A dynamical transition in urban systems Marc Barthelemy CEA, Institut de Physique Théorique, Saclay, France EHESS, Centre d ’ Analyse et de Mathématiques sociales, Paris, France marc.barthelemy@cea.fr http://www.quanturb.com Co CoSyDy Lo London 2016
Outline n Urban science: state of the art n Polycentricity: empirical results n Modeling: from urban economics to statistical physics q Krugman’s model q The Fujita-Ogawa model q A physicist variant n Discussion and perspectives Co CoSyDy Lo London 2016
Importance of cities: urbanization rate 60% 50% 40% Urban rate 30% 20% 10% 0% 1500 1600 1700 1800 1900 2000 2100 Year Projection: in 2050: 70% of the world population lives in cities Data from: HYDE historical database CoSyDy Lo Co London 2016
Importance of cities City Population Growth Rate 500 - 750 thousand <1% 750 - 1000 thousand 1-3% 1-5 million 3-5% 5-10 million 5% + 10 million or more Heterogeneous distribution of growth rates Co CoSyDy Lo London 2016
Many ‘theories’ of urbanism but nevertheless, we observe a large number of problems ! n Social and economical problems (spatial income segregation, crime, accessibility, …) n Traffic problems; pollution n Sustainability of these structures ? => Necessity of understanding these phenomena and to achieve a science of cities and quantitative urbanism validated by data (in particular, for large-scale projects) Co CoSyDy Lo London 2016
Science and cities: state of the art Number of Nu pa parameters Urban economics: Complex Very abstract simulations models, empirical (LUTI models): tests ? Validity ? Large perturbation ? Minimal model: the smallest number of parameters and the largest number of verified predictions Loop: theory-empirical data n Open problem: Existence of (phase) transition in urban systems ?? CoSyDy Lo Co London 2016
Towards a (new) science of cities n Game changer ? Always more data about cities ! n Different scales, different phenomena Scaling Evolution of networks - Spatial structure (with population) (roads and transportation) of cities (polycenters) - Mobility patterns (congestion, commuting, …) Co CoSyDy Lo London 2016
Spatial structure of cities n Theoretical framework (Alonso-Muth-Mills): - - Monocentric organization: One center (the central business district) - The population density is decreasing with r (exact form depends on the utility !) Co CoSyDy Lo London 2016
I. Polycentric structure: empirical results Co CoSyDy Lo London 2016
Polycentric structure n Activity centers (# of employees per zip code, USA) San Antonio (TX), USA Winter Haven (FL), USA n In general: existence of local maxima (‘hotspots’) of the density Co CoSyDy Lo London 2016
Local maxima identification n State of the art q No clear method q Density larger than a given threshold is a hotspot q Problem of the ρ i threshold choice ? ρ i > ρ c ⇒ i is a Hotspot Louail, et al, Sci. Rep. 2014 Co CoSyDy Lo London 2016
Local maxima identification n Our proposal q Discussion on the Lorentz curve q Identify a lower and upper threshold F average ( → ρ ) Louail, et al, Sci. Rep. 2014 Co CoSyDy Lo London 2016
Scaling for the number of centers n We can count the number of hotspots (employment density data) n The fit (9000 US cities, 1994-2010) gives n Sublinear ! Co CoSyDy Lo London 2016
Mobile phone data: urban structures Zaragoza Bilbao Co CoSyDy Lo London 2016
Scaling for the number of centers (Urban areas -Spain) Hotspots for residence density and ’activity’ density Exponent value is smaller for work/school/daily activity hotspots à The number of activity places grows slower than the number of major residential places. CoSyDy Lo Co London 2016
Summary: empirical results n We have a polycentric structure, evolving with P n We can count the number H of centers H ∼ P β β ≈ 0 . 5 − 0 . 6 n Mobility is the key: we need to model how individuals choose their home and work place n Problem largely studied in geography, and in spatial economics: Edge City model (Krugman 1996), Fujita-Ogawa model (1982) n Revisiting Fujita-Ogawa: predicting the value of β Co CoSyDy Lo London 2016
II. Polycentric structure: Urban economics modeling Co CoSyDy Lo London 2016
Naive scaling: Total commuting distance ρ = P/A area A Monocentric Nearest neighbor √ A √ ` 1 ∼ 1 / √ ⇢ ∼ ` 1 ∼ A √ P √ √ L tot / A ∼ P A ∼ P 1 / 2 L tot / n We obtain L tot ∼ P β β ∈ [0 . 5 , 1] √ A β ' 0 . 66 (Samaniego, Moses, 2008) CoSyDy Lo Co London 2016
