Strong coupling between scales in a multi-scalar model of urban dynamics
SStrong coupling between scales in a multi-scalar model of urbandynamics
Juste Raimbault , , , ∗ Center for Advanced Spatial Analysis, University College London UPS CNRS 3611 ISC-PIF UMR CNRS 8504 G´eographie-cit´es ∗ [email protected] Abstract
Urban evolution processes occur at different scales, with intricate interactions between levels andrelatively distinct type of processes. To what extent actual urban dynamics include an actual strongcoupling between scales, in the sense of both top-down and bottom-up feedbacks, remains an open issuewith important practical implications for the sustainable management of territories. We introduce inthis paper a multi-scalar simulation model of urban growth, coupling a system of cities interactionmodel at the macroscopic scale with morphogenesis models for the evolution of urban form at thescale of metropolitan areas. Strong coupling between scales is achieved through an update of modelparameters at each scale depending on trajectories at the other scale. The model is applied and exploredon synthetic systems of cities. Simulation results show a non-trivial effect of the strong coupling. As aconsequence, an optimal action on policy parameters such as containing urban sprawl is shifted. Wealso run a multi-objective optimization algorithm on the model, showing showing that compromisebetween scales are captured. Our approach opens new research directions towards more operationalurban dynamics models including a strong feedback between scales.
Keywords:
Urban dynamics; Systems of cities; Urban morphogenesis; Multi-scalar modeling; Strongcoupling
The modeling of urban growth and more generally the dynamics of urban systems is central to thedesign of sustainable territorial policies, through the understanding of past urbanisation processes andthe anticipation of future urban trajectories. The design of sustainable future cities requires an historicalknowledge of how past cities came to be and evolved [Batty, 2018]. Several models have been proposed atdifferent scales and integrating different dimensions of urban systems, such as models of land-use changeat a mesoscopic scale or systems of cities models at a macroscopic scale [Pumain and Reuillon, 2017].At the scale of a metropolitan area, Land-use Transport Interaction models [Wegener and F¨urst, 2004]are for example a widely used tool to estimate the dynamics of spatial distributions of activities (mostlyresidential location and economic activities) in response to an evolution of the accessibility landscapecaused by new transportation infrastructures [Raimbault, 2019a]. In a similar context, cellular automatamodels of urban growth or land-use change study more generally land-use transitions with a high spatialresolution, and are mostly data-driven [Clarke et al., 2007]. At the smaller scale of the system of cities,macroscopic models of urban growth have focused on reproducing the distribution of city sizes, eitherthrough economic processes as e.g. [Gabaix, 1999], or from a geographical point of view focusing oninteractions between cities [Favaro and Pumain, 2011].Territorial dynamics, and more particularly urban dynamics, have according to [Pumain, 1997] anintrinsic multi-scalar nature, with successive autonomous levels of emergence from individual microscopic1 a r X i v : . [ phy s i c s . s o c - ph ] J a n gents to the mesoscopic scale of the city and the macroscopic scale of the system of cities. Whilemodels at each scale with distinct ontologies are useful to answer their own questions, an explicit ac-count of inter-scale feedbacks, both top-down and bottom-up, would allow testing policies and inter-ventions distributed and differentiated across scales while not neglecting the interactions between scales[Wegener and Spiekermann, 2018]. Indeed, the need for sustainable territorial policies would imply theconstruction of multi-scalar models to take simultaneously into account issues associated to each relevantscale [Rozenblat and Pumain, 2018, Raimbault, 2019b].Multi-scalar models of urban dynamics are however still