A new aggregation model. Application to town growth

PHYSICA ELSEVIER

Physica A 219 (1995) 13-26

A new aggregation model. Application to town growth L. B e n g u i g u i Solid State Institute and Department of Physics, Technion-Israel Institute o f Technology, 32000 Haifa, Israel

Received 8 March 1995; revised 28 April 1995

Abstract A variant of the Eden model with noise reduction is presented. A site is occupied if it is visited p times but the neighboring sites are added to the potential sites not after occupancy but immediately after the first visit. It results in aggregates with a very different pattern from that of Eden aggregates. If p is large enough, scaling properties are observed, Possible application to the morphology of large towns is proposed.

1. Introduction Aggregation models [1, 2] are receiving increasing interest because of their own intrinsic properties and the variety of applications in different fields: in physics, in chemistry and even in biological systems [2]. Recently it was proposed to apply one of the most known aggregation models, diffusion limited aggregation (DLA) [-3], to the problem of town growth [-4]. Batty and his collaborators [-4] tried to compare the pattern of a D L A aggregate with the m o r p h o l o g y of a small town near London. This approach is new in urban geography and it is certainly a promising way to consider the problem of town growth. But straightforward application of D L A cannot give the expected results, i.e. an aggregate with the same properties like the town considered as an aggregate of buildings. Two elements which characterize towns are lacking in DLA: (a) Very often the central region of the town is compact: (b) the town is not compound of one continuous cluster but rather is made of a main central cluster surrounded by a large number of much smaller islands. In search for a more realistic aggregation model, I defined a new model which has remarkable scaling properties. This paper is devoted to the description of the model and its properties. Application to town growth needs further refinements and only some directions are mentioned in the conclusion. 2. The model The model is a variant of the well known Eden model [5]. As in the original model, a seed particle is located at one of the square lattice sites (the present model is 0378-4371/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 5 ) 0 0 1 4 5 - X

14

L.Benguigui / Physica A219(1995)13-26 +.,

a_

m.

•

b ..p&g~:j.h. I.i

" Z "

. •

W

"1"" L'P"r

C

Fig. 1. Aggregate with N = 3000 for different values of p: (a) p = 2; (b) p = 10; (c) p = 20.

restricted to two dimensions). This site is the center of the a g g r e g a t e a n d the four n e i g h b o r i n g sites are the p o t e n t i a l sites for a n e w particle. O n e of these p o t e n t i a l sites is c h o s e n at r a n d o m : it is n o t o c c u p i e d b u t o n l y visited. T h e n e i g h b o r i n g sites of the visited site are a d d e d to the list of the p o t e n t i a l sites. T h e next step is to pick u p a g a i n at r a n d o m a new site in the list (including the visited site), a n d o n l y if the chosen site has been visited p times, it is occupied. If not, it r e m a i n s a visited site. p is c h o s e n to be a n integer b u t it is p o s s i b l e to include n o n - i n t e g e r values of p b y a slight m o d i f i c a t i o n of the rules. If p is n o n - i n t e g e r , c o n s i d e r the t w o integers p~ a n d P2 which are Pl = int(p) a n d P2 = int(p) + 1. A site will be o c c u p i e d after Pt visits with p r o b a b i l i t y p - p~ a n d after P2 visits with p r o b a b i l i t y P2 - P-

L. Benguigui / Physica A 219 (1995) 13-26 RG

15

R6 /

N=500 ]

15 12

II IO 9'

22 21

20

./,

171

/I

/

N=I000

161 151 141

i i

r

it1

i

i

N=2000

i

r

N=3000

26F

5f /

24

19

231-

18

~-2~L~gg

o

Ib ' I'4

Fig. 2. R a d i u s o f g y r a t i o n RG as a f u n c t i o n p. N o t e the a n o m a l y f o r p = p~.

