Vicsek and Szalay proposed a cellular automaton model to study the fractal distribution of galaxies [14]. In their lattice model, a cell i which represents a mass element, would become part of a galaxy based on two parameters; the potential of belonging to a galaxy at position i, and a given threshold that regulates such a potential. Fujita and Thisse in their study of economic agglomeration theory [15] say that “… just as matter in the solar system is concentrated in a small number of bodies (the planets and their satellites); economic life is concentrated in a fairly limited number of human settlements (cities and clusters). Furthermore, paralleling large and small planets, there are large and small settlements with very different combinations of firms and households.” Following these ideas, a percolation model to study the fractal distribution of galaxies based on [16] is applied here to model the development of urban settlements.

We consider a potential for urbanization P_i(t) in cell i. As development takes place more cells develop the potential to urbanize and the decision on whether cell i becomes urbanized or not depends on an external parameter, the development threshold T. If P_i(t) > T and cell i has not already undergone urbanization, then cell i becomes urbanized and is consolidated with U_i(t) = 1. Once a migrant unit is attached to an urban settlement at i, it could be the case that for this particular i, P_i(t) is not greater than T; if that is the case, the cell i, occupied by the migrant unit would became an urbanized non-consolidated urban settlement with U_i(t) = 2; this reflects a pattern of development in many of the urban belts formed around metropolitan areas. If after recalculation of P_i(t) a migrant unit arrives at a cell with U_i(t) = 2 and P_i(t) > T, then the value of U_i(t) becomes equal to 1.The potential P_i(t + 1) is defined as: Pi(t+1)=∑j∈ΩPj(t)5+K⋅Ci(t)+εi(t) (8) where Ω represents the cell, i, plus its von Neumann neighbourhood (The north, south, east and west cells of the central cell i) and ε_i(t) is either 1 or −1 with equal probability. The inclusion of random events prevents any tendency towards equilibrium since noise is continually being introduced into the system. Once cell i becomes urbanized and consolidated, it cannot reverse this process, even if its potential P_i(t) falls below the value of T.

The function C_i (t) calculates the current number of urban units (buildings for example) that a cell i has at a particular time t. The constant K > 0 which is a damping factor, is fixed at the beginning of the simulation and represents the influence that the local settlements have over the regional potential (K = 0.1 in our simulations). The formulation for C_i(t) is given in Eq 11.

As discussed in the introduction, the conditions for migration to occur are fulfilled by a diffusion limited aggregation-like process [17]. At each time t (Eq 11), we select a fixed number of random positions over the study area and at each of these positions we create a migrant unit that begins a walk (diffusion) over any configuration that is emerging at that particular t until reaches some cell i, being in the nearest neighbourhood of an already occupied cell j. The nature of the walk is address in section 3.3. The number of migrant units N(t) created is not fixed in all the simulations but is defined from: N(t)={0fort<50tmod10for50≤t≤500 (9) so N(t) increases over time.

Self-Organized Criticality (SOC) is a characteristic possessed by spatially extended dynamical systems and other complex systems [11, 18]. When a phenomenon changes state, the resulting reactions are distributed across time and space at all levels in a way that can range from a simple isolated movement, to chain reactions involving all activities in the system. A good example is the sand pile model, as presented in [18, 19] and, based on this we introduce SOC into our model. However, instead of sand grains, we have urban units that migrate and in doing so, may trigger a chain reaction in which other units also relocate. To model this, two parameters are required for each cell i, a maximum capacity Cimax and a current capacity C_i(t). The Cimax parameter represents the maximum number of urban units that cell i can hold and is defined as Cimax(r)=Cseedmax.r−α (10) where r is the Euclidean distance from the spatial position of the seed to cell i and, Cseedmax is the maximum capacity for the historical accident, which is set here to 100 to ensure that its capacity is never constrained, and α is a parameter of the distance distribution which we define as a power law. Eq 10 constrains the density of an area by how far it is from the seed. The exponent α (which is set to 0.3 in our simulations) can be thought as a density gradient held constant over time. In reality appears to decrease gradually as the city grows [16]. In terms of the sand pile model, the critical slope of the distribution of the urban units over an area is controlled by α.

Once we have calculated the Cimax for every cell i, (except the seed cell which has fixed capacity) we need to set up the current capacities, C_i(t), for the whole lattice excepting the seed cell which is equal to 1, as we assume that at least one urban unit is already there.

Capacities are then defined as follows in Eq 11 which represents the sand pile process. If Ci(t)>Cimax for a cell i, then an avalanche begins (as in the sand pile model) around cell i by its losing 4 urban units to its von Neumann neighbourhood. The damping constant now defined as κ > 0, similar to that introduced in Eq 8, is fixed at the beginning of the simulation with a value of 0.1 and represents the influence that the regional settlements have over the local capacities. The value of K and κ ensures that the analysis of asymmetric regional and local influence is not biased by a varying damping factor. The idea of using damping factors as a parameters which can help us to control and explain the interactions between regions and people is explored in [20, 21]. Cell capacities are updated using the following algorithm Ci(t+1)={0fort=0Ci(t)+1fort≠0andPi(t+1)>Pi(t)Ci(t)−4andCΩ(t+1)=CΩ(t)+1+κPi(t)for{t≠0andCi(t)>Cimax (11)

Table A in S1 File summaries all the parameters used in our model along with its initial values and a brief description.

