Networks and Spatial Economics

, Volume 9, Issue 3, pp 401–425 | Cite as

Co-evolution of Density and Topology in a Simple Model of City Formation

  • Marc BarthélemyEmail author
  • Alessandro Flammini


We study the influence that population density and the road network have on each others’ growth and evolution. We use a simple model of formation and evolution of city roads which reproduces the most important empirical features of street networks in cities. Within this framework, we explicitly introduce the topology of the road network and analyze how it evolves and interact with the evolution of population density. We show that accessibility issues -pushing individuals to get closer to high centrality nodes- lead to high density regions and the appearance of densely populated centers. In particular, this model reproduces the empirical fact that the density profile decreases exponentially from a core district. In this simplified model, the size of the core district depends on the relative importance of transportation and rent costs.


Networks City formation and evolution Urban economics 



We thank G. Santoboni for many discussions at various stages of this work, and two anonymous referees for several important comments and suggestions. MB also thanks Indiana University for its warm welcome where part of this work was performed.


  1. Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows. Prentice Hall, New JerseyGoogle Scholar
  2. Amaral LAN, Scala A, Barthélemy M, Stanley HE (2000) Classes of small-world networks. Proc Natl Acad Sci (USA) 97:11149CrossRefGoogle Scholar
  3. Ball P (1998) The self-made tapestry: pattern formation in nature. Oxford University Press, OxfordGoogle Scholar
  4. Banavar JR, Colaiori F, Flammini A, Maritan A, Rinaldo A (2000) Topology of the fittest transportation network. Phys Rev Lett 84:4745CrossRefGoogle Scholar
  5. Banavar JR, Maritan A, Rinaldo A (1999) Size and form in efficient transportation networks. Nature 398:130–132CrossRefGoogle Scholar
  6. Barthélemy M (2003) Betweenness centrality in large complex networks. Eur Phys J B 38:163CrossRefGoogle Scholar
  7. Barthélemy M, Flammini A (2008) Modeling urban streets patterns. Phys Rev Lett 100:138702CrossRefGoogle Scholar
  8. Barthélemy M, Gondran B, Guichard E (2003) Spatial structure of the Internet traffic. Physica A 319:633–642CrossRefGoogle Scholar
  9. Batty M (2005) Cities and complexity. MIT, CambridgeGoogle Scholar
  10. Bejan A, Ledezma GA (1998) Streets tree networks and urban growth: optimal geometry for quickest access between a finite size volume and one point. Physica A 255:211–217CrossRefGoogle Scholar
  11. Bern MW, Graham RL (1989) The shortest-network problem. Sci Am 260:66–71CrossRefGoogle Scholar
  12. Bettencourt LM, Lobo J, Helbing D, Kuhnert C, West GB (2007) Growth, innovation, scale and the pace of life in cities. Proc Natl Acad Sci (USA) 104:7301–7306CrossRefGoogle Scholar
  13. Brueckner JK (1987) The structure of urban equilibria: a unified treatment of the Muth-Mills model. In: Mills EJ (ed) Handbook of regional and urban economics, vol 2. North Holland, Amsterdam, pp 821–845Google Scholar
  14. Buhl J, Gautrais J, Reeves N, Sole RV, Valverde S, Kuntz P, Theraulaz G (2006) Topological patterns in street network of self-organized urban settlement. Eur Phys J B 49:513–522CrossRefGoogle Scholar
  15. Cardillo A, Scellato S, Latora V, Porta S (2006) Structural properties of planar graphs of urban street patterns. Phys Rev E 73:066107CrossRefGoogle Scholar
  16. Christaller W (1966) Central places in Southern Germany. English translation by CW Baskin. Prentice Hall, LondonGoogle Scholar
  17. Coniglio A (1989) Fractal structure of Ising and Potts clusters: exact results. Phys Rev Lett 62:3054CrossRefGoogle Scholar
  18. Crucitti P, Latora V, Porta S (2006) Centrality measures in spatial networks of urban streets. Phys Rev E 73:036125CrossRefGoogle Scholar
  19. Derrida B, Flyvbjerg H (1987) Distribution of local magnetisations in random networks of automata. J Phys A Math Gen 20:5273–5288CrossRefGoogle Scholar
  20. Dixit AK, Stiglitz JE (1977) Monopolistic competition and optimum product diversity. Am Econ Rev 67:297–308Google Scholar
  21. Doyle G, Snell JL (1989) Random walk and electric networks. American Mathematical Society, ProvidenceGoogle Scholar
  22. Duplantier B (1989) Statistical mechanics of polymer networks of any topology. J Stat Phys 54:581CrossRefGoogle Scholar
  23. Freeman LC (1977) A set of measuring centrality based on betweenness. Sociometry 40:35CrossRefGoogle Scholar