at their beginnings. [Murcio et al., 2015]consider population flows at different spatial ranges from the urban area to the country, but does notincorporate distinct ontologies and processes for the different scales. [Batty, 2005] however suggests thata similar formalism can be applied to urban processes at different scales. Multi-level statistical modelscapture some information at imbricated scales [Shu et al., 2020], although they can not be used as dynam-ical simulation models. Similarly, multi-level cellular automata (CA) models for urban growth includefactors influencing urban expansion at multiple scales [Xu and Gao, 2019]. [Cheng and Masser, 2003]propose a general framework for such approaches. [Torrens and O’Sullivan, 2001] suggest that hybridmodels coupling CA with other formalisms is a crucial development in the field. [White, 2006] introducesa CA with variable grid size to account for heterogeneities across scales. [Zhu and Tian, 2020] couplean agent-based model with a CA at multiple scales. [Yu et al., 2018] embed a local CA into a regionalintercity model and a macroscopic potential model. [Ford et al., 2019] couple at different scales an urbandevelopment model with a flooding risk model to forecast the future impact of extreme climate eventson the London metropolitan region. [Xu et al., 2020] develop an agent-based model of urban expansionwith both macro and micro agents. [Raimbault, 2019c] suggests that integrating network dynamics atthe link level in a macroscopic urban system models is a way to implement a multi-scale model, as doneby [Raimbault, 2020b] which explores hierarchy properties of cities and networks in this context.In disciplines neighbour to urban modeling, methods have been developed for multi-scale models. Forexample in spatial epidemiology, [Banos et al., 2015] combines agent-based modeling for local diffusiondynamics with differential equations at the population scale. The NetLogo software for agent-basedmodeling includes a specific extension for multi-scale modeling [Hjorth et al., 2020]. Multi-scale modelshave also been used for the simulation of crowd dynamics [Crociani et al., 2016]. The study of trafficis also made more accurate by coupling macroscopic and microscopic models [Boulet et al., 2020]. Themanagement of ecosystems requires integrating across actors and scales [Belem and M¨uller, 2013]. Thesenon-exhaustive illustrations highlight how the integration of scales is a crucial feature and issue in theunderstanding of complex systems [Chavalarias et al., 2009].This paper contributes to the open question of multi-scalar models of urban dynamics by introducinga new simulation model which integrates a strong coupling between the mesoscopic scale and the macro-scopic scale. The dynamics within each scale influence the other and reciprocally in an iterative way.More precisely, the model is simple in its components as we focus on the spatial structure of processesrather than on their multi-dimensionality. Therefore, we take into account only population variables, butboth at the macroscopic scale of the system of cities and at the mesoscopic scale of the metropolitan areawith an urban morphogenesis model. Our contribution is novel regarding previous works in particularregarding the following points: (i) the stylised model explicitly couples distinct scales and ontologies ina strong manner, most models operating only a weak coupling between scales (i.e. no reciprocal anddynamical feedbacks); (ii) the behavior of the model is systematically studied on synthetic systems ofcities using model exploration methods.The rest of this paper is organised as follows: we first describe the model; we then develop itsexploration on synthetic systems of cities, and optimization using a genetic algorithm; we finally discussdevelopments and implications of this work. 2 eso scale Localmorphogenesismodels
Macro scale
Interactionmodels
Figure 1: Schematic representation of model architecture. Urban areas with their own urban form areinserted into the macroscopic spatial interaction model, and strong coupling is achieved through top-downand bottom-up feedbacks.
