AA comparison of simple models for urbanmorphogenesis
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
The spatial distribution of population and activities within urbanareas, or urban form at the mesoscopic scale, is the outcome of multipleantagonist processes. We propose in this paper to benchmark differentmodels of urban morphogenesis, to systematically compare the urban formsthey can produce. Different types of approaches are included, such as areaction-diffusion model, a gravity-based model, and correlated percolation.Applying a diversity search algorithm, we estimate the feasible space ofeach model within a space of urban form indicators, in comparison ofempirical values for worldwide urban areas. We find a complementarity ofthe different types of processes, advocating for a plurality of urban models.
Introduction
Understanding the dynamics of cities is an increasing issue for sustainability,since the proportion of the world population expected to live in cities willgrow to a large majority in the next decades, and that cities combine bothpositive and negative externalities on most aspects. Their complexity impliesthat quantitative and qualitative predictions are not relevant, but planners can invent future cities [1], what requires though a knowledge of key urban processeswhich can be acted upon. In that context, the growth of urban form in itsdifferent definition and scales, is essential [2]. Considering urban form at amesoscopic scale, i.e. roughly the scale of urban areas, it can be understoodas the spatial distribution of activities. More particularly the distribution ofpopulation density has a strong impact on commuting, energy consumption1 a r X i v : . [ phy s i c s . s o c - ph ] A ug nd emissions [3]. Being able to link microscopic processes with the growthof different types of urban form is thus important for a long term planning ofsustainable urban systems.Urban modeling at the mesoscopic scale is the subject of diverse approachesand disciplines. Intra-urban urban economic models, building on classic workssuch as the Alonso-Mills-Muth model or the Fujita-Ogawa model, propose modelslinking land-use with land and building markets, which are spatially explicitto different degrees [4]. Transportation and Urban Planning also have a longhistory in urban dynamics models, including Land-use transport interactionmodels [5]. Spatial interaction models can also be used in a similar manner tostudy urban dynamics and as a by-product urban form [6]. Cellular automatamodels of urban growth are also a privileged approach to study the growth ofurban form from a data-driven perspective [7].At the interface of physics, artificial life and quantitative geography, a fewapproaches propose simple models to explain the growth of urban form, andgenerally rely on an unidimensional description of urban form, namely thedistribution of population or of the built environment. In that context, thecorrelated percolation model introduced by [8] was a precursor. Such models canrely on abstract physical processes but also on agent behavior, such as in theSugarscape model which according to [9] can be considered as a model for humansettlements. [10] use migration between cities at multiple scales to simulate urbangrowth. Diffusion-limited aggregation (DLA) is an other approach transferredfrom physics to urban modeling [11] and has shown relevant to reproducefractal urban structures and urban migration processes [12]. [13] combinesDLA with percolation to obtain more realistic urban forms. Closer to the ideaof urban morphogenesis, [14] proposes a reaction-diffusion model to capturefundamental urban growth processes. [15] describes an urban growth model basedon geographical processes, namely an aggregation of population driven by spatialinteraction. All these works have in common to model urban growth in syntheticsettings, at a mesoscopic scale, considering population distribution only, and ina stylized way. They furthermore consider diverse processes, remaining simplein their structure although they lead to the emergence of a complex behavior.We will in this paper focus on such models, referring to them as models of urbanmorphogenesis .Exhibiting models with a few number of parameters and processes is usefulfrom an explanative viewpoint, when these can reproduce real world configura-tions. Having multiple concurrent models which include diverse, complementaryor contradictory processes, is furthermore useful for the construction of integratedurban theories, since concurrent explanations can be benchmarked, comparedand possibly integrated into multi-modeling approaches. This plurality in urbanmodeling is intrinsic to a literature with multiple disciplines focusing on a sameobject of study [16].We propose thus in this paper to benchmark several simple models of urbanmorphogenesis, in order to understand the potentialities of some of these modelsto exhibit a complex behavior and reproduce existing urban forms, and comparethem in a systematic way. More precisely, our contributions are the following:2i) we integrate four different models (correlated percolation, reaction-diffusion,gravity and exponential mixture) into a single software framework; (ii) we com-pute measures of urban form for urban areas worldwide; (iii) we apply a noveltysearch algorithm to the models in order to determine their feasible morphologicalspace, and compare these to real urban form values. This contributes to a generalunderstanding of the complementarity of urban models, more particularly forurban morphogenesis at this scale.The rest of this paper is organized as follows: we first describe the modelsbenchmarked and the quantitative measures used for urban form; we describeempirical values of urban morphology indicators for urban areas worldwide; theseare then compared to the feasible space of each model obtained with a diversitysearch algorithm. We finally discuss the implications of these results for theoriesof urban morphogenesis and possible developments. Materials and methods
