Fig 1.
The figures in the top panel represent the evolution of the road clusters for growing values of the threshold τ.
The figures in the bottom panels show the distribution of the equilibrium configuration defined in Eq (3), weq(α, β), for growing α’s and fixed β = 0.8. The size of the circles are proportional to the total floorspace of the retail cluster. In both cases the colours indicate the rank of the clusters. There is a striking similarity both in the types of dynamics, and in the spatial distribution.
Fig 2.
The top panels refer to the retail dynamics with β = {0.4, 0.8, 1.0}, while the bottom panels to the hierarchical percolation.
(a)-(d) The evolution of the size of the giant cluster, is presented, for increasing values of τ and α. (b)-(e) We show the evolution of the spatial entropy. Both quantities show a very similar behaviour in the two approaches. (c)-(f) We can see the distribution of the cluster (β = 0.8 for the retail model) that have an exponential form as shown in the insets.
Fig 3.
(a) This figure represents the evolution of the fraction of nodes of the road network allowed in the system for the given value of τ, nin(τ) (black curve). Moreover we show the same quantity considering only the giant cluster (blue curve) and every cluster but the giant cluster (yellow curve). (b) Here we show the fraction of the total retail floorspace belonging to the nodes that form the clusters fin(τ) where the colors have the same meaning. (c) We compare the two curves by showing their difference Θ(τ) = (nin − fin).
Fig 4.
(a) We have overlapped a snapshot of the road network clusters for τ = 130 with the retail clusters {weq} for α = 1.5. The colours indicate the size of the clusters in a logarithmic scale. We can see how most of retail clusters fall on road clusters, and there is good agreement between the spatial distribution of the ranks. In (b)-(c) We show how Θ(α, τ) varies with τ for several values of α, on the full network and not considering the giant cluster. The dashed black line indicates the values obtained from the data. We can see that the model for α ≤ 1.9 constantly produces higher values, with and without the giant cluster, while for α = 2.3 we can see how the giant cluster plays a fundamental role. This is because for that value of α the floorspace is mostly concentrated in the giant cluster (see Fig 2). (d)-(e) We show the fin(α, τ) with and without the giant cluster. The results are in line with that said for the previous figures.
Fig 5.
A diagram of the effect of the model’s dynamics on the retail floorspace distribution.
We see how the retail floorspace distribution in the data falls outside the road cluster, and how after the dynamics, the floorspace is all concentrated inside the cluster.