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Urban substrates as computation

Machine Learning methods based on pedestrian-scale morphological metrics doubtlessly hold potential. However, a broader question remains: what is the role of machine learning as a theoretical tool? Moreover, how compatible is it really with a complex systems perspective of cities? As is particularly evident for supervised deep neural networks, machine learning methods are capable of embodying complex dynamics but, for the same reasons, do not necessarily tell us much about what these dynamics are or how they might operate. In this sense, ML methods may ‘short-cut’ theoretical principles even if they still capture the complexities; at the same time, this is not necessarily a bad thing because neural networks do not, by nature,‘shoehorn’ or ‘dumb-down’ patterns and relationships as may be required for more conventional mathematical and computational models.

As an exercise in briefly exploring the potential ramifications of these questions, two dynamic toy models presented below show how a first-principles approach might differ from a machine learning approach and how these might play out in simulating land-use dynamics at a granular scale. Both models constitute road networks parcelled into 20m segments, with each of the intervening nodes assigned a population value and a land-use state. For the sake of simplicity, the populations are held steady throughout the experiments; however, land-use states are allowed to fluctuate between either no land-use or one of three land-uses. Distances are computed using network distances, and the models subsequently update land-use locations in response to surrounding population densities and land-use accessibilities.

While not strictly cellular automata (CA), the models follow several principles that strike at the underlying nature of CA. CA are generally attributed to John von Neumann, who, in considering the theoretical implications of ‘self-replicating robots’ crafted ‘The General and Logical Theory of Automata’ (Neumann et al. 1963); however, it would be an enigmatic figure, Nils Aall Barricelli, who would create amongst the most interesting early examples of CA. A somewhat eccentric figure (known for forfeiting his PhD out of a refusal to reduce the page count!), Barricelli was motivated to simulate the evolution of biological systems and pursued this line of inquiry with few restraints due to being independently wealthy. With the blessing of Von Neumann, Barricelli was given access to one of the earliest computers at the Institute for Advanced Study in Princeton and proceeded to simulate evolution using ‘molecule shaped numbers’: in effect, casting the natural world and the conception of life itself as a chain of local computational events. The evolution of these theoretical life forms took the form of binary numbers that jostled and evolved over subsequent time-steps and was visualised by plotting — quite literally — the sequence of time-steps in binary numbers. The patterns revealed how local computational rules gave rise to emergent regimes that expressed homogenous or chaotic states and, under certain conditions, a dance between these (Alexander R. Galloway n.d.; Robert Hackett n.d.; Dyson 2012). Over time, CA has provided fertile ground for theoretical experiments leading, for example, to Conway’s Game of Life and, ultimately, to Wolfram’s ‘A New Kind of Science’ (Wolfram 2002) in which computation is proposed as the basis for scientific inquiry and the universe is fundamentally recast as simple distributed computational processes out of which all complexity arises.

Cellular Automata are helpful because they embody a fundamentally ‘spatial’ form of computation with an important implication: the unfolding of simple local rules from multiple locales — in parallel — can lead to complex emergent behaviour. CA thus allow potentially complex information processing to take root at an emergent level, superseding the apparent simplicity of the underlying local rules. It is this characteristic that makes CA well suited for exploring the nature of cities as computational substrates, leading to conceptual explorations (Couclelis 1985) that were, as computational advances permitted, increasingly followed by application to real-world urban systems (White & Engelen 1993; Batty & Xie 1994) and broader scale adoption (Pinto & Pais Antunes 2007; Santé et al. 2010).

The two models here presented are CA in the sense that the land-use states update in successive steps in response to ongoing changes in the location’s ‘neighbourhood’, but the concept of a ‘neighbourhood’ is otherwise more diffuse than ordinarily implied. In the case of the first toy model (see Appendix for links to code) representing dynamics from first principles: each land-use is assigned a distance threshold representing the distance pedestrians are willing to walk in order to reach the land-use; a ‘capacitance’ representing the momentum or rate of reaction for the land-use to respond to positive or negative changes; and a population threshold representing the minimum number of patrons for a location to be successful. These rules allow for the simulation of different types of land-uses; for example, a more volatile land-use such as a coffee shop or eatery may command a relatively small distance threshold combined with a small momentum, meaning that they are relatively quick to appear or disappear in response to opportunities or challenges. More stable land-uses, on the other hand, such as a bank or a post-office, may command a greater distance threshold and more significant momentum, meaning that they take longer to appear but are also more resistant to minor perturbations. The model initialises by selecting a random starting location for each land-use, and a singly-constrained model is then applied at each time-step to apportion flows from all dwelling units to each of the land-uses. Existent land-use locations exceeding the required population threshold will have their capacitance boosted, whereas those that do not meet this level will decrease their capacitance. The potential for new land-use locations to emerge is based on the available population divided into the required population threshold for a selected land-use and multiplied by a competition factor. This represents latent demand for new locations and is apportioned to vacant land-use sites by boosting capacitances based on the most significant existing footfall. Land-use states are non-linear in that they ‘turn-on’ once the capacitance reaches an activation threshold and ‘turn-off’ once a location’s capacitance has been exhausted. The model includes a ‘spillover’ effect whereby a percentage of footfall reaching a location will ricochet into a subsequent time step, thus simulating pedestrians combining visits to multiple land-uses and allowing for synergistic clustering of land-uses to occur. Stochasticity is added to the calculated flows and capacitances to represent natural variability affecting trips and locations, and a small ‘death-rate’ randomly culls locations.