Urban morphogenesis models
We study and compare four different models of urban morphogenesis. Weconsider a population grid of size N = W × H (not necessarily square), each cellbeing characterized by its population P i with 1 ≤ i ≤ N . Gravity-based model
Following the so-called “first law of geography”, entities in space have interactionpatterns which can be described with spatial interaction models [17], includingthe gravity model. [15] proposed an urban growth model including this processwithin an iterative growth with population aggregation processes, extending amore simple model introduced by [18]. We generalize this model by adding (i) ahierarchy parameter regarding population aggregation and (ii) seeding multipleinitial sites to allow the emergence of polycentric urban forms.Formally, an initial grid is seeded with P ( G )0 sites with population 1, randomlyselected. Then, iteratively, one unit of population is added to each cell at eachtime step with a probability proportional to p i ∝ (cid:80) j (cid:54) = i P γ ( G ) P j · d − γ ( G ) D ij (cid:80) d − γ ( G ) D ij (1)where probabilities are rescaled such that the cell with the larger value has aprobability of g ( G ) . This last parameter allows modifying the speed of growth.The model is stopped when a total population P ( G ) M is reached. Reaction-diffusion
In his attempt to understand embryogenesis, Alan Turing proposed to use chem-ical partial differential equations (PDEs) to model morphogenesis, introducing3he nowadays famous reaction-diffusion equations [19]. In such systems, chemicalsubstances react together and diffuse in space, leading to the emergence ofcomplex geometrical patterns. The concept of morphogenesis has been sincewell used in urban studies [20], but very few models have actually implementedreaction-diffusion equations, [21] being a notable exception. [14] proposes tocapture the fundamental processes of agglomeration economies (positive exter-nalities) leading to aggregation and of congestion (negative externalities) leadingto sprawl, as an “aggregation-diffusion” model of urban morphogenesis. Themodel yields indeed in certain limits reaction-diffusion PDEs. Formally, startingfrom an empty grid, N ( R ) units of population are added at each time step,and attributed independently to cells with a probability proportional to P α ( R ) i (probabilities are rescaled to obtain a probability distribution over all cells).Population is then diffused in space d ( R ) times with a strength β ( R ) . The modelis stopped when a maximum population P ( R ) M is reached. Correlated percolation
The first two models presented are iterative and can in theory be used dynamically.Other approaches, closer to procedural modeling [22], do not simulate theprogressive growth of population. They can however capture processes at playin the growth of urban form. The correlated percolation model described by [8]integrates for example clustering processes in cities. A method to generate aspatial field exhibiting long range correlations was introduced for problems inphysics by [23]. It is combined to a monocentric density profile in [8] to produceurban forms. In practice, a correlated field p i is generated by (i) generating arandom spatial field; (ii) compute its spatial Fourier transform; (iii) introduce acorrelation by multiplying it with a spectral density function with a power-lawexponent α ( C ) ; (iv) retrieve a long-range correlated spatial field by taking theinverse Fourier transform. This field is combined to a density field ρ i to determinea binary value for the cell: it is populated if p i > θ i with ρ i = (cid:82) θ i −∞ d P ( p i ). Wegeneralize the initial model by taking a polycentric density field with ρ i ( (cid:126)x ) = n ( C ) (cid:88) j =1 exp ( (cid:107) (cid:126)x − (cid:126)x j (cid:107) /r j ) (2)where the kernel centers are chosen at random and kernel sizes r j are takensuch that kernel populations follow a rank-size law of hierarchy β ( C ) and thelargest kernel has a fixed size r ( C )0 . Kernel mixtures
Finally, to provide some kind of null model to understand the advantages of eachapproach compared to a simple description of population distribution, we alsoinclude urban forms generated as kernel mixtures. We consider in particularexponential mixtures [24], where population density is written as previously inEq. 2. Parameters for this model are the number of kernels n ( E ) , the rank-size4ierarchy α ( E ) , the size of the largest kernel r E . Contrary to previous modelsin which total population had an influence by controlling the speed of growth,density can here be rescaled arbitrarily (morphological indicators used are notchanged through rescaling, see below), and we set the maximal density for onekernel to one. Measures of urban form
Quantitative measures of urban form are multiple and depend on the scaleconsidered [25]. [26] for example introduces measures for buildings at the districtscale. The field of Landscape Ecology has its own metrics similar to urbanform measures [27]. For the scale we consider and considering populationdistribution only, metrics of urban form have been proposed for example toquantify sprawl [28]. These can be related to fractal approaches to urbanform [29]. The effective dimension when applied to real cities is reasonablylow [30], and a few complementary indicators can be used. We thus follow [14]and consider urban form measures which are: (i) Moran index to capturespatial autocorrelation and the existence of centers; (ii) average distance betweenindividuals which captures a level of aggregation; (iii) distribution entropy(aspatial) to capture the uniformity of the distribution; and (iv) rank-size slopewhich captures the hierarchy of population distribution.