T h e model is different from the Eden model with noise reduction, since once a site is visited, the neighboring sites are added to the list of the potential sites. The aggregate formed in this w a y is m a d e of one central cluster which is continuous in this sense that one can pass from a site to a n o t h e r by j u m p s from a site to its neighbors, and is m a d e of m a n y very small clusters. The central cluster has m a n y holes inside and a very complicated surface. F o r low p, the aggregate is similar to that of the Eden model (p = 0); however for large p, the pattern is different. O n e can speak of Eden type aggregates and p-type aggregates. In Fig. 1, the influence of p is shown in the case of three aggregates with N -- 3000 particles and p -- 2, 10 and 20. The distinction between the two types of aggregates can be m a d e m o r e precise in studying the three following quantities: - The radius of gyration of the aggregate, R~. - The time t necessary to build the aggregate. This time is equal to the n u m b e r of times a site is chosen in the list of the potential sites. - The local density at the center of the aggregate. If N(r) is the n u m b e r of occupied sites in a circle of radius r, the local density at distance r is defined as p(r) = 2~(dN/dr)/r; and Po = p(0).

L. Benguigui / Physica A 219 (1995) 13-26

16

RG

//

16

15

14

15

J ~

0

..,/

r

r

l

l l 5

r

l

l

l

~

I0

l

p

15

Fig. 3. RG versus p, in the vicinity of Pc (N = 1000).

These three quantities have been studied as functions of two parameters: p and N, the total number of particles in the aggregate. The interesting result is that these three quantities exhibit anomalies only when p is varied and N kept constant. This gives a precise distinction between the Eden-like aggregates and p-type aggregates. H o w ever, when these quantities are studied by varying N and keeping p constant, there are no anomalies. In Fig. 2, the variation of RG with p is shown for N = 500, 1000, 2000 and N = 3000. One sees clearly a change of the slope at a value of p we call Pc. We made a detailed study of RG(p) for N -- 1000 around Pc = 6, using p as a continuous variable as defined above. The results are shown in Fig. 3. Effectively, a small discontinuity is observed for po = 6 associated with a change in the slope. F r o m a systematic study of Re(p) for different N, one can fix the critical value Pc for which the anomalies are observed. The curve pc(N) is shown in Fig. 4. Pc depends on N as Pc ~ N ~, e ~ 0.45 as indicated by the line drawn in Fig. 4. For p lower than Pc one has R e ~ p2. But above Pc, the variation of RG is best represented by R e ~ p~, fl ~ 0.18 (Fig. 5). As mentioned above, the variation of R e with N (for p kept constant) does not exhibit any anomaly. In Fig. 6, the variation of RG with N in a log-log plot is given, for p = 6. RG can be expressed as RG ~ N 7, with 7 -- 0.43. Only the prefactor is p dependent. This exponent is identical to that of the Eden model [6] in the case of small samples. For large sample the exponent is expected to be 0.5. This confirms that Re(N) has the same dependence with N, whatever the value of p.

L. Benguigui/ Physica A 219 (1995) 13-26

17

Pe

,o

~ ~

0

i

;

i

,

I

IxiO3

i

'

N

5xlO 3

Fig. 4. Pc versus N. The continuous line is p~ ~ N °*s.

R~

3o

2o

II

.

.

.

.

.

.

.

.

.

.

.

p

Fig. 5. RG versus p. For p above Pc, Re ~ pO.ls as shown by the continuous line (N = 1000).

Similar results are obtained when studying the variation of t and Po at different p, for N constant. Figs. 7 and 8 show t(p) and Po(P) for N = 1000. O n e observes again a change in the slope for Pc = 6. However, the small discontinuity can be observed only in the curve t(p). It seems that the discontinuity is too small to be observed in Po(P). In Figs. 9 and 10, t(N) and po(N) are drawn. Here also a n o m a l y is absent confirming that all the three quantities behave "normally" with N but not with p. However, the shape of the curve is dependent on p.

L. Benguigui / Physica A 219 (1995) 13-26

18

FOG

i

f

i

i

,

I0 2

i

r,I

,

i

i

,

i

i

i~l

I0~

N

104

Fig. 6. Log-log plot of RG versus N, for p = 6. Note that for N < 1000, p > po, and for N > 1000, p < Pc.

t 4.104

3-1~

2 llO4

I.Io4

/ ;

~ l l l l l l l ~ l p I0 15

Fig. 7. t, t h e t i m e n e c e s s a r y t o b u i l d a n a g g r e g a t e o f N = 1000 p a r t i c l e s , a s a f u n c t i o n o f p.