The main characteristic of our model is a strong coupling between the macroscopic scale of a systemof cities with the mesoscopic scale of metropolitan areas. We consider simple urban dimensions withpopulation variables only to describe cities, and use stylised processes. The system of cities evolvesfollowing a spatial interaction model as described by [Raimbault, 2020c], assuming that interaction flowsbetween cities will increase their attractivity and thus their population growth rate. Population updatesare not done directly following migration flows as in stochastic urban models such as in [James et al., 2018],but with the hypothesis of hidden variables (generally scaling with population) capturing for exampleeconomic dimensions, which influence population growth rate. At the mesoscopic scale, we consider thespatial distribution of population within an urban area and focus on the growth of urban form. Toachieve this, the reaction-diffusion model of urban morphogenesis described by [Raimbault, 2018] allowscapturing concentration forces and dispersion forces, and has been shown to reproduce a large range ofexisting urban forms. Furthermore, [Raimbault, 2020a] showed that this model had a good coverage ofthe morphological space when compared to other models of urban morphogenesis.To couple the scales, we assume that (i) macroscopic performance of a city will influence the choicesmade by planers in terms of land-use, and thus parameters of the morphogenesis mode locally - empiricalevidence of such links have been suggested in the literature [Joy and Vogel, 2015] and we stylise them here;(ii) urban form influences the inner workings of a city through congestion and the fostering of exchangesbetween agents (individuals, firms), and finally the global insertion of the city - thus spatial interactionparameters are updated as a function of local urban form. This last link is the most discutable in termsof empirical evidence and implies many other dimensions than population only. We however work on astylised explicative model which must keep a simple structure but on which a systematic knowledge ofmodel behavior can be developed. We show in Fig. 1 a schematic description of model structure.3 .2 Formalization
We consider N urban areas, represented at the macroscopic scale by their population P j ( t ) at time t , andat the mesoscopic scale by a population grid p ( j ) kl ( t ). The model runs for a total number t f of time steps,and we will assume that ∆ t = 1 for the sake of simplicity (the formulas can be generalized for arbitraryvalues of the time step, for example when running on real data with irregular time sampling).The simulation model operates iteratively, and at each time step goes sequentially through the follow-ing steps:1. populations of urban areas are evolved at the macroscopic scale with the spatial interaction model(Equations 1 to 3 below);2. for each urban area, parameters of the moprhogenesis model are updated given parameters capturingtypical scenarios: transit-oriented development or sprawl for the diffusion parameter, metropoliza-tion or uniformization for the aggregation parameter (Equations 4 to 7);3. local urban form are evolved conditionally to a fixed population increase and following the mesco-scopic parameters;4. congestions and interactions within areas are synthesised following a cost function, which is used toupdate the macroscopic interaction parameters (Equations 8 and ?? ).Details of each stage are described below Following [Raimbault, 2020c] and [Favaro and Pumain, 2011], aggregated populations variables P i ( t ) areevolved according to P i ( t + 1) = P i ( t ) t · g i + w i N · (cid:88) j V ij < V ij > (1)The gravity interaction potential V ij between cities is given by V ij = (cid:18) P i P j ( (cid:80) k p k ) (cid:19) γ G · exp (cid:18) − d ij d i (cid:19) (2)We write the population variations as∆ P i ( t ) = P i ( t + 1) − P i ( t ) (3) The population variations at the macroscopic scale then influence the mesoscopic parameters to capturethe top-down feedback. Therefore, the parameters of [Raimbault, 2018], namely β i of diffusion and α i ofaggregation are both updated. First, the mesoscopic growth rate is adjusted to the population growthuniformly over the time interval such that N ( i ) G ( t + 1) = ∆ P i /t m , where t m is a global model parametercontrolling the temporal granularity of local population increases.The sprawl parameter evolves according to a fixed multiplier and the relative population increasefollowing β i ( t + 1) = β i ( t ) · (cid:18) δβ · ∆ P i ( t )max k ∆ P k ( t ) (cid:19) (4)4able 1: Summary of model parametersType Parameter Process RangeMacro g i = g Endogenous growth [0; 0 . w i = w G Interactions weight [0; 0 . γ i = γ G Interactions hierarchy [0; 5 . d i Interactions decay [0; 1000]Meso α i Aggregation [0; 5] β i Diffusion [0; 0 . t m Urban growth speed [0; 20] n d Diffusion [0; 5]Multiscale δα Downward feedback [ − .
5; 0 . δβ Downward feedback [ − .
5; 0 . δd Upward feedback [ − .
