes the following process: in thefirst stage, with fixed values for T and D , with H = 0,the system evolves allowing both internal and externalrelocations, until a stationary state is reached with bothagent populations settled inside clusters, as we can seein Fig. 5. Now, we proceed to increase H gradually, inorder to provide an economical advantage to one typeof agents over the other. Thus, a gentrification processwill start. As a consequence one of the agent types willtend to leave the system, while the other will fill thesevacancies, giving rise to avalanches. From this momenton, we will not allow internal relocations, because ouraim is to characterize migratory movements inside andoutside the city.
1. Types of borders
Starting from a random configuration, we set a finitevalue for T and D , with H = 0. Given enough time,the system reaches an equilibrium state where the bor-ders have both straight and curved segments. In thisscenario, N s can be either 4 or 5 for agents at the bound-ary, depending on whether the agent stands at a straightor a curved point of the boundary, as we can readily seein Fig. 5. The dissatisfaction index of each agent dis-minishes when the number of similar neighbors increases(see Eq. (2)). Thus, any agent on a curved spot will leave (a) (b)(c) (d)Figure 5. For T = 1 /
4, detail of the border types: C (a), 0(b), I (c) and II (d). the lattice with higher probability than one at a straightspot. This difference is the key to the creation of differ-ent types of borders. Notice that, for a given value of T , an agent for which D > T ( N s + N d ) − N d becomesunsatisfied and is transferred out of the lattice. Thus,different types of borders can arise, associated with theirvalue for N d . • N d = 4. If D <
C-type border , see Fig. 5 (a). • N d = 3. Vacancies appear at straight segments sep-arating both types of agents, in addition to cornersites (Fig. 5 (b)). This kind of interface locationwill be termed a . • N d = 2. Straight segments of vacancies are com-pleted, but contacts along diagonals between dis-tinct clusters are allowed (I-type border [11]), as inFig. 5 (c). • N d = 1. Diagonal contacts between clusters are notallowed any more (II-type border [11]), see Fig. 5(d). • N d = 0. This is the usual situation in economicallyhandicapped systems, where D >
0. When D isabove this treshold the system expels all agents.In previous work [9] borders of types I and II weredescribed and characterized by means of geometric con-structions. In this paper we have chosen to determinethem via threshold values, including borders of type Cand 0, which have not been reported previously.
2. Avalanche processes
At this point, we proceed to increase H gradually bya fixed amount ∆ H (cid:38) T / µ b = D + H . Meanwhile, redagents become more and more prosperous, µ r = D − H ,thus enlarging the economic gap. Both expressions ac-count for the dissatisfaction or unhappiness of an agentdue to their economic conditions. Moreover, if we definethe happiness associated to being in a specific neighbor-hood as λ = T ( N s + N d ) − N d , the satisfaction condition,Eq. (2), can be expressed now as µ r,b ≤ λ. (14)The interpretation of Eq. (14) is straightforward:when the satisfaction of being into a neighborhood, λ compensates the unhapiness arising from the economicsituation, µ r,b , the agent remains in the system. As wekeep increasing H , the gap between µ r and µ b opens up.The system becomes more hostile towards blue agents,which increase µ b , and some of them are forced to leavewhen Eq. (14) ceases to hold. Therefore, there appearsome new vacancies in the lattice. At the same time,the effect is opposite for red agents: µ r decreases andthe lattice is more satisfactory for them, because theyare getting more and more economic advantages. Thus,red agents come from outside and occupy the vacanciespreviously created. Now, some of the blue agents closeto these occupied locations cease to verify Eq. (14) be-cause λ has decreased for them due to the arrival of redagents. Thus, they may be transferred out of the systemon subsequent time-steps. This gives rise to further newvacancies that are filled again with red agents and theprocess goes on in a self-sustained way. This is what wecalled a blue avalanche , because it originates with blueagents leaving the lattice.Yet, there is another way to generate an avalanche.We depart from an equilibrium situation with some va-cancies, which requires a mid-ranged economic environ-ment, check Fig. 5 (b), (c) and (d). Before H is strongenough to force blue agents out, red agents may be ableto fill these vacancies up. Blue agents next to these lo-cations may cease to verify Eq. (14), as in the previoussituation, and are forced out of the system. These newvacancies are occupied by red agents from outside, mak-ing further blue agents leave. The process goes on, givingrise to what we will call a red avalanche . The value of