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- 1. Lake Como 2016 Spatial network Theory and applications 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
- 2. Lake Como 2016 Outline n I. Introduction: space and networks n II. Tools q Irrelevant tools q Interesting tools n Typology (street patterns) n Simplicity n Time evolution (Streets, subway) n Mobility networks: Extracting mesoscale information n III. Some models q “Standard” models n Random geometric graph, tessellations n Optimal networks q “Non-standard” n Road networks q Scaling theory n Subway and railways
- 3. Lake Como 2016 (Short) Bibliography n Review article and books q Spatial Networks, MB, Physics Reports 499:1-101 (2011) (and arXiv) q Morphogenesis of spatial networks, MB, Springer, to appear (2017) n Books and articles (more mathematical) q Spatial tessellations, A Okabe (2009) q Papers by D. Aldous n Older (but excellent) material: q Models in geography, Chorley and Haggett q Network analysis in geography, Haggett and Chorley n More specialized (urban systems and transportation) q The geography of transport systems, Rodrigue, Comtois, Slack q Geography of transportation, Taaffe and Gauthier q Planning for place and plexus, Levinson and Krizek q A new science of cities, M Batty (2013) q The Structure and dynamics of cities, MB, Cambridge Univ Press, to appear (2016)
- 4. Lake Como 2016 Space and networks n Network science: space is not important n But for many networks, space does matter ! q Biological (neural networks, veins) q Technological (power grids, Internet) q Transportation (roads, railways, airline network) ⇒ Nodes in space Questions: - Effect of space on the traffic and on the shape of (transportation) networks - Principles of formation/evolution of spatial networks
- 5. Lake Como 2016 Spatial networks n A network is ‘spatial’ if q The nodes are located in space q The network is embedded in a space: There is a ‘cost’ associated to the length of links: a long link must be compensated by something else (large degree, traffic, etc.) n Adjacency matrix (NxN) + the position of nodes q Very rich object q Time evolution: difficult problem n Many examples q Transportation systems q Power grids q Social networks q …
- 6. Lake Como 2016 Transportation networks Transporting energy, goods or individuals - formation and evolution - congestion, optimization, robustness - disease spread, urban evolution
- 7. Lake Como 2016 Transporting water Nodes: intersections, auxins sources" Links: veins ! ! Example of a " planar network"
- 8. Lake Como 2016 Transporting goods State of Indiana (Bureau of Transportation statistics)
- 9. Lake Como 2016 Nodes: power plants, transformers, etc,…)" Links: cables" Transporting electricity New York state power grid"
- 10. Lake Como 2016 Transporting electricity US power grid"
- 11. Lake Como 2016 Transporting gas European pipelines"
- 12. Lake Como 2016 Transporting individuals: urban scale
- 13. Lake Como 2016 Intra urban movements: streets and roads
- 14. Lake Como 2016 TRANSIMS project" Nodes: locations (homes, shops, ofﬁces, …)" Links: ﬂow of individuals" Chowell et al Phys. Rev. E (2003) Nature (2004) Transporting individuals: intra city" Mobility networks
- 15. Nodes: airports Links: direct ﬂight" Transporting individuals: global scale " (air travel)
- 16. Neural Network Nodes: neurons " Links: axons !