We defined a set of four thresholds T equal to 4.0, 4.5, 5.0 and 6.0 relating to the difficulty of development actually occurring, which are determined by physical or policy factors. We fixed the length of the simulation to t = 500 iterations as this gives sufficient time for patterns to emerge. At time t = 0 all grid positions, except the seed, have a potential P_i(0) = ε_i. To represent the importance of the historical accident across time, its initial potential is P_seed(0) = 20 which remains constant through all the iterations, so Eq 8 is never applied to the seed cell and its potential always exceeds the threshold. A typical configuration obtained with this model is shown in Fig 1. As our approach is stochastic we performed 1000 runs per configuration in order to derive robust statistics. All the quantities and measures derived are then the averaged over each configuration. We refer the reader to [22, 23] for examples in which aspects of this model have been applied.

Migration units walk across the lattice (as described in section 3.1) until they fix their position in a cell i according to the rules set out above. This walk is achieved by selecting a subset of all possible trajectories that migrant units may take. We accomplish this in two different ways:

All migration units scan a defined area in search of the location with the highest potential, and when they find it, they walk a certain distance in terms of their units towards that location;

A ten point time series is defined for the configurations obtained by our model. This was accomplished by recording the pattern of urban structures generated at times t = {50, 100, 150… 500}. We label each point in the time series as t_j, with t₁ = 50, t₂ = 100, until t₁₀ = 500. The main problem here is how to assign them a unique value in order to apply Eq 4. We performed a classification analysis based on the number of clusters N detected at each t_j, applying a subtractive cluster algorithm [24] that strongly depends on a predefined cluster radius r_c which is used as a cut-off to define these time series at different geographical scales. The logic is as follows: at a local level, we can observe in great detail the urban structures that surround us and easily distinguish one structure from another, but our observation area is very limited (based on the small radius of r_c); as we begin to increase the geographical scale, much of the urban detail begins to disappear, as many structures become indistinguishable from each other but now our observation area covers many more structures (based on a larger radius r_c).

The subtractive cluster algorithm can be summarized in three steps:

Select the data point k with the highest potential G to became a cluster or group centre, according to Gk=∑j∈k−j≤rce−β‖k−j‖2 (12)

In our calculations, parameter r_a takes values in the range {1, 46.6, 92.3, 138, 183.6, 229.3, 275, 320.6, 366.36, 412}. This selection is constrained by the maximum (1) and minimum (412) Euclidean distance that a pair of cells i and j can have in our lattice. The intermediate values are calculated dividing 412 by 10, as we are defining 10 different geographical scales. For example, for T = 4.5 we obtained the time series shown in Table 1.

Using the previous method, three more tables were constructed giving a total of four tables, one for each T. These other three tables can be found in Tables B, C and D in S1 File.

In Eq 6 the joint probability calculation is key to obtaining the TE value. Estimating these probabilities has been proved to be a very difficult task. Several methods via probability density function estimation have been proposed to solve this problem [25]. In this research, representing the different scenarios, through the four tables mentioned above, we calculated the empirical joint probabilities for the emerging configurations, i.e., we calculated the probability of having a certain number of clusters between scales, assuming that if a particular configuration is not found in a particular table, and then the joint probability for that configuration is zero. Thus, if we take Table 1, in order to calculate the TE_{Scale 1→Scale 2} at step t₁, we need to find out the probability p(y2,y1,x1) as required by the first term of Eq 8. This probability is easily obtained inspecting the first and second rows of the mentioned table (as Scale 1 is represented by the first row and Scale 2 by the second row). The term y₂ corresponds to the value located in the second column, second row; y1 is the value at the first column, second row and x1 is located in the first column, first row position, i.e., p(y2,y1,x1) = (5,4,51). We then count how many combinations of these values exist in these two rows. For this example there is only one, so p(y2,y1,x1) = (5,4,51) = 1/10, as there are 10 positions in the time series. Now, if we take TE_{Scale 7→Scale 8} and t₄, then we examine rows 7 and 8 to find the values for y5,y4 and x4, which are 3, 3 and 4, so, p(y4,y3,x3) = (3,3,4) = 5/10, as there are seven identical (3,3,4) combinations between rows 7 and 8.

Taking Table 1 again as a typical cluster configuration, we observe that at time t₁, except for Scale 1 (the most local scale), the number of clusters at all scales is very low, because at this point almost no urban structures are formed; for at time t₁₀ the number of clusters reported is constant, which suggests that when the urban structures reach the point of consolidation, the probability of detecting new developments is low; on the other hand, the change between t₁ and the rest of the temporal sequence is dramatic, implying at this early stage, the full complexity of the urban structures is only measurable at a local scale.

To construct a coherent analysis for the TE results among all our scales and thresholds, they were categorised as:

TE between contiguous scales (scale 1 vs scale 2, scale 2 vs scale 3, etc.)