H remains constant while theseavalanches take place. After the avalanches are finished,if there are still some blue agents in the system the valueof H is increased further until a new avalanche takesplace.The similarity with a gentrification process is clear:one of the agent types gets richer while the other is be-coming economically handicapped. The latter is forcedout of the system, leaving vacancies on the borders be-tween the two clusters. Meanwhile, red agents withmore economic power enter the system and occupy thesevacancies. The process becomes self-sustained becauseother agents from the less favoured group are forced toleave the system due to their proximity to members of thewealthy class. Of course, not all processes in the systemgive rise to an avalanche.Avalanches will be characterized by their size s , de-fined as the total number of blue agents that have leftthe lattice as a consequence of the departure of the firstone. We have obtained the avalanche size histograms,and fitted them to a probability density function (PDF)of the form p ( s ) = Cx α exp( − x/x ) , (15)where C is the normalization constant, α accounts forthe scaling exponent for the avalanches and x acts as amaximal cutoff value. For reasons of numerical stabilitywe focus on the complementary cumulative distributionfunction (CCDF) [17], which is fitted by the expression C ∗ x α +1 exp − ( x/x ). Data from 100 complete extinc-tions of the blue agents in the system have been mea-sured for each D value. For more numerical details, seeAppendix A.The next sections are devoted to the analysis ofavalanche distributions for three values of T : low ( T =1 / T = 1 /
2) and high ( T = 3 / T we have characterized the behavior of the sys-tem for a wide range of fixed values of D , ranging fromthe situation where no vacancies are present to the ex-tinction point in which, due to the high hostility of themedium all agents leave. We will vary H in order toobserve the avalanche processes for each D value consid-ered.Let us introduce a convenient notation to describeneighborhoods, using s and d to denote similar and dif-ferent neighbors, respectively, e.g. 3 s + 2 d means that acertain agent has 3 similar and 2 dissimilar neighbors.
3. Avalanche distributions for T = 1 / We begin our study in a city with a high economicinterest, D = − .
125 and low tolerance value T = 1 / D . Now, we increase H gradually, µ b becomes higher, and some blue agentsmay abandon the lattice. In fact, these blues agents arethe ones which do not verify Eq. 14, and have 4 s + 4 d neighbors. It must be noted that knowing the value of T and the number of different and similar neighbors, λ is fixed. Meanwhile, the situation for red agents keepsimproving: µ r becomes lower, so the vacancies createdby the blue agents are filled by red agents coming fromoutside. This is what we defined previously as a blueavalanche (Sec. III B 2).The system is so interesting economically for blueagents that they will adopt different strategies in orderto remain, such as the formation of special cluster con-figurations: triangles and rectangles in contact with thesystem border. Thus, up to three avalanche processescan be needed to deplete the system of blue agents, ascan be seen in Fig. 6 (a), where D = − . λ n,k , where n denotesthe order and k takes value r for red and b for blue ones.The fitted values for α and x in all these cases are shownin Table I.For D = − .
125 and increasing H gradually, we findthat the first three avalanches start at treshold values λ ,b = − . λ ,b = − .
75 and λ ,b = − .
0. Extendingour notation, we may say that each avalanche set hasa dominant process, which is characterized by the typi-cal neighborhood of the expelled blue agents and their λ value. The values previously calcultaed correspond withcertain neighborhood types: 4 s + 4 d , 2 s + 3 d and 5 s + 3 d ,respectively. Of course, no avalanche consists of a sin-gle type of process in practice. We must note that whenblue agents with a higher λ start their expulsion (suchas 2 s + 3 d ) those with an inferior treshold (for example4 s + 4 d ) can be still be in process of leaving. Thus, itis possible for various processes to take place simultane-ously in the same avalanche. This is the situation for thefirst and second avalanche sets, λ ,b and λ ,b in Fig. 6(a). The process 4 s + 4 d dominates both avalanches sotheir curves are close to each other. As the avalanche for λ ,b is the last one for this D value, it has a smaller sizedue the small number of blue agents remaining on thelattice.The social meaning is clear: the less favoured agentswill try to stand on an economic advantageous environ-ment despite their income gap with the other group. Toachieve this goal diverse neighborhood structures will becreated ( ghettos ). This is the case scenario for neighbor-hoods as Harlem or Clinton Hills [18].Now, let us focus our attention on higher values of D : − . − .