- 17. Social Networks and space Nodes: individuals " Links: ‘relation’! Distance distribution of your friends ? P(r) ⇠ 1 r
- 18. Lake Como 2016 Planar networks Most of the interesting spatial networks are planar
- 19. Lake Como 2016 Planar networks Graph G=(V,E) and can be represented in the plane without intersections n Does not contain K5 or K3,3 n Euler’s formula: n Any face has at least 3 edges and every edge touches at most two faces (2E≥3F): Exact bound
- 20. Lake Como 2016 Planar networks n Other consequence: n Important for null models ! Don’t compare to Emax=N(N-1)/2
- 21. Lake Como 2016 n A graph has many representations (in 2d) n But it can be drawn in 2d in such a way that no edges cross each other: planar embedding, plane graph, planar map n The planar map is the well-defined object for mathematicians -used for the four color theorem for example and in combinatorics (Tutte, Schaeffer, …) Planar maps
- 22. Lake Como 2016 Irrelevant measures
- 23. Lake Como 2016 Uninteresting measures n Average degree n Degree distribution n Clustering and assortativity properties n Edge length distribution
- 24. Lake Como 2016 Degree distribution P(k) n Physical constraints limit the degree (planar networks: <k> <6) n Road network and commuters in sardinia Lammer et al, Physica A (2006) De Montis et al, Env Plan B (2007)
- 25. Power grids Sole et al (2008) Map Topological network Degree distribution Europe, Italy, UK &Ireland
- 26. Power grids Albert et al (2004) North America
- 27. Lake Como 2016 Airline network n For spatial non-planar networks, we can have large degrees (worldwide airline network, inter-urban) Barrat, MB, Pastor-Satorras, Vespignani (2005) Airline network
- 28. Lake Como 2016 n In general: Spatial networks are usually more clustered (>> 1/N) n In general Almost ﬂat assortativity ( ) Clustering and assortativity properties "
- 29. Lake Como 2016 North American Airline network Clustering and assortativity properties " Barrat et al, 2005
- 30. Lake Como 2016 Power grid (Western US) Clustering" Ravasz et al, 2003
- 31. Lake Como 2016 Internet (router level) Clustering" Ravasz et al, 2003 C(k)=const. (BA: C(k)=1/k)]
- 32. Lake Como 2016 Empirical fact: Length of links n Edge link limited by cost Gastner et Newman EPJ (2006) Barrat, MB, Vespignani, JSTAT (2005)
- 33. Lake Como 2016 Empirical fact: Length of links n London street length Masucci et al., 2009
- 34. Lake Como 2016 Road networks: Total length Data from: Cardillo et al, PRE 2006 Total length vs. N
- 35. Argument for the total length n At this point, a perturbed lattice is a good model: q Scaling of length with N: q Peaked distribution of degrees: n Cell area and shape ?
- 36. Lake Como 2016 Interesting tools - Various indicators - Typology of street patterns - Simplicity - Time evolution of networks
- 37. Lake Como 2016 Space has an important impact on the structure Low information content measures: n Generally planar (nodes, edges, faces or cells) n Degree distribution: peaked (if planar) n Clustering: large n Assortativity spectrum: flat n Edge length distribution: peaked n Total length: N1/2 More interesting: n Geometry statistics of cells (for planar graphs) n Betweenness centrality properties n Communities n Traffic, correlations: degree-distance, degree-traffic n …
- 38. Specific tools for spatial networks § Indices alpha, gamma, etc…
- 39. Specific tools for spatial networks § Indice for street networks § If r close to 1: ‘organic’ city § Small r<<1: organized city (N(4) dominant)
- 40. Specific tools for spatial networks § Indice for transportation networks § Q=1: straight line § Q>>1: large detour (see D. Aldous)
- 41. Specific tools for spatial networks § Indice for transportation networks
- 42. Specific tools for spatial networks § Indice for transportation networks
