Multifractal to monofractal evolution of the London street network

We perform a multifractal analysis of the evolution of London's street network from 1786 to 2010. First, we show that a single fractal dimension, commonly associated with the morphological description of cities, does not suffice to capture the dynamics of the system. Instead, for a proper characterization of such a dynamics, the multifractal spectrum needs to be considered. Our analysis reveals that London evolves from an inhomogeneous fractal structure, which can be described in terms of a multifractal, to a homogeneous one, which converges to monofractality. We argue that London's multifractal to monofractal evolution might be a special outcome of the constraint imposed on its growth by a green belt. Through a series of simulations, we show that multifractal objects, constructed through diffusion limited aggregation, evolve toward monofractality if their growth is constrained by a nonpermeable boundary.

Figure 1

The street network of Cardiff, UK, along with a zoom of two particular areas.

Figure 2

GLA street intersection maps over time. In red we highlight the street intersection maps (SIPPs in the text) for the urbanized area as defined by the methodology employed in the text.

Figure 3

Schematic representation for the distribution of the $\alpha -f\left(\alpha \right)$ pair. Each set of ${\alpha }_i$ has its own dimension. In this hypothetical example, the dimension A should be the highest, as it covers more space (four tiles) than the others. In contrast, the dimension C is zero, as it corresponds to a single point.

Figure 4

Multifractal analysis for all the years studied. The differences in value between these measures follow a pattern that imply that a single dimension is enough to characterise the SIPP, or, in other words, the structure is undergoing a transition from a multifractal to a monofractal structure.

Figure 5

Evolution of the constrained DLA model.

Figure 6

(a) $q-D_q$ plot. Until point 6, when the structure begins to interact with the green belt, all the points for both models depicts the same similar behavior. From point 7 the effect of the barrier over the structure is clear. (b) $\alpha \left(q\right)$ vs. $f\left[\alpha \right(q\left)\right]$ plot. The spectrum at points 8 and 9 are almost completely collapsed to a single point, which is an indicator of a monofractal structure.

Figure 7

$Z_q$ behavior for integer $q$ values $\in \{0,10\}$. Each plot shows the $\epsilon -Z_q$ for four particular years. We can observe how as the street network is consolidating, the larger $Z_q$ are collapsing into a single value. For (a) 1786 we have 11 distinguishable values, for (b) 1920 we have 10, for (c) 1965 we have 9, and for (d) 2010 we have only 7. For each $q$, the linear regression confirms that $Z_q$ decays following a power law regardless of the year selected.