625 and − . D = − . µ r (cid:46) − .
0. That points at a 4 s + 4 d redavalanche with λ ,r = − .
00 (Sec. III B 2). The next one, D = − . purple avalanche : a si-multaneous red avalanche with λ ,r = − .
00 as in the for-mer case, and a blue avalanche, 4 s +3 d with λ ,b = − . D ± H ). We must note that the 4 s + 3 d is the vecindaryassociated with a 0-type border (Fig. 5(b)). So, whilethe red agents coming from outside are filling the vacan-cies with 4 s + 4 d which correspond to a C-type border,blue agents with 4 s + 3 d are leaving the system creatinga 0-type border. Finally, we have blue avalanches for thefirst and second avalanche sets with D = − . s + 3 d and 3 s + 1 d , re-spectively. The behavior of blue avalanches is similar tothe ones explained for D = − . x .The last value that we analyze for T = 1 / D =0 . predomi-nant vacancy state is verified, D/T > H = 0, and Eq. (3) the conditionfor an agent to remain on the system can be written as N d ≤ ( N s − D/T ) / (1 /T − T is lower than unity,the denominator stays positive, so the numerator of theformer equation dictates the agent behaviour. Any agentmust fulfill the condition N s ≥ D/T to remain on the lat-tice, even if no different agent is in the vecindary. As thesystem initial configuration is random and both kindsof agents and vacancies are equally likely, the probabil-ity for an agent to have more than three similar neigh-bors is small. Therefore, when
D/T > x (Fig 7 a). The border between clusters is not relevantanymore: red clusters grow with a vecindary as (3 s + 0 d )and blue ones become smaller when agents have a neigh-borhhod (4 s + 0 d ) (Fig 7 (b) and (c).Socially, the situation for blue agents might be com-pared to the Chicago suburbs where the population couldincrease their personal ties via a community network [9].These ties may prevent a massive exodus despite thelack of attractiveness of the environment. However, inour model, the evolution of the system departs from thissituation and develops in two opposite directions: blueagents are removed from the system, in contrast to redagents, which increase their population. This could beunderstood as the unrelated behavior of two communi-ties. One of them chooses to cooperate and the ensuinggrowth overcomes economical hardships. The other onedoes not strength their links and is forced to leave thecity.Finally, we show the values of the fitted parame-ters in Table I, being x c the choosen size for the fit-ting (Appendix A). We can find α values in the range − − + Avalanche size
CCD F D=-2.125 λ =-1.75D=-2.125 λ =-2.00D=-2.125 λ =-1.00 − − − Avalanche size
CCD F −−− λ λ D= − λ =-2.00 λ = ±0.375 =0.00 (a) (b)Figure 6. CCDF of the avalanche distribution sizes for T = 1 /
4. For each curve the D value is specified. Treshold valuesare given as λ n,k where n is the avalanche set index and k its kind: r for reds, b for blues and p for purple ones. For purpleavalanches upper and lower tresholds are expressed as D ± λ ,p . The fitted power-law cutoff functions are depicted with lines.(a) (b) (c)Figure 7. From left to right: snapshot of the system evolution for T=1/4 and D=0.875. Equilibrium (a), 6 MC steps (b) and13 MC steps (c). [ − . , − .
98] previously reported in references [19–23],concerning self-organized criticality in different systems.
B A D α x x c C N b − . − . ± .
002 16 . ± . . ± .
03C r − . − . ± .
02 15 . ± . . ± .
02C p − . − . ± .
01 24 . ± . . ± .
03C b − . − . ± .
02 23 . ± . . ± .
030 r − . − . ± .
03 3 . ± .
04 20 1 . ± .
040 p − . − . ± .
008 42 . ± . . ± .
020 b − . − . ± .
004 140 ± . ± .
02I r − . − . ± .
02 18 . ± . . ± .
02I p − . − . ± .
004 79 . ± . . ± .