W e c a n consider pC as the critical value of p, at the t r a n s i t i o n between the two k i n d s of aggregates. T h e physical m e a n i n g of this t r a n s i t i o n is given by Po, the local density at the center of the aggregates. As long as Po ~ 1, the center has its m a x i m u m density. This a p p e a r s when p ~< Pc. F o r p > Pc the local density is everywhere less t h a n the

L. Benguigui / Physica A 219 (1995) 13 26

19

Po 1.0

0.9

0.8

o

. . . .

~, . . . .

ib . . . .

1'5

P

Fig. 8. Po, the local density at the center of the aggregate for N = 1000 particles, as a function of p.

t(x~O41 P =20 P=9

P=5

I

2

5

4

N(xlO~)

Fig. 9. t versus N, for various p.

m a x i m u m v a l u e . T h i s t r a n s i t i o n c a n be s e e n as a c o m p a c t - d i l u t e t r a n s i t i o n of t h e a g g r e g a t e . T h e o r d e r p a r a m e t e r o f t h e t r a n s i t i o n m a y be c h o s e n as t h e c h a r a c t e r i s t i c d i s t a n c e r l , w h i c h is d e f i n e d t h r o u g h t h e l o c a l d e n s i t y p(r). W h e n p < Pc, p(r) is c o n s t a n t a n d e q u a l t o 1 for r ~< r l . F o r l a r g e r r, p(r) is a d e c r e a s i n g f u n c t i o n (Fig. 11).

L. Benguigui / Physica A 219 (1995) 13-26

20

1.0

0.~

5~)0

0

~ 1000

~ 1500

N

Fig. 10. Po versus N, for p = 6. Below N = 1000, the continuous curve is P0 = 2.85 N °is.

[

P

5-;>0.5

0 ........

"\,\

rl

I ....

rl

Fig. 11. Local density p(r) versus r (N = 3000). p = 0 and p = 3 correspond to p < Pc.

I n Fig. 12, r l as a f u n c t i o n o f p is d r a w n for s o m e v a l u e s o f N. It is c l e a r f r o m t h e results of Fig. 8 t h a t r l g o e s t o z e r o for a v a l u e o f p s l i g h t l y less t h a n Pc, since for P = Pc, Po is a l r e a d y s m a l l e r t h a n o n e . W h e n r l is p l o t t e d as a f u n c t i o n of (po - p), it a p p e a r s t h a t t h e v a r i a t i o n s o f r ~ a r e a u n i v e r s a l f u n c t i o n . F r o m a l o g - l o g p l o t , o n e gets r l ~ (Pc -- P)o. W e use Pc as t h e v a l u e o f p l for w h i c h r l = 0, b e c a u s e p l s e e m s so n e a r to Pc t h a t it is n o t m e a n i n g f u l t o m a k e this d i s t i n c t i o n . T h e e x p o n e n t 6 is a b o u t 1.7. C o n t r a r i l y t o t h e g e n e r a l t r e n d o f p h a s e t r a n s i t i o n , t h e e x p o n e n t 6 is l a r g e r t h a n one. O n e n o t e s t h a t t h e b e h a v i o r o f t h e m o d e l n e a r Pc is v e r y c o m p l e x . T h e a n o m a l i e s in R e , t a n d Po a p p e a r s a p p a r e n t l y at t h e s a m e v a l u e of P(Pc) b u t r l g o e s to z e r o for Pl
L. Benguigui / Physica A 219 (1995) 13-26

0

21

5

Fig. 12. r l as a function of p for different values of N. I n the i n s e r t a l o g - l o g p l o t of r l versus (po - p).