Results
Implementation
The models are implemented in scala and integrated into the spatialdata library for spatial sensitivity analysis [31]. The library is bundled as an Open-MOLE plugin for the numerical experiments. OpenMOLE is an open sourcesoftware for model exploration and validation [32] combining model embeddingwith state-of-the-art exploration methods (including for example sensitivity analy-sis, design of experiments, calibration with genetic algorithms) and a transparentdistribution of computations on high performance computing environments. Inour case, we use its workflow system and an integrated algorithm to determinethe feasible space of models.
Empirical data
We compare model simulation results to urban form measures computed onworldwide urban areas. We use therefore the Global Human Settlement Layerdatabase (GHSL), which provides an exhaustive worldwide population rasterwith a 1km resolution [33]. [34] has shown the relevance of using this database forworldwide simulation models. It is available at four dates from 1975 onwards, butas not all models are dynamical we use the most recent population configurationonly (2015). The database provides a layer of urban areas, within which weextract for the 1000 largest areas in terms of population a covering window ( ± ig 1. Maps of urban morphology indicators for worldwide urbanareas. (Top) Rank-size hierarchy; (Bottom) Moran spatial autocorrelationindex. Area of circles gives population.of extent on each side) from the population raster (note that some windows maybe overlapping, as in the case of Hong-Kong which is separated from the maincluster of the Pearl River Delta mega-city region). Morphological indicators arecomputed on these extracted areas.We show in Fig. 1 maps of indicators. More particularly, we map the rank-sizehierarchy and Moran spatial autocorrelation which have meaningful geographicalvariations. Rank-size hierarchy will tell if the metropolitan area is stronglydominated by one center or if it more balanced. We retrieve the fact that inEurope, Paris and London are known for such a strong monocentricity, comparedto cities in Germany for example. Similarly, mega-city regions in East Asia(Pearl River delta, Yangtze River delta, Beijing-Tianjin) are more polycentricand thus balanced than Wuhan or Seoul. Regarding spatial autocorrelation,we also observe a strong variation in East Asia (Tokyo compared to Chinesemegacities for example), and within India (agglomerations in the Gange plainmaking a cluster of non-correlated, thus highly sprawled areas).The areas can be clustered following a non-supervised approach. We proceedto a k-means clustering on normalized indicator values, and find k = 4 clusters6 ig 2. Urban form typology. Using an unsupervised k-meansclustering algorithm, we find clusters among urban areas. (Top) Mapof cluster belonging; (Bottom) Statistical distribution of indicator values, withineach cluster and in time for all dates in the GHSL dataset.as meaningful regarding the derivative of within-cluster variance. A map ofcluster belonging and cluster profiles are shown in Fig. 2. We retrieve thevariation within East Asia (Tokyo, Seoul and Shanghai being each in a differentcluster) but less in Europe. The main cluster (light blue) corresponds to stronglymonocentric urban areas. We see in density distributions of indicator values thatclusters have clearly different profiles, which correspond to different typologiesof urban morphology that the models try to approximate. For example, cluster 3(light blue) and 1 (red) have the same level of hierarchy, but the latter has a much7igher autocorrelation and entropy, and corresponds thus to more polycentricconfigurations. Fig 3. Examples of generated urban forms for a square world ofwidth W = 50 . From Left to Right and Top to Bottom: (i) Correlatedpercolation model for r ( C )0 = 50, α ( C ) = 0 . n ( C ) = 3; (ii) Exponential mixturemodel for n ( E ) = 10, r ( E )0 = 40, α ( E ) = 1; (iii) Gravity model for g ( G ) = 0 . γ ( G ) D = 2 . γ ( G ) P = 0 . P ( G )0 = 3, P ( G ) M = 20000; (iv) Reaction-diffusion modelfor α ( R ) = 0 . β ( R ) = 0 . d ( R ) = 1, N ( R ) = 100, P ( R ) M = 5000. Generated urban forms