What is wrong with the naive scaling q Assume k secondary centers: area A A = kA 1 A 1 A 1 L tot = k P Total commuting length: p A 1 A 1 k ⇒ L tot = P √ √ A k q Can change scaling exponents if k varies with P ! q We have to understand the polycentric structure of cities CoSyDy Lo Co London 2016
Spatial economics: the edge city model (Krugman 1996) n The important ingredient is the ‘market potential’ R Π ( x ) = K ( x − z ) ρ B ( z )d z n Describes the spillovers due to the business density in z n Specifically K ( x ) = K + ( x ) − K − ( x ) n The average market potential is Π = 1 R Π ( x ) ρ B ( x )d x A CoSyDy Lo Co London 2016
Spatial economics: the edge city model (Krugman 1996) n The equation for the evolution of business density is d ρ B ( x,t ) � � = γ Π ( x, t ) − Π d t n Linearize around flat situation ρ B ( x ) = ρ 0 + δρ B ( x ) ρ B ( k ) ∼ e γ ˜ K ( k ) t δ ˜ n At least one maximum at k=k*; the number of hotspots is then: H ∼ Ak ∗ 2 n Scaling with the population ? Individual’s choices ? CoSyDy Lo Co London 2016
Spatial economics: Fujita-Ogawa (1982) n A model for the spatial structure of cities: an agent will choose to live in x and work in y such that Z 0 ( x, y ) = W ( y ) − C R ( x ) − C T ( x, y ) is maximum Home x - W(y) is the wage (‘attractiveness’) at y Office y - C R (x) is the rent at x - C T (x,y) is the transportation cost from x to y C T ( x, y ) = td ( x, y ) Co CoSyDy Lo London 2016
Spatial economics: Fujita-Ogawa (1982) n And a similar equation for companies (maximum profit) P ( y ) = Π ( y ) − C R ( y ) − L ( y ) W ( y ) - W(y) is the wage at y - C R (y) is the rent at y Home i - L(y) number of workers (N=ML 0 ) - is the benefit to come to y: Π ( y ) Office j Agglomeration effect ! (market potential) R Π ( y ) = K ( y − z ) ρ B ( z )d z K ( u ) = k e − α | u | CoSyDy Lo Co London 2016
Spatial economics: Fujita-Ogawa (1982) RA BD RA -x 1 -x 0 x 0 x 1 x 0 n Main result: monocentric configuration stable if t k ≤ α - t: transport cost - 1/ α interaction distance between firms n Effect of congestion: larger cost t CoSyDy Lo Co London 2016
Spatial economics: Fujita-Ogawa (1982) n This model is unable to predict the spatial structure and the number of activity centers…. n We have to simplify the problem ! Co CoSyDy Lo London 2016
Spatial economics: Fujita-Ogawa (1982) n There are many problems with this model: q Not dynamical: optimization. We want an out-of- equilibrium model q No congestion (!) We want to include congestion (for car traffic) q No empirical test. Extract testable predictions (see the book: Spatial Economics, by Fujita, Krugman, Venables) Co CoSyDy Lo London 2016
A physicist’s variant of Fujita-Ogawa n Assumptions and simplifications: q Assume that home is uniformly distributed (x): find a job ! Z 0 ( x, y ) = W ( y ) − C T ( x, y ) q We have now to discuss W and C T Co CoSyDy Lo London 2016
A physicist’s variant of Fujita-Ogawa n Assumptions and simplifications: q Add congestion (BPR function, t=cost/distance) and the generalized cost reads: ⌘ µ i h ⇣ T ( x,y ) C T ( x, y ) = td ( x, y ) 1 + c q Wages: a typical physicist assumption (s: typical salary) W ( y ) = s η ( y ) The ‘ attractivity ’ is random (in [0,1]) (cf. Random Matrix Theory) W can be seen as a the ‘quality’ of the job, encoding many factors CoSyDy Lo Co London 2016
Summary: the model n Every time step, add a new individual at a random i n The individual will choose to work in y (among N c possible centers) such that ⌘ µ i h ⇣ Z ( x, y ) = η ( y ) − d ( x,y ) T ( y ) 1 + ` c is maximum - W(y) is the wage at y --> random - C T (x,y) is the transportation cost from x to y: depends on the traffic from x to y --> congestion effects Lo Louf, MB, PRL L 2013 CoSyDy Lo Co London 2016
Results n Depending on the values of parameters, we see three type of mobility patterns: 1. Monocentric: one activity center Co CoSyDy Lo London 2016
Results n Depending on the values of parameters, we see three type of mobility patterns: 2. Attractivitydriven polycentrism: many activity centers, attractivity dominates η Co CoSyDy Lo London 2016
Results n Depending on the values of parameters, we see three type of mobility patterns: 3. Spatial polycentrism: many activity centers, basins spatially coherent Co CoSyDy Lo London 2016
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