This document provides an overview of spatial network theory and applications. It discusses how space impacts network structure and introduces several tools for analyzing spatial networks. These include indices to characterize street and transportation networks, typologies of planar graphs based on block shape and area statistics, and methods for studying the time evolution of networks using old map digitization. Specific examples analyzed include road, power grid, airline and neural networks.

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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

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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

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(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)

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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

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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 …

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Transportation networks
Transporting energy, goods or individuals
- formation and evolution
- congestion, optimization, robustness
- disease spread, urban evolution

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Transporting water
Nodes: intersections, auxins
sources"
Links: veins !
!
Example of a "
planar network"

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Transporting goods
State of Indiana (Bureau of Transportation
statistics)

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Nodes: power
plants, transformers,
etc,…)"
Links: cables"
Transporting electricity
New York state power grid"

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Transporting electricity
US power grid"

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Transporting gas
European pipelines"

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Transporting individuals: urban scale

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Intra urban movements: streets and roads

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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

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Planar networks
Most of the interesting spatial networks
are planar

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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

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Planar networks
n Other consequence:
n Important for null models ! Don’t compare to
Emax=N(N-1)/2

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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

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Irrelevant measures

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Uninteresting measures
n Average degree
n Degree distribution
n Clustering and assortativity properties
n Edge length distribution

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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

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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

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n In general:
Spatial networks are usually more clustered (>> 1/N)
n In general
Almost flat assortativity ( )
Clustering and assortativity properties "

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North American
Airline network
Clustering and assortativity properties "
Barrat et al, 2005

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Power grid
(Western US)
Clustering"
Ravasz et al, 2003

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Internet
(router level)
Clustering"
Ravasz et al, 2003
C(k)=const.
(BA: C(k)=1/k)]

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Empirical fact: Length of links
n Edge link limited
by cost
Gastner et Newman EPJ (2006)
Barrat, MB, Vespignani, JSTAT (2005)

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Empirical fact: Length of links
n London street length
Masucci et al., 2009

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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 ?

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Interesting tools
- Various indicators
- Typology of street patterns
- Simplicity
- Time evolution of networks

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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

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Typology:
“classification” of
planar graphs

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Typology of planar graphs
Many applications:
n Botanics (classification of leaves)
n Urban morphology: street network (“Space syntax”)

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Typology of
street networks
S. Marshall (2005)

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Typology of
street networks
Classification
Attempt
S. Marshall (2005)

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“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)

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Typology of planar graphs
From the graph to the statistics of blocks

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Lammer et al, Physica A (2006)
Shape of blocks
Lammer et al, Physica A (2006)

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- Lammer et al, Physica A (2006)
- MB, Physics Reports (2011)
Area of blocks
⌧ ' 2.0

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Broad block area distribution
n Area of blocks
n Simple argument: density fluctuations
n Assumption: density random

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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)

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Combining shape and area of blocks: a
“fingerprint” of planar graphs:
Louf & MB, RS Interface (2014)
P( |A)

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A “fingerprint” of planar graphs
Tokyo
NYC

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Typology of street patterns
Louf & MB, RS Interface (2014)
n Distance constructed on
n Clustering -> classes of planar graphs (131 cities)
P( |A)

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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)

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Time evolution:
Characterization ?
Too many things to measure…

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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

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Road network evolution
Groane region, Italy 1833-2007
Strano, Nicosia, Latora, Porta, MB, Nature Scientific Reports (2012)

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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)

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Road network evolution
Central Paris, France 1789-2010
MB, Bordin, Berestycki, Gribaudi (2013)
1789

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Road network evolution
Central Paris, France 1789-2010
MB, Bordin, Berestycki, Gribaudi (2013)
1826

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Road network evolution
Central Paris, France 1789-2010
MB, Bordin, Berestycki, Gribaudi (2013)
1836

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Road network evolution
Central Paris, France 1789-2010
MB, Bordin, Berestycki, Gribaudi (2013)
1888

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Road network evolution
Central Paris, France 1789-2010
MB, Bordin, Berestycki, Gribaudi (2013)
1999

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Road network
China
1600(BC)- 1900 (AC)
Wang, Ducruet,
Wang (2015)

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1. Simple measures

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Road network evolution (Groane region, Italy)

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Importance of central planning
n N follows the population evolution
n The “good” clock here is the number of nodes

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Time evolution (Paris 1789-2010)
n Standard indicators versus time or N

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Lammer et al, Physica A (2006)
Faces (blocks): shape and area
P(A) ⇠?

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Haussmann effect: shape factor

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Haussmann effect: angle distribution

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Road network evolution

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2. The betweenness
centrality

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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

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Betweenness centrality and space
Lammer et al, 2006

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Betweenness centrality
n Backbone of stable central roads

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Haussmann effect
n Spatial distribution of centrality (most central nodes)

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Characterization of new links: BC impact
n Average BC of the graph at time t:
n BC impact of new edge e*:

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Evolution: two processes
n Two different categories of new links: ‘densification’ and
‘exploration’ clearly identified by the BC impact

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3. The simplicity

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!
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)

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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)

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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

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Lammer et al, Physica A (2006)
Simplicity
Spectrum
Viana, Strano,
Bordin, MB
Scientific reports
(2013)
Length of simplest path

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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)

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4. Template:
the subway case
Too many things to measure: a template as
a guide

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The subway evolution: not a new problem
n Cope (1967): Stages of the London underground rail

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All large cities have a subway system

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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)

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“Universal” template
Algorithm to
identify the
core and
the branches
(non-ambiguous)

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Measures on this universal structure
n Characterizing the core
NC: number of nodes in the core
EC: number of links in the core

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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

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Evolution
fraction of
branches
stations

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Average degree
Percentage f2

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Spatial extension of branches

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“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%

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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 ?

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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

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Spatial organization of the core and branches
n Interstation spacing at distance r
n Natural explanation of the Benguigui-Daoud
result

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Number of branches
n If the spacing between two branches is constant:
n For a lattice of size N

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Number of branches

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“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 (?)!
!

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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)

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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

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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

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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

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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 !!!

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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 ?

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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