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