  24. Fujita M, Krugman P, Venables AJ (1999) The spatial economy: cities, regions, and international trade. MIT, CambridgeGoogle Scholar
  25. Gastner MT, Newman MEJ (2006) The spatial structure of networks. Phys Rev E 74:016117CrossRefGoogle Scholar
  26. Gerke S, McDiarmid C (2004) On the number of edges in random planar graphs. Comb Probab Comput 13:165CrossRefGoogle Scholar
  27. Goh K-I, Kahng B, Kim D (2001) Universal behavior of load distribution in scale-free networks. Phys Rev Lett 87:278701CrossRefGoogle Scholar
  28. Goodman AC (1988) An econometric model in housing price, permanent income, tenure choice and housing demand. J Urban Econ 23:327–353CrossRefGoogle Scholar
  29. Graham RL, Hell P (1985) On the history of the minimum spanning tree problem. Ann Hist Comput 7:43–57CrossRefGoogle Scholar
  30. Itzykson C, Drouffe J-M (1989) Statistical field theory, vol 2. Cambridge University Press, CambridgeGoogle Scholar
  31. Jaromczyk JW, Toussaint GT (1992) Relative neighborhood graphs and their relatives. Proc IEEE 80:1502CrossRefGoogle Scholar
  32. Jensen P (2006) Network-based predictions of retail store commercial categories and optimal locations. Phys Rev E (R) 74:035101CrossRefGoogle Scholar
  33. Jiang B, Claramunt C (2004) Topological analysis of urban street networks. Environ Plan B 31:151–162CrossRefGoogle Scholar
  34. Kalapala V, Sanwalani V, Clauset A, Moore C (2006) Scale invariance in road networks. Phys Rev. E 73:026130CrossRefGoogle Scholar
  35. Lammer S, Gehlsen B, Helbing D (2006) Scaling laws in the spatial structure of urban road networks. Phys A 363:89CrossRefGoogle Scholar
  36. Levinson D (2008) Density and dispersion: the co-development of land use and rail in London. J Econ Geogr 8:55–77CrossRefGoogle Scholar
  37. Levinson D, Yerra B (2006) Self-organization of surface transportation networks. Transp Sci 40:179–188CrossRefGoogle Scholar
  38. Makse H, Andrade JS, Batty M, Havlin S, Stanley HE (1998) Modeling urban growth patterns with correlated percolation. Phys Rev E 58:7054CrossRefGoogle Scholar
  39. Makse HA, Havlin S, Stanley HE (2002) Modelling urban growth patterns. Nature 377:608CrossRefGoogle Scholar
  40. Maritan A, Colaiori F, Flammini A, Cieplak M, Banavar JR (1996) Universality classes of optimal channel networks. Science 272:984–986CrossRefGoogle Scholar
  41. Porta S, Crucitti P, Latora V (2006) The network analysis of urban streets: a primal approach. Environ Plan B 33:705CrossRefGoogle Scholar
  42. Price ND, Reed JL, Palsson BO (2004) Genome-scale models of microbial cells: evaluating the consequences of constraints. Nat Rev Microbiol 2:886–897CrossRefGoogle Scholar
  43. Rodriguez-Iturbe I, Rinaldo A (1997) Fractal river basins: chance and self-organization. Cambridge University Press, CambridgeGoogle Scholar
  44. Roswall M, Trusina A, Minnhagen P, Sneppen K (2005) Networks and cities: an information perspective. arXiv:cond-mat/0407054Google Scholar
  45. Runions A, Fuhrer AM, Lane B, Federl P, Rolland-Lagan A-G, Prusinkiewicz P (2005) Modeling and visualization of leaf venation patterns. ACM Trans Graph 24(3):702–711CrossRefGoogle Scholar
  46. Samaniego H, Moses ME (2007) Cities as organisms: allometric scaling as an optimization model to assess road networks in the USA. Presented at the Access to Destinations II Conference, Minneapolis, August 2007Google Scholar
  47. Scellato S, Cardillo A, Latora V, Porta S (2006) The backbone of a city. arXiv:physics/0511063Google Scholar
  48. Schwartz M (1986) Telecommunication networks: protocols, modelling and analysis. Addison-Wesley Longman, BostonGoogle Scholar
  49. Stevens PS (1974) Patterns in nature. Little, Brown, BostonGoogle Scholar
  50. Toussaint GT (1980) The relative neighborhood graph of a finite planar set. Pattern Recogn 12:261CrossRefGoogle Scholar
  51. UN Population Division (2008) UN Population Division homepage.
  52. von Thünen JH (1966) Von Thünen’s isolated state. Pergmanon, OxfordGoogle Scholar
  53. West GB, Brown JH, Enquist BJ (1997) A general model for the origin of allometric scaling laws in biology. Science 276:122CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Département de Physique Théorique et AppliquéeCEA-Centre d’Etudes de Bruyères-le-ChâtelBruyères-Le-ChâtelFrance
  2. 2.Centre d’Analyse et Mathématique Sociales (CAMS, UMR 8557 CNRS-EHESS)Ecole des Hautes Etudes en Sciences SocialesParis Cedex 06France
  3. 3.School of InformaticsIndiana UniversityBloomingtonUSA

Personalised recommendations