Spatial network, Theory and applications - Marc Barthelemy

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

[email protected] 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

Nodes: airports Links: direct flight"

Transporting individuals: global scale "(air travel)

Neural Network Nodes: neurons "Links: axons !

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)

Power grids

Sole et al (2008)

Map Topological network

Degree distribution Europe, Italy, UK &Ireland

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

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  …

Specific tools for spatial networks

§ Indices alpha, gamma, etc…


§  Indice for street networks

§  If r close to 1: ‘organic’ city §  Small r<<1: organized city (N(4) dominant)


§  Indice for transportation networks

§  Q=1: straight line §  Q>>1: large detour (see D. Aldous)






§  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

Lake Como 2016 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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1826

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1836

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1888

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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


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

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


Simplicity Spectrum


Length of simplest path

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10 Km

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.41833191419131955198019942007

S(d)

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

1.5

17891826183618881999

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 11

1.1

1.2

1.3

1.4h04h08h10h15h20

d/dmax

S(d)


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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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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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n  Natural explanation with the universal template

: core density Nb : number of branches : Interstation spacing at distance r

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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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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 ?

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

Hotspots: local maxima of density

City structure (mono- vs. polycentric)

Aire urbaine de Zaragoza Aire urbaine de Bilbao

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

0.1

0.2

0.3

0.4

0.5

106P

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 [email protected]

Spatial network, Theory and applications - Marc Barthelemy

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