5; 0 . λ Cost of congestion [0; 2]where the multiplier parameter δβ allows testing different scenarios: a negative value corresponds totransit-oriented development while a positive value corresponds to an uncontrolled sprawl.The aggregation parameter evolves in a similar way but as a function of accessibility increase, followingthe rationale that more accessible place with concentrate more activities, as α i ( t + 1) = α i ( t ) · (cid:18) δα · ∆ Z i ( t )max k ∆ Z k ( t ) (cid:19) (5)where the multiplier parameter δα allows switching between a metropolisation scenario (more aggre-gation) and an uniformisation scenario (less aggregation). In that context, the accessibility is computedas Z i = (cid:88) j P j (cid:80) k P k · exp( − d ij /d i ) (6)Change in the level of sprawl ( β i ) depends on the population pressure only, while aggregation dependson accessibility since it is linked to metropolisation processes. Mesoscopic grids population grids are evolved separately within each urban area, using the updatedmorphogenesis model parameters. We follow here the same exact steps described in [Raimbault, 2018],with parameters N ( i ) G total population increase, β i diffusion level, α i aggregation level, t m time steps and n d number of diffusions. The model adds iteratively population to grid cells, by distributing it with aprobability following a preferential attachment to population, and diffusing it.In terms of implementation, small population differences in the end between macroscopic and meso-scopic levels (due to rounding in computing the number of steps) is corrected by adjusting the macroscopicincrements by the effective mesoscopic increments (which are assumed to be more precise). Finally, bottom-up feedback is captured by updating the macroscopic parameters. For the sake of sim-plicity, only interaction decays are updated, assuming that patterns of urban form play a role in theglobal insertion of the city. More precisely, we compute gravity flows within the area, and aggregate theirvalue as an economic activity with a squared negative externality interpreted as a congestion with a costparameter λ , following 5 i = (cid:88) kl (cid:32) P k P l P · d kl − λ (cid:18) P k P l P · d kl (cid:19) (cid:33) (7)This utility U i is used to update the interaction decays following d i ( t + 1) = d i ( t ) (cid:18) δd · U i max k | U k | (cid:19) (8)where the multiplier parameter δd allows controlling for the influence of local performance on globalinsertion. The Table 1 summarizes model parameters. For clarity, we do not include parameters linked to syntheticmodel setup, and distinguish them by the scale to which they are attached.
The system is initialized with synthetic systems of cities. A number N = 20 cities are distributedrandomly into a regional urban system, as a square world of width w = 500 km (reference unit for thedecay parameter). City populations follow a Zipf law with hierarchy parameter α = 1 (may be modifiedin experiments), and P (0) = 100 ,
000 the initial size of the largest city. Within each city, we considerinitial population grids as monocentric (grid of size W = 50 and center cell density P m = 1000 populationunits, following an exponential kernel of width one fifth of the world). The model is simulated for 20macroscopic time steps (order of magnitude of half a century), each representing t m mesoscopic time steps(parameter changed in experiments). Model behavior is characterized using the following indicators: at the macroscopic scale: distributionsof population, accessibilities, and centralities (summarized by average, hierarchy, entropy), following[Raimbault, 2020d]. At the mesoscopic scale, urban form is captured following the indicators used by[Raimbault, 2018], i.e. Moran index, average distance between individuals, hierarchy of population dis-tribution, and entropy of population distribution.