This shows how much of the information generated at one scale i is responsible for the information obtained at scale j and how the middle scales filter such information. As in the analysis of clusters above, the TE values obtained reflect the differences between different thresholds. This suggests that as settlements become harder to generate, as a result of policy or physical factors, the probability that a migrant unit settles, decreases since it becomes harder for it to find a consolidated area to attach to. In reality, these migrant units would still settle somewhere, regardless of the urban policy, but in our model this fine grain detail is not taken into account.

In general, the TE in the local→regional direction dominates that for the regional→local, as is show in Fig 2. Information flows less easily from regional to local scales, a situation that might be expected as the information generated at the higher scale of the urban system is not ultimately responsible for the creation of the local structures; policies established at a regional level, unless they are very restrictive and impose a particular local development pattern, would tend to become diluted (and be less effective) at the more local scales. One notable aspect of all the plots in Fig 2 is that the TE rises initially with the length of the rise decreasing as the threshold increases. At the early stages of urban structure formation, the system always increases the transfer of information in both directions. But this tendency does not hold as the system continues to grow. As the urban systems get larger, the information transfer tends to equalise with, in the higher threshold cases, the regional→local direction dominating that for the local→regional as the scale increases. This equalisation may simply represent the convergence of the two r_a values at each mark as the mark number increases but it suggests that the information transfers at the higher scales are less significant in both their absolute size and in their directional difference. Where the dominance reverses, the difference in information flows is small suggesting a similarity in the cluster pattern. This might be expected if growth was concentrated mainly at the cluster edges and the increase in r_a left the number of clusters substantially unchanged. The peaks in TE may therefore be seen as the points at which the growth pattern changes from one of many new separated clusters to fewer larger clusters caused initially by edge accretion of cells and in the later stages by amalgamation of larger clusters. This would account for the multiple peaks observed in the local→regional TE. Interestingly enough, for T = 4.5, mark 4, TE values, in both directions equal zero, meaning that the number of clusters detected from scale 4 to scale 5 is exactly the same. There was no particular change in the overall structure of the system from one scale to the other. Further research is needed concerning this effect.

For the threshold T = 5.0, representing a moderate urban policy at mark 4, TE values for the regional→local direction are greater than the local→regional until mark 8 (except for mark 6 in which both values are practically equal). At these points, the structure of the information changes in a way that allows the information generated at higher scales to flow to lower ones. From mark 6 this is not that surprising, because we are operating at the higher scales and the information generated at a regional level is mainly responsible for the structures observed at this level. At mark 4 (TE between scales 4 and 5, in both directions) the net direction of the information is indistinguishable in practical terms. This suggests that the regional pattern of information is leading the local but this may simply be the effect of scale with the regional scale identifying as clusters those settlements at a local scale which are growing by accretion. In Fig 3 we show a section from one typical configuration generated for scales 4 and 5, T = 5.0. Although the latter contains a greater proliferation of urban structure, it does not seem to contain more clusters, thereby reinforcing the view that the TE equality reflects growth by accretion and amalgamation with little change in the cluster numbers. The reversal of dominance and the near equality of TEs from thresholds 4.0, 4.5 and 5.0 reflect a change in the distribution of the settlements.

Finally, for T = 6.0, this represents a more restrictive urban policy. Once the scales begin to increase, the number of clusters generated at this threshold begin to stabilize, so when we calculate the TE between series it tends to zero. As stated above, when T → ∞ the number of clusters tends to zero so the TE would also tend to zero in the process. This unrealistic scenario implies restrictive urban policies, to the point where no urban structures could be created. Alternatively, when T → 0, (no restrictions at all) this would lead to same result since each location can be urbanized, giving one big cluster with zero TE.

The last TE measure performed was over the non-contiguous scales. Fig 4 presents 10 plots at T = 5.0. First, notice that from Scale 2, all plots, in both directions, have a zero value. This point represent the TE (Scale i, Scale i), so it should not be considered as part of the analysis per se. We kept this in order to generate a continuous sub-plot.

These plots show us in one single frame the behaviour between directions of information flow. From sub-plot Scale 3 onwards, the regional→local direction TE values dominate where they are to the left of TE = 0. Conversely to the right of TE = 0, the local→regional tend to dominate. The division where we lose the non-consolidated structures from the analysis is very clear: until Scale 5, there is a constant gap at the right part of the TE zero value, while from Scale 6, this gap can be found on the left part. The fact that the TE value between directions is kept constant is evidence that the amount of information that is flowing from one scale to the rest does not get amplified or decreased at these bands. From Scale 6, the situation is similar, but mirroring at the left part of the TE zero value and at the right part, the values between directions are very close to each other, as we have already established at this scale.

The other somewhat unusual result is that the TE between Scale 1 every other, in the regional→local direction, is practically zero. All the information generated at the regional scale never reaches the local level, a situation that is also reflected in all the other thresholds. This situation is observed on a daily basis in our cities, where political decisions, imposed at the regional level, never reach or find their way down to improve the lives of citizens at local level. The current debate in British politics and government for example, is largely about these issues of regionalism and localism.