We show in Fig. 3 examples of generated urban forms for each model included inthis study. Visually, these urban form look rather different. To what extent theyare statistically distant for the morphology indicators can only be determined8y systematic experiments. The gravity and percolation results look similar,although at a slightly different scale. This particular configuration of the reaction-diffusion model corresponds more to a rural or peri-urban configuration, whilethe exponential mixture is fuzzy and would resemble a blurred polycentric urbanconfiguration.
Fig 4. Feasible morphological space for each simulation model.
Thelower part of the plot matrix gives a scatterplot between each indicator. We alsoplot real urban areas in purple. The diagonal gives statistical distribution, whilethe correlation matrix is given in the upper part, conditionally to each model.
Feasible morphological spaces
We now turn to the main experiment of this paper: using a diversity searchalgorithm to determine the full feasible morphological space for each model.Therefore, we use the Pattern Space Exploration (PSE) algorithm [35] embeddedin OpenMOLE. Diversity search was introduced in the field of artificial life asgenetic algorithms with the aim to maximize diversity of the population [36]. Inthe case of the PSE algorithm, a novelty criteria leads the search towards newregions of the indicator space, and results are stored in a hitmap. When runninglong enough, convergence in terms of number of solutions found is generallyreached, and one can consider the final population as the feasible space of themodel. We run here the algorithm up to 100,000 generations for each model,9ith grid of step 0.05 in the indicator space. Convergence was reached separatelyfor each model.We show in Fig 4 a scatterplot of the final population in the morphologicalspace. First of all, we observe that correlation are very different for each model(for example opposite value for gravity and reaction-diffusion between entropyand slope), confirming that the way urban form is produced is very differentalthough the final form may be the same. Then, we observe that the pointclouds slightly intersect but also have their own proper space that which no othermodel can reach. For example, in the entropy-slope space, the reaction-diffusionmodel is very flexible and covers a large part of space, while gravity is restrictedto a small region and correlated percolation and exponential mixture exhibitmuch less variability with a strong correlation between the two dimensions. Themodels cover a similar region for the Moran-entropy space, but all fail to capturean important part of real points (in purple), corresponding to configurationswith a high Moran index but a relatively low entropy. Models systematicallyassociate a high moran with a high entropy. In other dimensions, real pointsare reasonably covered. Thus, we can conclude that (i) the different modelsare complementary in terms of urban forms produced and (ii) that a part ofreal configurations are approached by some of the models, but other can not bereproduced.
Fig 5. Overlap between point cloud hypervolumes.
To better quantify how each model are similar and how they approach realconfigurations, we compute the hypervolume (in the four dimensional indicatorspace) corresponding to each point cloud and the intersections between thesehypervolumes. We use the hypervolume R package [37] with a gaussian kerneldensity estimation with adaptable bandwidth. Then, we compute for each coupleof model the ratio between the intersection of hypervolumes and the secondmodel. The non-symmetric relative overlap matrix is plotted in Fig. 5 (diagonal10as removed for a better readability). We find that the closer model are thecorrelated percolation and reaction diffusion, with close to 50% overlap. Thenthe gravity point cloud is mostly contained by reaction-diffusion and correlatedpercolation, but not the other way around. Thus models produce similar pointsin the parameter space, but most of their output cloud is original. Finally, wefind regarding the proximity to real configurations that the best model is theexponential mixture. At the price of producing blurry urban forms, the flexibilitytherein is the best to approach the best existing configurations. Then comes thereaction-diffusion model, and the least flexible to approach real values is thecorrelated percolation model.