Model implementation is done under some performance constraints: N mesoscopic morphogenesis modelsmust be simulated in parallel at each macroscopic time step. On the contrary, macroscopic interactionsare efficient to implement as they are based on matrices computations. The model is implemented in scala and integrated into a broader library for spatial sensitivity analysis and dynamical models forsettlement systems, namely the spatialdata library [Raimbault et al., 2020]. This library in particularalready includes implementations of the models used in [Raimbault, 2020c] and [Raimbault, 2018], butalso of other urban interaction models such as [Favaro and Pumain, 2011]. The relative large numberof parameters and output indicators imposes the use of appropriate model exploration methods to geta grasp of model behavior. Therefore, the parameter space is explored with the OpenMOLE modelexploration software [Reuillon et al., 2013]. Standalone source code for the model is available on an opengit repository at https://github.com/JusteRaimbault/UrbanGrowth-model , while the repository forresults is at https://github.com/JusteRaimbault/UrbanGrowth . Simulation data used in this paper isavailable on the dataverse repository at https://doi.org/10.7910/DVN/IRHMQK .6igure 2: Statistical distribution of indicators for a sample of 10 parameter values. The first numerical experiment is aimed at checking the statistical consistency of model output acrossstochastic repetitions. We run a one-factor-at-a-time sampling on all parameters with 5 steps for eachwithin the domain provided in Table 1, and 500 stochastic repetitions for each parameter.Obtained sta-tistical distribution are shown in Fig. 2 for a sample of 10 parameters. Different indicators exhibit variousdistribution shapes, but are always unimodal and generally rather localized. This means different param-eter values can be discriminated when considering summary statistics. To estimate the role of noise, wecompute Sharpe ratios (rate between average and standard deviations) for each indicators and parametervalues. Median values of these across parameters are always larger than 1.6, confirming a reasonableinfluence of noise on model outputs. We run experiments with 50 replications for each parameter valuein the following.
We then turn to a grid exploration of the parameter space. Results show an important role of the strongcoupling between scales. This is shown for example by the variation of macroscopic indicators trajectorieswhen switching from a “transit-oriented development” scenario (negative feedback of population growthon diffusion δβ ) to a “sprawl” scenario (positive feedback δβ ) as shown in Fig. 4 below.We show in Fig. 3 that both macroscopic and mesoscopic indicators have a U-shaped behavior asa function of the bottom-up feedback parameter δd . In terms of macroscopic trajectories, we see thathierarchies of accessibilities, closeness centrality and population are maximal in negative value when themechanism is deactivated, what means that the feedback process moderates the hierarchisation of the7igure 3: Variations of indicators as a function of δd , when all other parameters are fixed to their nominalvalue.global urban system. For the urban form, this feedback also mitigates the aggregation of population.The strong coupling of scales, in particular the interaction between bottom-up and top-down feed-backs, has a non-trivial influence on model behavior, as shown in Fig. 4 for the macroscopic hierarchy ofpopulations. This allows investigating the influence of the macroscopic interaction decay and the meso-scopic evolution speed, under stylised scenarios for the top-down feedback. We exhibit interaction effectsbetween δβ and δd , as quantitative and qualitative behavior as a function of t m changes across panels.Interaction range d G has on the contrary no significant effect. We find that increasing t m decreases macro-scopic hierarchy when top-down feedback is deactivated ( δβ = 0), but a cancelling of this effect when itis activated. This may have policy implications in terms of controlling urban sprawl by acting or not on δβ , and its influence on the overall urban system. We then proceed to a targeted experiment by detailing more the grid for δβ and δα . The aim is toinvestigate the influence of policies taken at the level of urban areas, but which can also be constrained bya higher level of governance. We show in Fig. 5 the influence of δβ on urban form. In particular, the averagevariation of Moran index captures the centralisation of urban areas. This centralisation exhibits a minimalvalue when δα is low, consistent across values of λ . This implicates that when decentralisation policiesare implemented by acting on aggregation ( δα ), the policy on urban sprawl will have an unexpected effecton centralisation. This is an illustration of complex and possibly negative interaction between policies.When investigating population hierarchy the mesoscopic scale, i.e. inequalities of population distri-bution within the urban area, we find in Fig. 6 a U-shaped behavior as a function of δβ for negative δα values, but has a plateau for positive values of δα , showing also