In the range p > Pc, we propose that the local density p(r) can be expressed the position of the most remote particle)

p(r) = Po

a s (r m is

(1)

with Po ~ NaP b and/'max ~'~ NCp a. a, b, c and d are exponents which are not known and have to be determined. First, the validity of (1) can be checked in plotting P/Po versus r/rmaxfor different values of N and p, as shown in Fig. 13. The collapse of the points onto one curve shows that the assumption (1) is justified. The determination of the exponents can be made by fitting the measured values of Po and rmax. It appears that the exponents a and d are small (about 0.15) and difficult to get with precision. The following procedure is adopted: c and b are determined directly and the two other exponents (a and d) are determined directly from the relation Ymax

N ..~ ~ p ( r ) r d r . o

(2)

L. Benguigui / Physica A 219 (1995) 13-26

22

N

P

8

[] - 2 0 0 0 zx - 3 0 0 0

-;o~,O~ o o _ _

~-'~'e~'e%%~3 ~

I0

• - 4000

~K.

o

-

5000

12 14

%, % r / r max.

F i g . 13.

P/Po v e r s u s r/rm s h o w i n g

the collapse of the points onto a unique curve.

I n t r o d u c i n g (1) i n (2) o n e gets rmax

(3)

N ~ f Nap-bf\l~ P / dr (~-~23_a']r 0 or

1

N ~ N a+ 2cp2d- b f f (U) IAdu.

(4)

0 (4) is m e a n i n g f u l o n l y if a + 2c = 1 a n d b = 2d. A n o t h e r c o n s i s t e n c y of (1) was c h e c k e d b y t h e d e t e r m i n a t i o n of RG, rmax

R 2 ~ f p(r)r3dr/g

(5)

0 or

R e ~ N (4C+a- 1)/2p(4d-b)/2

f (u) u 3 du,

(6)

w i t h t h e r e l a t i o n s a + 2c = 1 a n d b = 2d, o n e gets t h a t R~ ~ rm,~. So the e x p o n e n t 7 d e f i n e d a b o v e is i d e n t i c a l w i t h the e x p o n e n t c. T h e o t h e r e x p o n e n t s are: a = 0.15 -t- 0.02, b = 0.32 _ 0.01, a n d d = 0.16 _ 0.02. a a n d d are v e r y n e a r b u t n o t n e c e s s a r i l y equal. O n e c a n find the v a l u e of the e x p o n e n t ~(Pc ~ N ") f r o m Eq. (1). W h e n p = Pc, p(0) = Po = 1 a n d o n e gets Nap~-b ,-~ 1, o r NaN ~b ~ 1. T h i s gives ~ = a/b = 0.47, in a g r e e m e n t w i t h the d i r e c t d e t e r m i n a t i o n .

L. Benguigui / Physiea A 219 (1995) 13-26

23

The exponent fl can be also found from Eq. (1), when one calculates the dependence of RG with p. It is found fl = 2d - b/2 = 0.16, also in agreement with the results presented above. However the exponent ~ is defined in the range p < Pc, for which Eq. (1) is not applicable. The values of the different exponents were determined from relatively small samples and therefore must be taken with some caution. To resume the properties of the model, one can say that it is possible to distinguish two well defined regions depending on the values of N and p: the Eden type aggregates for p < Pc and the p-type aggregates. The properties of these two types are different and this can be seen through various quantities characteristic of the aggregates, in particular RG and t, the time necessary to form the aggregate. The difference can be also seen from the very different behavior of the function p(r). Below Pc, it is not possible to give a simple expression for p(r). However, for p > Pc, scaling behavior is observed.

3. Application to town growth The growth of towns is a very complex phenomena. However, there are very few studies which consider the growth m o r p h o l o g y very accurately. Since the introduction of the notion of fractal, it becomes possible to study more carefully this aspect of the town growth [8]. As mentioned in the introduction, Batty a n d his collaborators were the first to consider a town as an aggregate and to look for an adequate aggregation model. To compare our model with real towns, we shall use two ways: (a) qualitatively, we check by simple visual inspection, the agreement between the results of the model and the town; (b) quantitatively, we compare the function N(r) in the both cases. As we show below, it is easy to fill the quantitative criterion but much more difficult to reach the first goal. The basic idea for the application of the model to the geographical problem is to have p dependent on n, the number of particles already in the aggregate. We choose to look for a function p(n) which gives the function N(r) very similar to that of the towns. C o m p a r i s o n is made with the data of three towns: Baltimore, Paris and L o n d o n for which it is possible to determine N(r) [9]. The function p(n) must have the following properties: (a) F o r n small, (n ~ N), p ~ 0 since the central part of a town is very often compact. This means that the town begins to growth as an Eden-like aggregate. (b) F o r large n, p is always much larger than Pc. The transition between these two behaviors can be smooth or sharp depending on the town we want to study. We present the results for three large towns: Baltimore, Paris and London. The data have been taken from Refs. [-7, 8]. We show in Fig. 14 the curve N(r) from our model, with the function p(n) chosen for each city. N is taken equal to 500. The points are the experimental data. We see that the agreement is very good, showing that we can easily reconstruct the structure of the town. F o r example, London is characterized by