Discussion
We have shown that different simple urban morphogenesis models are stronglycomplementary in the morphological space, confirming that a complementarityof processes also leads to a complementarity in patterns produced. This result re-joins the results obtained by [38] in the case of transportation network generativemodels at a similar mesoscopic scale, the results of [26] for generating buildingconfigurations, and the results of [39] in the case of dynamical models for systemsof cities. Both found a complementarity of approach including different types ofprocesses. We argue that this is further evidence of the multi-dimensionality ofurban systems and for a necessity of a plurality of urban models to capture bothdiverse processes but also outcomes.This work is a first step towards a systematic benchmark of simple urbanmorphogenesis models. Future work should include the search for an explanationof the unreachable real urban configuration, and possibly alternative modelsapproaching these. Other urban form indicators should also be tested. Finally,dynamical calibration of models remains an open question, investigated in thecase of the reaction-diffusion model by [40]. An issue is that some models such asthe correlated percolation model are not dynamical and should thus be adaptedto be calibrated between successive points in time.
Conclusion
We have implemented and systematically compared four very different simplemodels for urban morphogenesis, including reaction-diffusion, correlated percola-tion, exponential mixture and a gravitational aggregation model. We applieda diversity search algorithm to obtain the feasible morphological space. Theresults confirm a complementarity between the models and the relevance of aplurality in urban modeling approaches.
References
1. Batty M. Inventing future cities. MIT press; 2018.11. Williams K, Burton E, Jenks M. Achieving sustainable urban form: anintroduction. Achieving sustainable urban form. 2000;2000:1–5.3. Le N´echet F. Urban spatial structure, daily mobility and energy con-sumption: a study of 34 european cities. Cybergeo: European Journal ofGeography. 2012;.4. Vigui´e V, Hallegatte S. Trade-offs and synergies in urban climate policies.Nature Climate Change. 2012;2(5):334–337.5. Wegener M, F¨urst F. Land-use transport interaction: state of the art.Available at SSRN 1434678. 2004;.6. Milton R, Roumpani F. Accelerating Urban Modelling Algorithms with Ar-tificial Intelligence. In: Proceedings of the 5th International Conference onGeographical Information Systems Theory, Applications and Management.vol. 1. INSTICC; 2019. p. 105–116.7. Batty M. Cellular automata and urban form: a primer. Journal of theAmerican Planning Association. 1997;63(2):266–274.8. Makse HA, Andrade JS, Batty M, Havlin S, Stanley HE, et al. Modelingurban growth patterns with correlated percolation. Physical Review E.1998;58(6):7054.9. Batty M. Cities and complexity: understanding cities with cellular au-tomata, agent-based models, and fractals. The MIT press; 2007.10. Murcio R, Morphet R, Gershenson C, Batty M. Urban transfer entropyacross scales. PLoS One. 2015;10(7):e0133780.11. Batty M, Longley P, Fotheringham S. Urban growth and form: scaling,fractal geometry, and diffusion-limited aggregation. Environment andplanning A. 1989;21(11):1447–1472.12. Murcio R, Rodr´ıguez-Romo S. Colored diffusion-limited aggregation forurban migration. Physica A: Statistical Mechanics and its Applications.2009;388(13):2689–2698.13. Murcio R, Sosa-Herrera A, Rodriguez-Romo S. Second-order metropolitanurban phase transitions. Chaos, Solitons & Fractals. 2013;48:22–31.14. Raimbault J. Calibration of a density-based model of urban morphogenesis.PLOS ONE. 2018;13(9):1–18. doi:10.1371/journal.pone.0203516.15. Li Y, Rybski D, Kropp JP. Singularity cities. Environment and PlanningB: Urban Analytics and City Science. 