UNIVERSALITY IN CITY MORPHOLOGY AND THE MORPHOLOGY ...

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Jan 2, 2008 - IMPLICATIONS FOR CITY EVACUATION PLANS .... the city of London, but similar curves are found for the cities of Paris, Rome and Modena.
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Advances in Complex Systems, Vol. 10, Suppl. No. 2 (2007) 373–377 c World Scientific Publishing Company 

UNIVERSALITY IN CITY MORPHOLOGY AND THE MORPHOLOGY OF A CITY AND ITS IMPLICATIONS FOR CITY EVACUATION PLANS

MASSIMO PICA CIAMARRA∗ and ANTONIO CONIGLIO† Dipartimento di Scienze Fisiche, Universit´ a di Napoli Federico II, Via Cintia, 80126, Napoli, Italy ∗[email protected][email protected] Received 12 June 2006 Revised 1 September 2006 Street networks play a crucial role in the emergence of vehicular traffic. Here we present an experimental analysis of the street networks of several European cities (London, Paris, Rome, Modena). This analysis evidences the existence of universal properties in the morphology of different cities, which is captured by a model for the dynamics of city growth. Then, we discuss the implications of our findings for the development of city evacuation plans, whose efficiency is of paramount importance for cities threatened by natural hazards, such as volcanoes. Keywords: Cluster growth; traffic.

1. Introduction Several kinds of transportation infrastructures form clusters near “vital” centers. For instance, the heart is the cluster of the cardiovascular system, while large cities are clusters of transportation networks. When this is the case the topology of the network near the cluster influences the dynamics of the transportation processes which occur over it [1]. Here we investigate, both experimentally and theoretically, the properties of the street networks of large cities. This analysis is useful for the development of evacuation plans needed for cities threatened by natural hazards, such as Plinian (explosive) volcanoes, possibly flooding rivers, or even hurricanes. 2. Morphology of Large Cities We have conducted an experimental study of the street network of several European cities: London (UK), Paris (France), Rome (Italy) and Modena (Italy). These cities, which are not located by the sea or near mountains, have been free to grow acquiring a round shape. In order to determine the spatial properties of the street network of a city, we have downloaded from the Web [2] very-high-resolution maps (1 pixel 373

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corresponds to around 3.3 m2 ) covering the city. Via the use of these maps we have constructed a gridded representation of the street network. This is a very large square lattice ρ whose elements ρij may assume two values: ρij = 1 if the corresponding pixel covers a street, ρij = 0 otherwise. A coarse-grained representation of ρ is shown in Fig. 1 for the cities of Paris and London. By exploiting the radial symmetry of the investigated cities we have extracted from ρij the radial density of the street, ρs (r). This is defined in such a way that ρs (r)dr is the fraction of the area of the circular annulus with radii r and r + dr (r = 0 identifies the center of the city) which corresponds to a street. Figure 2 shows that ρs (r) behaves in a similar way for all considered cities: it has a maximum in the city center, and decays to a constant value at large distances. Interestingly, the density of streets of all the investigated cities is well described (plain lines in the figure) by the functional form ρ(r, N )  ρ∞ +

ρmax − ρ∞ Erfc 2



r − rN √ 2σN

 ,

(1)

pointing out the existence of a universality in the morphology of all cities. This universality is related to the way cities grow, as Eq. (1) can be derived [8] by assuming that cities are growing as compact clusters, and that the probability that a new element of the city is added at a distance r from the city center is   (r − rN )2 , P (r, N ) ∝ r2 exp − 2σrN

(2)

where N is the number of city elements already present, rN N 1/2 is the radius of the city, and σrN ∝ rN . Equation (2) is a simple generalization of the growth probability distribution of the diffusion limited aggregation [3–6] and of the Eden model [7, 8].

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Fig. 1. The density of streets ρ of the city of Paris (left panel) and of the city of London (right panel). Each cell covers an area of size 1.9 × 1.7 km2 and indicates, according to the color code on the right, the percentage of the cell surface which corresponds to a street. The center of the city (x = y = 0) is located on the Ile de la cit´e for the case of Paris, and near Westminster for the case of London.

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0.5

ρs(r) 0.4

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shifted of 0.1

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r (Km) Fig. 2. The density of streets as a function of the distance from the city center ρs (r) for the cities of Modena, Rome, Paris and London. This last curve is shifted vertically of 0.1 for clarity. Plain lines are fitted to Eq. (1).