- 43. Specific tools for spatial networks § An interesting null model: the minimum spanning tree (MST) § Cost and efficiency
- 44. Lake Como 2016 Typology: “classification” of planar graphs
- 45. Lake Como 2016 Typology of planar graphs Many applications: n Botanics (classification of leaves) n Urban morphology: street network (“Space syntax”)
- 46. Lake Como 2016 Typology of street networks S. Marshall (2005)
- 47. Lake Como 2016 Typology of street networks Classification Attempt S. Marshall (2005)
- 48. Lake Como 2016 “Census of planar graphs” n BDG bijection between a rooted map and a tree (Bouttier, Di Francesco, Guitter, Electron J Combin, 2009) n Approximate tree representation of a weighted planar map (Mileyko et al, PLoS One, 2012 Katifori et al, PLoS One 2012)
- 49. Lake Como 2016 Typology of planar graphs From the graph to the statistics of blocks
- 50. Lake Como 2016 Lammer et al, Physica A (2006) Shape of blocks Lammer et al, Physica A (2006)
- 51. Lake Como 2016 - Lammer et al, Physica A (2006) - MB, Physics Reports (2011) Area of blocks ⌧ ' 2.0
- 52. Lake Como 2016 Broad block area distribution n Area of blocks n Simple argument: density fluctuations n Assumption: density random
- 53. Lake Como 2016 Shape versus area n But even with the same P(Φ) the networks can be very different ! Importance of P(A) n The visual impression of a map is given by both the distribution of shapes and areas n We thus use P(Φ|A) and bin the area (small, medium, large)
- 54. Lake Como 2016 Combining shape and area of blocks: a “fingerprint” of planar graphs: Louf & MB, RS Interface (2014) P( |A)
- 55. Lake Como 2016 A “fingerprint” of planar graphs Tokyo NYC
- 56. Lake Como 2016 Typology of street patterns Louf & MB, RS Interface (2014) n Distance constructed on n Clustering -> classes of planar graphs (131 cities) P( |A)
- 57. Lake Como 2016 Not the end of the story n Attempt to classify planar maps n Taking into account both topological and geometrical features n Correlations ? (existence of neighborhoods)
- 58. Lake Como 2016 Time evolution: Characterization ? Too many things to measure…
- 59. Lake Como 2016 An old problem in quantitative geography n Kansky (63-69) Evolution of the Sicilian railroad network n Morrill (1965) Railway network Growth n New data sources: Digitization of old maps
- 60. Lake Como 2016 Road network evolution Groane region, Italy 1833-2007 Strano, Nicosia, Latora, Porta, MB, Nature Scientific Reports (2012)
- 61. Lake Como 2016 Road network evolution: Importance of central planning n Evolution of the Paris street network 1789-2010 (1789, 1826, 1836, 1888, 1999, 2010-soon 1591, 1652, 1728) n Haussmann period (~1853-1870)
- 62. Lake Como 2016 Road network evolution Central Paris, France 1789-2010 MB, Bordin, Berestycki, Gribaudi (2013) 1789
- 63. Lake Como 2016 Road network evolution Central Paris, France 1789-2010 MB, Bordin, Berestycki, Gribaudi (2013) 1826
- 64. Lake Como 2016 Road network evolution Central Paris, France 1789-2010 MB, Bordin, Berestycki, Gribaudi (2013) 1836
- 65. Lake Como 2016 Road network evolution Central Paris, France 1789-2010 MB, Bordin, Berestycki, Gribaudi (2013) 1888
- 66. Lake Como 2016 Road network evolution Central Paris, France 1789-2010 MB, Bordin, Berestycki, Gribaudi (2013) 1999
- 67. Lake Como 2016 Road network China 1600(BC)- 1900 (AC) Wang, Ducruet, Wang (2015)
- 68. Lake Como 2016 1. Simple measures
- 69. Lake Como 2016 Road network evolution (Groane region, Italy)
- 70. Lake Como 2016 Importance of central planning n N follows the population evolution n The “good” clock here is the number of nodes
- 71. Lake Como 2016 Time evolution (Paris 1789-2010) n Standard indicators versus time or N
- 72. Lake Como 2016 Lammer et al, Physica A (2006) Faces (blocks): shape and area P(A) ⇠?