This first toy model contains several key ingredients that would commonly be associated with the simulation of a complex system: competition for limited resources, parallel exploration, feedback processes, and randomness (Mitchell 2009) which allow for complex behaviour to emerge; for example, locations are at once competing with other land-uses of the same type but may simultaneously benefit from adjacency due to spillovers. The principles and the rules remain relatively clear-cut and straightforward, even if the dynamics are potentially complex. This approach is contrasted by the second toy model, which now discards most of the rules mentioned above and instead adopts a set of deep neural networks representing three land-uses. Three land-uses are selected from data for Greater London: commercial establishments based on a 400m walking tolerance; manufacturing locations based on an 800m tolerance; and eating establishments based on a 200m tolerance. Three deep neural networks are then trained: one for each of the selected land-use scenarios based on closeness centralities; betweenness centralities; the two ‘other’ non-selected land-uses; and dwelling densities, with each of these provided to the respective models at a multi-scalar range of pedestrian distances from 100m to 1600m. The models are trained using neural networks with three hidden layers and 10% dropout. For the given information, the models respectively attain r2r^2 values of 92%, 76%, and 71% based on a 10% test set split. Model training uses a further 25% validation set split, with all splits performed using a spatial splitting method to prevent information bleed-over between training/validation / and test sets. The three machine learning models are now supplied to the toy model, which proceeds by calculating network centralities, land-use accessibilities, and dwelling densities at each step, then providing this information to the respective machine learning land-use models. The iterative update rules are now more straightforward: predictions for land-use accessibilities are compared to actual measurements: land-use capacitances are boosted if predicted accessibilities exceed measured accessibilities and vice-versa. Land-use state changes retain the non-linearity of the capacitance mechanism, and some stochasticity is applied to capacitances at each time step.

The two models are compared across three scenarios: a ‘historic’ street network based on the Town of York’s historic core; a’mesh-like’ grid; and a ‘tree-like’ network.

The first toy model (first-principles) shows patterns in keeping with expectations: land-uses compete for locations with high levels of centrality and density. Dynamics related to clustering and distancing become evident and play out over time, particularly in the free-form context of the mesh-like substrate. Higher capacitance land-uses (purple) lay claim to the more prominent locations near the centres, with medium (orange) and low (cyan) capacitance locations adopting less prominent locations in keeping with demand. The second toy model (neural networks) shows similar patterns for the historic location but reveals interesting differences for the mesh-like substrate, which is now strongly geometric, and the tree-like substrate is now unable to gain traction.

The first-principles approach offers dynamics that are debatably more realistic while being more adaptable to different scenarios. This approach opens the door to wide-ranging experimentation: How do different urban substrates affect the expression of land-use dynamics? What happens if the population increases or decreases? What if we increase or decrease the number of available land-use locations or place constraints on where land-uses are permitted? The first-principles approach allows these questions to be resolved naturally, and the expression of the dynamics is logically consistent when parameters are modified. A drawback to this approach is that application to real-world land-uses requires either experimental approximations or calibration to dynamics for observed cities.

The opposite tends to be true for the deep-learning approach: It is advantageous in the sense that the land-uses are now modelled on observed real-world land-uses, and the embodied patterns are potentially complex, for example, the way manufacturing locations (purple) adopt remote locations around the periphery while food retail and commercial adopt central locations with the most significant access to population and footfall. It is disadvantageous because the embodied dynamics are now more fickle: the models can struggle to handle unrecognised morphologies. They cannot decipher broader patterns and how they might generalise to unknown situations that they have not learned from the data, making it harder to ask explorative questions.

More broadly, these forms of model hold tremendous potential for application to scenario planning and could be particularly interesting for the methodical theoretical exploration of different urban substrates. They potentially lay bare the correspondence between flows of information and the network and land-use structures that embody, harness, and amplify this information over successive iterations to reveal higher-order properties such as resilience and self-organisation. How exactly do these processes unfold, and what morphological forms might enable them to proceed unimpeded? This question is left to subsequent research.

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