an interaction effect between policies. Interms of hierarchy, a maximal negative value may be wanted to limit inequalities (slopes are negative),8igure 4: Macroscopic hierarchy of populations as a function of mesoscopic time steps t m , for differentvalues of interaction decay d G (colour), and of feedback parameters δd (columns) and δβ (rows).and this optimum is reached for a slightly positive value of δβ (left columns in Fig. 6). This means thatsprawl may be needed in some configurations to optimise this objective. We finally turn to an optimization objective to investigate if compromises have to be made when targetsare simultaneously aimed for at different scales. We therefore use a NSGA2 algorithm implementedin OpenMOLE, with a population of 200 individuals, run for 100,000 generations. Objectives are theaverage mesoscopic hierarchy which can be minimized for more compact and thus sustainable cities, atthe macroscopic population hierarchy. Convergence is reached in terms of hypervolume and points in thePareto front.We show in Fig. 7 the Pareto front obtained after convergence of the algorithm. This confirms that thetwo objectives are indeed contradictory and that compromises between scales have to be made when tryingto optimise both simultaneously. Most points correspond to policies favouring sprawl, at the exception ofa small set of points with low macroscopic hierarchy values, which discourage sprawl ( δβ < δβ , fordifferent values of feedback parameter δα (colour), and of feedback parameters δd (columns) and λ (rows).with the most equalities between cities within the system of cities correspond to the most hierarchicalurban form, implying inequalities at the mesoscopic scales. Our model effectively captures an interaction between downward and upward feedback. In that sense, itillustrates weak emergence in its proper sense [Bedau, 2002]. This suggests this view can be effectivelyoperationalised into models of urban dynamics.In terms of model complexity and in relation to urban complexity in itself, we show that couplingsimple models (each having only three parameters) already yields a complicated and complex simulationmodel. It also requires additional ontologies to achieve the coupling (in our case the policy processeson one side and the congestion-performance process on the other side). Therefore, we suggest this isadditional evidence for a necessity of complexity and simulation models to understand urban complexity.This implies in some way the failure of reductionist epistemologies, at least to grab the multi-scalar natureof urban systems.Although a stylised and highly simplified model, we were already able to integrate parameters linked topolicies. Indeed, governance processes are intrinsically multi-scalar [Liao and Gaudin, 2017]. Therefore,developing such models is a first step towards a progressive integration models for policy and sustainablemanagement of territories. 10igure 6: Relative evolution of average mescoscopic population hierarchy, as a function of feedbackparameter δβ , for different values of feedback parameter δα (colour), and of feedback parameters δd (columns) and λ (rows). This model is only a first structural sketch with very restrictive assumption, in particular regarding thedownward and upward feedbacks on submodel parameters. There may be no link between urban formand global insertion, or it may be due to other processes, be expressed as an other functional form. Animportant stage before shifting to robust knowledge will consist in (i) reviewing and making a typology ofsuch potential processes across scales; (ii) including most in a multi-modeling fashion to compare possibleconcurrent mechanisms.Future work also implies the use of more elaborated model exploration and validation methods. Forexample, using the diversity search integrated in OpenMOLE [Reuillon et al., 2013] would allow unveil-ing the diversity of possible future urban trajectories, and possibly unexpected novel pathways towardssustainability, finding for example regimes with the strongest effect of feedback parameters on indicatorsrelated to sustainability.Regarding an application of the model on real systems of cities, a calibration on empirical trajectorieswould be needed. However, such data is difficult to obtain, and some investigations should be made onthe relevant calibration targets (in particular at which scale).Our model is a first step towards multi-scalar models of urban dynamics which effectively includea strong coupling between processes at different scales. While remaining highly stylised, we showed animportant role of the coupling and unveiled for example interaction effects between different types ofpolicies. This type of approach will need in the future to be further developed, towards multi-scalarmodels for sustainable policy making [Rozenblat and Pumain, 2018]. This works paves the way for suchmore complex models. 11igure 7: Pareto front obtained with a NSGA2 genetic algorithm, to minimize simultaneously hierarchyat the macroscopic and mesoscopic scales. Point size gives the number of stochastic samples and colourthe value of δβ . Acknowledgments
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