3. Implications for Vehicular Traffic Empirical evidence shows that there is a transition from free traffic flow to congested traffic flow as the car density ρcar increases. For instance, highway traffic becomes congested when ρcar ≥ ρcong car  30 vehicles/km [9]. The knowledge of the density of streets allows an estimate of the car density. Given N cars in a region A one can  estimate ρcar = βN/SA , where SA = A ρs (x, y)dxdy is the area occupied by the streets in the region A, and β a coefficient of proportionality. Let us consider what happens when N cars, initially located in a circular annulus of radius r and width ∆ around the city, move outward. As their mean distance from the city center r increases, the car density varies as ρcar (r) = βN/S(r), where  S(r) = 2π

r

r+∆

   r − rN ρmax − ρ∞ Erfc √ r ρs (r )dr 2πr∆  ρ∞ + 2 2σN

(3)

is the area occupied by the streets in the circular annulus we are considering. The car density is higher where S(r) is smaller, and increases in the regions in which ∂S/∂r < 0. These regions are those which most probably act as traffic bottlenecks. For the city of London, Fig. 3 shows that S(r) increases for r  17 km and for r  36 km, where the density of streets varies very slowly. In the range 17  r  36 km of fast variation of the density of streets, S(r) decreases. The nonmonotonic behavior of S(r) also characterizes Paris, Rome and Modena. It follows that in this range of distances from the city center, i.e. at the city boundary, traffic is maximum. This is therefore the area where new transportation infrastructures and strategies for traffic optimization should be concentrated [10].

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2

S(r) (Km )

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Expansion

Compression

Expansion

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r (Km) Fig. 3. Area occupied by the streets in a circular annulus of width ∆ = 1 km centered on the city center, S(r) = 2πr∆ρs (r), as a function of the distance from the city center r. The curve refers to the city of London, but similar curves are found for the cities of Paris, Rome and Modena. The nonmonotonic behavior of S(r) influences vehicular traffic: cars moving outward are compressed in the region 17  r  36 km, where S(r) decreases.

4. Conclusion In this paper, we have introduced and studied a new model of cluster growth. This model is inspired by numerical results relative to the DLA and to the Eden model, but it takes into account the fact that the probability of building a cluster unit at a given point depends on the local density. In this model the probability of building a cluster unit at a point at a distance r from the cluster center is a Gaussian centered on the radius of the cluster, with variance proportional to the radius, modulated by the local density ρ(r). An explicit solution of the cluster properties in d = 2 dimensions, which is not available for other cluster growth models, shows that the cluster density decays as a complementary error function. We have validated the model via a high resolution study of the density of streets of several European cities, and we have discussed the relation between the cluster topology and the dynamics over the cluster.

References [1] Trusina, A., Rosvall, M. and Sneppen, K., Communication boundaries in networks, Phys. Rev. Lett. 94 (2005) 238701–238704. [2] Digital maps provided by www.maporama.com [3] Meakin, P., Coniglio, A., Stanley, H. E. and Witten, T. A., Scaling properties for the surfaces of fractal and nonfractal objects: An infinite hierarchy of critical components, Phys. Rev. A 34 (1986) 3325–3340. [4] Ossadnik, P. and Lee, J., Power law tail in the radial growth probability distributions for DLA, J. Phys. A: Math. Gen. 26 (1993) 6789–6796. [5] Plischke, M. and R´ acz, Z., Active zone of growing clusters: Diffusion limited aggregation and the Eden model, Phys. Rev. Lett. 53 (1984) 415–418.

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[6] Meakin, P. and Sander, L. M., Comment on “Active Zone of growing clusters: Diffusion-Limited Aggregation and the Eden model”, Phys. Rev. Lett. 54 (1985) 2053–2053. [7] Eden, M., in Proc. 4th Berkeley Symp. on Mathematical Statistics and Probability, ed. Neyman, F. (University of California Press, 1961). [8] Ciamarra, M. P. and Coniglio, A., Random walk, cluster growth and the morphology of urban conglomerations, Physica A 363 (2006) 551–557. [9] Helbing, D., Traffic and related self-driven many particle system, Rev. Mod. Phys. 73 (2001) 1067–1141. [10] Ciamarra, M. P., Optimizing on-ramp entries to exploit the capacity of a road, Phys. Rev. E 72 (2005) 066102–066109.