- 73. Lake Como 2016 Haussmann effect: shape factor
- 74. Lake Como 2016 Haussmann effect: angle distribution
- 75. Lake Como 2016 Road network evolution
- 76. Lake Como 2016 2. The betweenness centrality
- 77. Lake Como 2016 More interesting: Betweenness Centrality (Freeman ‘77) σst = # of shortest paths from s to t σst(ij)= # of shortest paths from s to t via (ij) i j k ij: large centrality jk: small centrality Measures the importance of a segment in the shortest paths flow
- 78. Betweenness centrality and space Large BC: distance to barycenter Large BC: large degree
- 79. Lake Como 2016 Betweenness centrality and space Lammer et al, 2006
- 80. Lake Como 2016 Betweenness centrality n Backbone of stable central roads
- 81. Lake Como 2016 Haussmann effect n Spatial distribution of centrality (most central nodes)
- 82. Lake Como 2016 Characterization of new links: BC impact n Average BC of the graph at time t: n BC impact of new edge e*:
- 83. Lake Como 2016 Evolution: two processes n Two different categories of new links: ‘densification’ and ‘exploration’ clearly identified by the BC impact
- 84. Lake Como 2016 3. The simplicity
- 85. Lake Como 2016 ! Statistical comparison of the length of shortest and simplest paths (with the minimal number of turns) Another measure: Simplicity of planar networks MP. Viana, E. Strano, P. Bordin, MB (Sci. Rep. 2013)
- 86. Lake Como 2016 Perspective: the Simplicity of planar netwoks ! n Statistical comparison of the length of shortest and simplest paths MP. Viana, E. Strano, P. Bordin, MB (2014)
- 87. Lake Como 2016 Simplicity of paths Viana, Strano, Bordin, MB Scientific reports (2013) S(d) = 1 N(d) X i,j/d(i,j)=d `⇤ (i, j) `(i, j) `(i, j) `⇤ (i, j) Length of shortest path Length of simplest path For small d: and increases For large d: ⇒ There is a (at least one) maximum at d=d* Meaning of d*: typical size of ‘domains’ not crossed by long straight lines S(d ! 0) ⇡ 1 S(d ! dmax) ⇡ 1
- 88. Lake Como 2016 Lammer et al, Physica A (2006) Simplicity Spectrum Viana, Strano, Bordin, MB Scientific reports (2013) Length of simplest path
- 89. Lake Como 2016 10 Km 0 0.2 0.4 0.6 0.8 1 1 1.1 1.2 1.3 1.4 1833 1914 1913 1955 1980 1994 2007 S(d) 0 0.2 0.4 0.6 0.8 1 1 1.1 1.2 1.3 1.4 1.5 1789 1826 1836 1888 1999 d/dmax S(d) 1955 1980 2007 ¯Groane evolving street network 4 KmParis evolving street network 2 CmPhysarum evolving vascular network a b c d/dmax 1789 h 4 h 8 h 10 h 15 h 20 19991836 0 0.2 0.4 0.6 0.8 1 1 1.1 1.2 1.3 1.4 h04 h08 h10 h15 h20 d/dmax S(d) Viana, Strano, Bordin, MB Scientific reports (2013)
- 90. Lake Como 2016 4. Template: the subway case Too many things to measure: a template as a guide
- 91. Lake Como 2016 The subway evolution: not a new problem n Cope (1967): Stages of the London underground rail
- 92. Lake Como 2016 All large cities have a subway system
- 93. Lake Como 2016 World subway networks We focus on large networks (N>100 stations) Time evolving spatial networks: too many things to measure ! Most large cities have a subway network (50% for P>106)
- 94. Lake Como 2016
- 95. Lake Como 2016 “Universal” template Algorithm to identify the core and the branches (non-ambiguous)
- 96. Lake Como 2016 Measures on this universal structure n Characterizing the core NC: number of nodes in the core EC: number of links in the core
- 97. Lake Como 2016 Measures on this universal structure n Characterizing the branches NB: number of stations in branches NC: number of stations in the core DB: average distance from barycenter to branches stations DC: average distance from barycenter to core stations
- 98. Lake Como 2016 Evolution fraction of branches stations
- 99. Lake Como 2016 Average degree Percentage f2
- 100. Lake Como 2016 Spatial extension of branches
- 101. Lake Como 2016 “Universal” template n Quantitative convergence q Fraction of branches stations of order 50% q Extension of branches/core extension of order 2 q Average degree of core of order 2.5 and f2>60%
- 102. Lake Como 2016 Spatial organization of the core and branches n Old result for Paris (Benguigui, Daoud 1991) N(r): number of stations at distance less than r from barycenter First regime: homogeneous distribution with df=2 Second regime ?
