Modeling Urban Street Patterns

Urban street patterns form planar networks whose empirical properties cannot be accounted for by simple models such as regular grids or Voronoi tesselations. Striking statistical regularities across different cities have been recently empirically found, suggesting that a general and detail-independent mechanism may be in action. We propose a simple model based on a local optimization process combined with ideas previously proposed in studies of leaf pattern formation. The statistical properties of this model are in good agreement with the observed empirical patterns. Our results thus suggest that in the absence of a global design strategy, the evolution of many different transportation networks indeed follows a simple universal mechanism.

Figure 1

(a) Number of roads versus the number of nodes (i.e., intersections and centers) for data from 1 (circles) and from 2 (squares). In the inset, we show a zoom for a small number of nodes. (b) Total length versus the number of nodes. The line is a fit which predicts growth as $\sqrt{N}$ (data from 1).

Figure 2

$M$ is the closest network point to both centers $A$ and $B$. The road will grow to point $M^{\prime }$ in order to maximally reduce the cumulative distance $\Delta$ of $A$ and $B$ from the network.

Figure 3

Snapshots of the network at different times of its evolution: (a) $t=100$, (b) $t=500$, (c) $t=2000$, (d) $t=4000$ (the growth rate is here $\eta =0.1$). At short times, we have almost a tree structure and loops appear for larger density values obtained at larger times (the number of loops then increases linearly with time).

Figure 4

Simulation results (averaged over 1000 configurations). (a) Total length of roads versus the number of nodes. The dotted line is a square root fit. (b) Structure factor distribution showing good agreement with the empirical results of 15. (c)–(d) Rescaled distributions of the perimeter (c) and of the areas (d) of the cells displaying an exponential behavior.

Figure 5

(a) Network obtained for an exponential distribution of centers (1000 centers and $r_c=0.1$). (b) In this case, the area distribution is a power law (obtained for 5000 centers and 100 configurations). The solid line is a power law fit with an exponent $\approx 1.9$ (for this size and with a smaller number of configurations, we observe fluctuations of this exponent of the order of 10%).