01I b 0 . − . ± .
005 181 ± . ± . . − . ± .
007 20 . ± . . ± . . − . ± .
03 69 . ± . . ± . . − . ± .
02 9 . ± . . ± . T = 1 /
4) for D values from (Section III B 3). B is the type of border, with ’V’meaning vacancy dominated regime. A is the avalanche type,(’b’ is blue, ’p’ is purple and ’r’ is red), α is the power lawexponent, x accounts for cutoff value and x c is the chosensize for the fitting (see Appendix A). − − + Avalanche size
CCD F D=1.625D=1.750D=1.875 λ =1.50 λ =2.00 λ =2.00 Figure 8. CCDF of the avalanche distributions for T = 1 /
4. Avalanche distributions for T = 1 / and T = 3 / Most of the processes that arise for these values of T are avalanches with similar parameter values and be-haviours to those explained in the previous section (Sec.III B 3).Nevertheless, as this T fixed value is larger than be-fore, the dissatisfaction index disminishes and Eq. 14 isverified for higher values of D . Thus, despite being in aless economically interesting system, agents populate thelattice. As a consequence the range of values of D forwhich the system is vacancy dominated increases.0The two last avalanches for T = 1 / D = 1 .
750 (shortdash) and D = 1 .
875 (long dash) are in the predominantvacancy state (Fig. 8). They are also really close toeach other, suggesting that the same mechanism takesplace and it is related to 4 s + 0 d neighborhoods with λ ,b = 2 .
00. This process implies that the blue agentswill be expelled when they are surrounded by vacancies.The system evolution for D = 1 .
750 is apparently sim-ilar to the one in Fig. 7. Nonetheless, the environmenthas become so hard that red agents are not able to ex-pand as the blue agents progressively leave the city, con-verting a purple avalanche into a blue one. From a socialperspective it can be explained as people leaving the citylooking for a better financial situation somewhere else,abandoning the neighborhood into obliteration.One of the most interesting processes in our work takesplace for D = 1 .
625 (continuous), and is also found in thepredominant vacancy state (Fig. 8). It is convenient todepict its evolution in Fig. 9. As it is shown, the firstprocedure that takes place is the red cluster expansion(Fig. 9 b)) for the neighborhood 3 s +0 d which takes placewhen µ r (cid:46) .
5. This mechanism proceeds until a bluecluster is found. As the red cluster grows, the populationincreases. When both kinds of agents are close, the 4 s +1 d red avalanche removes all blue agents. For this D value, α = − . ± . x = 17 . ± . T = 3 /
4, multiple D values are found inthe vacancy dominated regime, but all the situations arerelated to those previously explained. Type of borders 0,I and II are found to be in this situation.When the system is in the predominant vacancy state,there are only two processes that guarantee the com-plete removal of blue agents. The first one is associatedwith 3 s + 0 d neighborhoods and consist of the red clusterexpansion, which changes the environment. After that,other complementary processes take place (as in Fig. 9).The other one is related to 4 s + 0 d neighborhoods, whichimplies the total expulsion of blue agents surrounded byvacancies (Fig. 7). As a consequence, fitted parametersare close to the ones previously calculated for other val-ues of T in the same situations. IV. CONCLUSION
Summing up, interesting results which can be corre-lated to socio-economical situations emerge when a vari-ation of parameters such as the tolerance in the closedcity framework or H in the open city model are consid-ered.The generalization of the open city model provides a new framework for the study and understanding of abroad class of segregation processes. Besides analyzingits results from a social and economical perspective, themodel is also linked with the physics of the BEG modelunder the influence of an external magnetic field. Wehave considered this model under two different and com-plementary angles: the closed city (Sec. III A) and theopen city approximation (Sec.III B).In the closed city approximation the only relevant so-cial variable is the tolerance. Since agents can not enteror leave the city, economic environment variables do notplay a role in this case. We characterize the behaviourof the system when the tolerance falls in a continuousor a sudden way, after the system has reached equilib-rium for an intermediate or high tolerance value. For thecontinuous decay we have found that the final clusteriza-tion degree of the system, measured via the segregationcoefficient, depends on the drop rate: for slow rates thesystem has enough time to create clusters, so the segre-gation coefficient is close to unity (Fig. 1). The mainfeature is that once the system has reached equilibriumfor T = 1 / T = 1 /