L. Benguigui / Physica A 219 (1995) 13-26

24

N(r) I00%

BALTIMORE

--

•

P

60

. . . . . . . .

.

/ / ~

0

i

i

i

i

0.5

r

I

I

I

I

I

I

nlN

0.88

r

i

I

arbitrary uniis

NCr) I00%

PARIS

•

;

60

•

•

. . . . . . . .

u u I I I

] •

s

F

00.4

arbitrary

0,7~

I I I

I, n/N

1.0

units

N(r)

I00%

LONDON

60

----

i

r

,

~

,

o

oz

o16

LO -fl/N I

arbitrary

I

T r

unffs

Fig. 14. N(r) for different realizations of the model with p variable during the growth. In the inserts, the variation of p with n/N. The points correspond to Baltimore; Paris; London.

L. Benguigui / Physica A 219 (1995) 13-26

'

.',..

•

•

M •

•

• ,. .~,u.u.i.~. L , u

•

25

:..

a

"d m"'i" :-" talk ,, b " ,,mlmE~m.-:-.mm roll •

::"" ' . ; - . - - . v . . ~ [ i ~ . ~ - m] j . ' • .. reelm-m.'Ik~[~- ".... ..'. re,o . •

"11

,..;m i=

"

•

Fig. 15. Comparison between the model aggregate (bottom) and the built area of Paris (top).

a function N(r) which is in a large part linear and the model gives exactly the same behavior. Now, if we compare the aggregate itself with the built area of the towns, it appears that the model is less successful. In Fig. 15, we got the aggregates from the model and the m a p of Paris is shown. In spite of some similarity, it is difficult to conclude that the model reproduces the town pattern. We think that the main reason for this discrepancy is the fact that a town does not grow only from the center. There are also secondary centers which grow and are "eaten" by the main center during the growth. In other words, these secondary aggregates disappear in the main aggregate.

4. Conclusion One of the main features of the new model is the existence of two different types of aggregates. The distinction can be made by varying the parameter p. We relate the two

26

L. Benguigui / Physica A 219 (1995) 13-26

types to a c o m p a c t - d i l u t e t r a n s i t i o n . T h e p - t y p e a g g r e g a t e s are c h a r a c t e r i z e d b y f o u r e x p o n e n t s , b u t since t h e r e are t w o r e l a t i o n s b e t w e e n t h e m , this m e a n s t h a t t h e r e are o n l y t w o i n d e p e n d e n t e x p o n e n t s . S c a l i n g is also o b s e r v e d for these aggregates. W e p r o p o s e to a p p l y the m o d e l to t o w n g r o w t h b u t we c a n o n l y c l a i m a relative success.

References [-1] T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1994). [-2] P. Meakin, in: Phase Transitions and Critical Phenomena, Vol. 12, C. Domb and J.L. Lebewitz, Eds. (Academic Press, New York, 1987). [-3] T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. [-4] M. Batty and P. Longley, Area 19 (1987) 215; M. Batty and P. Longley, Environment and Planning B 14 (1987) 123; M. Batty, P. Longley and S. Fotheringham, Environment and Planning A 21 (1989) 1447. [-5] M. Eden, Proc. 4th Berkeley Symp. on Math. Statistics and Probability, Vol. 4, F. Neyman, Ed. (University of California Press, Berkeley, 1961). [-6] M. Plischke and Z. Racz, Phys. Rev. Lett. 53 (1984) 415. [-7] P. Frankhauser, Thesis, Paris (1992) (unpublished). [-8] J. Bastie and B. Dezert, L'espace urbain, PUF, Paris (1977).