- 103. Lake Como 2016 Spatial organization of the core and branches n Natural explanation with the universal template : core density Nb : number of branches : Interstation spacing at distance r
- 104. Lake Como 2016 Spatial organization of the core and branches n Interstation spacing at distance r n Natural explanation of the Benguigui-Daoud result
- 105. Lake Como 2016 Number of branches n If the spacing between two branches is constant: n For a lattice of size N
- 106. Lake Como 2016 Number of branches
- 107. Lake Como 2016 “Universal” template n Quantitative convergence! q Fraction of branches ! "stations of order 50%! q Extension of branches/core ! "extension of order 2! q Average degree ! "of core of order 2.5 ! "and f2>60%! ! q Number of branches! "! ! ! => Existence of a minimal model (?)! !
- 108. Lake Como 2016 5. Extracting mesoscale information from mobile phone data How can we get meaningful information from large dasets ?
- 109. Typology of mobility patterns (journey to work trips) Motivation: Compare the spatial structure of mobility patterns in many cities Question: How to build a quantitative typology of cities based on the spatial structure of the mobility patterns ? (Bertaud & Malpezzi 2003)
- 110. Lake Como 2016 How to compare OD commuting matrices of different cities? § The OD matrix is a large and complicated object § Difficult to compare different cities ! - Different sizes - Potentially different spatial resolutions § We need a simpler, clearer picture: coarse-grained information Fij i j
- 111. Hotspots: local maxima of density City structure (mono- vs. polycentric) Aire urbaine de Zaragoza Aire urbaine de Bilbao
- 112. ENPC-2016 Hotspot identification n State of the art q No clear method q Density larger than a given threshold is a hotspot q Problem of the threshold choice n A simple approach q Discussion on the Lorentz curve q Identify a lower and upper threshold Louail, et al, Sci. Rep. 2014 ⇢1 < ⇢2 < · · · < ⇢N
- 113. Lake Como 2016 How to compare OD commuting matrices of different cities? 1. Determine Residential and work hotspots (Louail et al, 2014) 2. Separate 4 categories of flows: I, C, D, R Integrated: Hotspot->Hotspot Convergent: Non hotspot->hotspot Divergent: Hotspot->non hotspot Random: non hotspot->non hotspot Louail, et al, Nature Comms 2015
- 114. 0.1 0.2 0.3 0.4 0.5 106 P Flows I C D RThe importance of Integrated flows decreases when population size increases, in favor of an increase of “Random” flows Weights of Divergent and Convergent flows are constant I and R alone seem enough to characterize cities Structure of flows versus population (30 largest urban areas in Spain) Louail, et al, Nature Comms 2015
- 115. Lake Como 2016 Structure des flots (Espagne) Vient des possibilité plus grandes dans les grandes villes de se deplacer (?) Structure spatiale “délocalisée” des grandes villes Cordoba Gijon Vitoria Zaragosa Malaga Valencia Sevilla Madrid Barcelona R 27% 36% 41% 46% I 43% 37% 31% 25% Population
- 116. Lake Como 2016 Numbers of hotspots vs. population size of the city Exponent value is remarkably smaller for work/school/daily activity hotspots à in Spanish urban areas, the number of activity places grows slower than the number of major residential places. Sublinear in both cases !!!
- 117. Lake Como 2016 Discussion n New results on new datasets usually imply to have new tools ! n Typology q Attempt to classify planar maps q Taking into account both topological and geometrical features q Correlations ? n Evolution of planar graphs q Simple measures usually not very helpful q Important structural changes: betweenness centrality distribution q Use of templates q Better characterization ? q Models ?
- 118. Lake Como 2016 Thank you for your attention. Students and Postdocs: Giulia Carra (PhD student) Riccardo Gallotti (Postdoc) Thomas Louail (Postdoc) Remi Louf (PhD student) Emanuele Strano (PhD student) Collaborators: M. Batty H. Berestycki P. Bordin S. Dobson M. Gribaudi P. Jensen JP. Nadal V. Nicosia V. Latora J. Perret S. Porta C. Roth S. Shay MP. Viana Funding: EUNOIA (FP7-DG.Connect-318367 European Commission) PLEXMATH (FP7-ICT-2011-8 European Commission) www.quanturb.com marc.barthelemy@cea.fr