4) createsa vacancy border between the two big remaining clusters(Fig. 3). The interface rugosity, W , scales with the sys-tem length N as W ∼ N α with α = 0 . ± . D is associated withthe mean economic city level, and H stands for the eco-nomic gap between both types of agents. The dynamicalrules are the following: once the system has evolved intoan equilibrium state, fixing T and D , H progressivelydecreases. In the economical sense, there are less finan-cial resources available for blue agents, meanwhile, redagents are becoming prosperous, so an economic sepa-ration is created. Some blue agents, generally the onesthat are closer to the red agents, begin to leave the city.These new vacancies are occupied by red agents comingfrom outside, so the process goes on in a self-sustainedway, and resembles gentrification.A power law with exponential cutoff expression hasbeen used to describe the avalanche size histograms.While the cutoff length depends on the system size, thepower law exponents are in the range [ − . , − . (a) (b) (c)Figure 9. From left to right: snapshot of the system evolution with T = 1 / D = 1 . values that can be found in the literature for diverseavalanche processes.In our modified Schelling model gentrification pro-cesses could also help understand the formation of ghet-tos, as special configuration of the less favoured class inorder to remain in the cities. Moreover, our results high-light the importance of tolerance and ties for people tobe satisfied despite harsh conditions. On one hand, col-laboration inside of a neighborhood implies an improv-ing in economic and trading exchanges, making growthpossible. On the other hand the lack of ties in a disfavor-able environemt results in the neighboorhood progressivedegradation.Further studies should focus on variants in which agents consider their future happiness perspectives [25],or the influence of altruistic behaviour [26] and the bal-ance between cooperative and individual dynamics [27].The transfer rules from these works combined with theextended open city model presented in this paper couldlead to a framework where evolution of a city could beanalyzed during several stages. ACKNOWLEDGMENTS
We acknowledge financial support from the SpanishGovernment through grants PGC2018-094763-B-I00 andPID2019-105182GB-I00. [1] Schelling, T., Dynamic Models of Segregation.
J MathSociol , 143 (1971).[2] Zhang, J. A., Dynamic Model of Residential Segregation. J Math Sociol , 147 (2004).[3] Dall’Asta, L., Castellano, C. and Marsili, M., Statisticalphysics of the Schelling model of segregation. J. Stat.Mech. , L07002 (2008).[4] Albano, E. V., Interfacial roughening, segregation anddynamic behaviour in a generalized Schelling model. J.Stat. Mech.: Theory Exp , 03 (2012).[5] Fossett, M. and Dietrich, D. R., Effects of city size, shape,and form, and neighborhood size and shape in agent-based models of residential segregation: are Schelling-style preference effects robust? Environment and Plan-ning B-Planning & Design , 149 (2009).[6] P. Bak, How Nature works , Springer (1996).[7] Bartolozzi, M., Leinweber D.B. and Thomas, A.W.,Scale-free avalanche dynamics in the stock market,
Phys-ica A , 132 (2006).[8] Gauvin, L., Vannimenus, J. and Nadal J.-P. Phase dia-gram of a Schelling segregation model.
Eur. Phys. J. B , 293 (2009).[9] Gauvin, L., Nadal, J.-P. and Vannimenus, J. Schellingsegregation in an open city: A kinetically constrainedBlume-Emery-Griffiths spin-1 system. Phys. Rev. E ,066120 (2010).[10] Barmpalias, G., Elwes, R. and Lewis-Pye, A. Unper-turbed Schelling Segregation in Two or Three Dimen-sions. J Stat Phys , 1460 (2016).[11] Gargiulo, F., Gandica, Y. and Carletti T. EmergentDense Suburbs in a Schelling Metapopulation Model: A Simulation Approach. Advances in Complex Systems ,1750001 (2017).[12] Smith, N., New globalism, new urbanism: Gentrificationas global urban strategy. Antipode , 427 (2009).[13] Blume, M., Emery, V. J. and Griffiths, R. B. Ising modelfor the λ transition and phase separation in He 3 -He 4mixtures. Phys Rev A , 1071 (1971).[14] Simon, B. The statistical mechanics of lattice gases,
Princeton University Press (1993).[15] Stauffer, D. and Aharony, A.