2019; p. 2399808319843534.16. Pumain D, Raimbault J. Conclusion: Perspectives on urban theories. In:Theories and Models of Urbanization. Springer; 2020. p. 303–330.127. Fotheringham AS, O’Kelly ME. Spatial interaction models: formulationsand applications. vol. 1. Kluwer Academic Publishers Dordrecht; 1989.18. Rybski D, Ros AGC, Kropp JP. Distance-weighted city growth. PhysicalReview E. 2013;87(4):042114.19. Turing AM. The chemical basis of morphogenesis. Bulletin of mathematicalbiology. 1990;52(1-2):153–197.20. Raimbault J. Co-evolution and morphogenetic systems. arXiv preprintarXiv:180311457. 2018;.21. Bonin O, Hubert JP. Mod´elisation morphog´en´etique de moyen terme desvilles: une sch´ematisation du mod`ele th´eorique de Ritchot et Desmaraisdans le cadre du mod`ele standard de l’´economie urbaine. Revue dEconomieRegionale Urbaine. 2014;(3):471–497.22. Parish YI, M¨uller P. Procedural modeling of cities. In: Proceedings of the28th annual conference on Computer graphics and interactive techniques;2001. p. 301–308.23. Makse HA, Havlin S, Schwartz M, Stanley HE. Method for generating long-range correlations for large systems. Physical Review E. 1996;53(5):5445.24. Anas A, Arnott R, Small KA. Urban spatial structure. Journal of economicliterature. 1998;36(3):1426–1464.25. Zhang M, Kukadia N. Metrics of urban form and the modifiable arealunit problem. Transportation Research Record. 2005;1902(1):71–79.26. Raimbault J, Perret J. Generating urban morphologies at large scales. In:Artificial Life Conference Proceedings. MIT Press; 2019. p. 179–186.27. Bosch M. PyLandStats: An open-source Pythonic libraryto compute landscape metrics. PLOS ONE. 2019;14(12):1–19.doi:10.1371/journal.pone.0225734.28. Tsai YH. Quantifying urban form: compactness versus’ sprawl’. Urbanstudies. 2005;42(1):141–161.29. Chen Y. Derivation of the functional relations between fractal dimensionof and shape indices of urban form. Computers, Environment and UrbanSystems. 2011;35(6):442–451.30. Schwarz N. Urban form revisited—Selecting indicators for characterisingEuropean cities. Landscape and urban planning. 2010;96(1):29–47.31. Raimbault J, Perret J, Reuillon R. A scala library for spatial sensitivityanalysis. arXiv preprint arXiv:200710667. 2020;.132. Reuillon R, Leclaire M, Rey-Coyrehourcq S. OpenMOLE, a workflowengine specifically tailored for the distributed exploration of simulationmodels. Future Generation Computer Systems. 2013;29(8):1981–1990.33. Melchiorri M, Florczyk AJ, Freire S, Schiavina M, Pesaresi M, KemperT. Unveiling 25 years of planetary urbanization with remote sensing:Perspectives from the global human settlement layer. Remote Sensing.2018;10(5):768.34. Raimbault J, Denis E, Pumain D. Empowering Urban Governance throughUrban Science: Multi-Scale Dynamics of Urban Systems Worldwide. Sus-tainability. 2020;12(15):5954. doi:10.3390/su12155954.35. Ch´erel G, Cottineau C, Reuillon R. Beyond corroboration: Strength-ening model validation by looking for unexpected patterns. PloS one.2015;10(9):e0138212.36. Lehman J, Stanley KO. Exploiting open-endedness to solve problemsthrough the search for novelty. In: ALIFE; 2008. p. 329–336.37. Blonder B, with contributions from David J Harris. hypervolume: HighDimensional Geometry and Set Operations Using Kernel Density Estima-tion, Support Vector Machines, and Convex Hulls; 2019. Available from: https://CRAN.R-project.org/package=hypervolume .38. Raimbault J. Multi-modeling the morphogenesis of transportation net-works. In: Artificial Life Conference Proceedings. MIT Press; 2018. p.382–383.39. Raimbault J. A systematic comparison of interaction models for systemsof cities. In: Conference on Complex Systems 2018; 2018.40. Raimbault J. Worldwide estimation of parameters for a simple reaction-diffusion model of urban growth. In: International Land-use Sym-posium 2019. Paris, France; 2019.Available from: https://halshs.archives-ouvertes.fr/halshs-02406539https://halshs.archives-ouvertes.fr/halshs-02406539