Introduction to PercolationTheory,
Taylor and Francis (1992).[16] Barab´asi, A. L. and Stanley, H. E.
Fractal Concepts inSurface Growth
Cambride University Press (1995).[17] Clauset, A., Shalizi, C. R. and Newman, M. E. J. Power-Law Distributions in Empirical Data.
SIAM Rev. , 661(2009).[18] Freeman, L. There goes the ’hood: view of gentrificationfrom the ground up,
Temple University Press (2006).[19] Bak, P., Tang, C. and Wiesenfeld, K., Self-organized crit-icality: An explanation of the 1/f noise.
Phys. Rev. Lett. , 381 (1987).[20] Munoz, M. A., Dickman, R., Vespignani, A and Zapperi,S. Avalanche and spreading exponents in systems withabsorbing states. Phys. Rev. E , 6175 (1999).[21] Beggs, J. M. and Plenz, D. Neuronal Avalanches in Neo-cortical Circuits. The Journal of Neuroscience , 11167(2003).[22] Batac, R., Paguirigan, Jr. A., Tarun, A. and Longjas,A. Sandpile-based model for capturing magnitude distri-butions and spatiotemporal clustering and separation inregional earthquakes. Nonlinear Processes in Geophysics , 179 (2017).[23] Zachariou, N., Expert, P., Takayasu, M. and ChristensenK. Generalised Sandpile Dynamics on Artificial and Real-World Directed Networks. PLOS ONE , e0142685(2015).[24] Thomas, W. L. (Ed). Man’s Role in the Changing theFace of the Earth,
University of Chicago Press (1956).[25] Houy, N. Forecasts in Schelling’s segregation model.arXiv:1911.08191 (2019).[26] P. Jensen, T. Matreux, J. Cambe, H. Larralde, and E.Bertin., Giant catalytic effect of altruists in Schelling’ssegregation model.
Physical Review Letters , ,208301 (2018).[27] S. Grauwin, E. Bertin, R. Lemoy, and P. Jensen. Compe-tition between collective and individual dynamics. Pro-ceedings of the National Academy of Sciences , ,20622 (2009). [28] Gillespie, C. S., Fitting Heavy Tailed Distributions:The poweRlaw Package. Journal of Statistical Software , Appendix A: Fitting the avalanche histograms
Some numerical difficulties associated with direct es-timation of α and x from the probability density func-tion (PDF) of the histograms are known to arise [17].In order to address them, we have resorted to the useof the complementary cumulative distribution function(CCDF). Our analytical form is given by Eq. (A1). P ( s ) = P r ( X > s ) = ˆ ∞ s Cx − α exp( − x/x ) dx = F ( α, x ) + G ( α, x ) s α +1 exp − ( x/x ) (cid:20) α sx + ... (cid:21) . (A1)The terms inside the brackets are an expansion of M (1 , α, s/x ), where M is the confluent hypergeo-metric function. Once α and x have been estimated,both F ( α, x ) and G ( α, x ) take fixed values. In fact, F ( α, x ) does not introduce significative changes in thefit, and it can be neglected. Under these circumstances,and retaining only the leading term of the series, we have P ( s ) ≈ G ( α, x ) x α +1 exp( − x/x ) . (A2)Consequently, we can infer the exponent α and the length x of an avalanche from the CCDF of the avalanche his-togram.The experimental CCDF data series in our avalancheshave been calculated from 100 complete extinctions of the blue agents. After that, data are plotted with a constantbin size by means of the poweRlaw R package [28]. Thenan expression of the type C ∗ x α +1 exp − ( x/x ), where C ∗ ≈ G ( α, x ) is fitted by the nonlinear least squaremethod from the stats subroutines [29]. Although thecurves have been depicted for a wide avalanche size range,they have been fitted inside the interval [1 , x c ), being x c a choosen value to uphold precision. Deviations betweendata and fit can appear for s (cid:29) x , due to the seriesexpansion approximation and the own nature of the taildistribution, but they are not relevant. As a practicalrule, if x <
60 then x c ≤ x (see Table I). Choosing x cc