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

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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 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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Models of spatial networks
n Large classes of ‘standard’ models
q 0. Tessellations
q 1. Geometric graphs (i and j connected if distance < threshold)
q 2. Spatial generalization of ER networks (hidden variables,
Waxman)
q 3. Spatial generalization of small-world (Watts-Strogatz) networks
q 4. Spatial growing networks (Barabasi-Albert)
q 6. Optimization (global and local)

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Some classical models
Could be useful null models

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Importance of models
n Choose the null model wisely
n Needs to satisfy constraints and should be ‘reasonable’
n MST, Voronoi tessellation, etc. Planar Erdos-Renyi ? (Masucci et al,
EPJ B 2009)

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Voronoi-Poisson tessellation
n Take N points randomly distributed
n Construct the
Voronoi tessellation
V (i) = {x|d(x, xi) < d(x, xj), 8j 6= j}

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Voronoi-Poisson tessellation
n Spatial dominance (Okabe): local centers
(1,20)
(2,18)
(3,15)
(4,3)
(5,7)
(6,11)
(7,1)
(8,16) (9,3) (10,15)
(11,5) (12,3)
(13,6)
(14,12)
(15,2)

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Voronoi-Poisson tessellation
n Spatial dominance (Okabe): local centers
(1,20)
(2,18)
(8,16)

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Voronoi-Poisson tessellation
n Spatial dominance (Okabe)
1
2 8

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Incidentally: 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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Random geometric graphs
n i and j connected if d(i,j)<R
n Large mathematical literature
n Continuum percolation: existence of a threshold
n Renewed interest: wireless ad hoc networks
q Existence of a giant component ?

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Random geometric graphs
Dall Christensen 2002

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Spatial generalization of Erdos-Renyi
n Erdos-Renyi random graph (1959)
n Spatial generalization
n Example the fitness model (Caldarelli et al, 2002)
F(x, y) = ✓(x + y z)
P(x) = e x
) P(k) ⇠
1
k2

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Spatial small-worlds
n Watts-Strogatz model (1998)
n Spatial generalization

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Spatial small-worlds
n Kleinberg’s result on navigability (Nature 2000)

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Growth models

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Models for growing scale-free graphs
§ Barabási and Albert, 1999: growth + preferential attachment
§ Generalizations and variations:
Non-linear preferential attachment : Π(k) ~ kα
Initial attractiveness : Π(k) ~ A+kα
Fitness model: Π(k) ~ ηiki
Inclusion of space
Redner et al. 2000, Mendes et al. 2000, Albert et al. 2000, Dorogovtsev et al. 2001,
Bianconi et al. 2001, MB 2003, etc...

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A model with spatial effects
• Growing network:
addition of nodes + distance
with:
Many other models possible, but essentially
one parameter η=d0/L : Effect of space
Interplay traffic-distance

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Optimal networks

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Optimal network design: hub-and-spoke
n Point-to-point vs. Hub-and-Spoke
…See paper by Morton O’Kelly

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Optimal network design: general theory
n Optimal network design: minimize the total cost
(usually for a fixed number of links)
Cost per user on
edge e
Traffic on e
on a given network

Global optimization: simple cases
Shortest path tree (SPT)

Global optimization: simple cases
Euclidean minimum spanning tree (MST)
Important null model: provides connection to all nodes at a
minimal cost
Average longest link (Penrose,97)
M ⇠
s
log ⇢
⇢

“Xmas” tree
Global optimization: simple cases

Optimal traffic tree (OTT)
Network which minimizes
the weighted shortest
Path
Global optimization: simple cases

Global optimization
n Resilience to attacks
to fluctuating load
Minimize the total
dissipation (total cost fixed):
where Pe is the total
power dissipation when
Edge e is cut
Corson, PRL 2010
Katifori, Szollosi, Magnasco, PRL 2010
R =
X
e
Pe
Pe
=
X
e06=e
C(e0
)(V (i) V (j))2
(
Istem = N
Ii6=stem = 1
X
e
C(e) = 1

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Optimization and growth

Local optimization
n Global optimization not very satisfying: limited time
horizon of urban planners; growing, out-of-equilibrium,
self-organizing cities
n However, locally, it is reasonable to assume that cost
minimization prevails

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§ Growing Networks+optimization: Fabrikant model
§ A new node i is added to the network such that
is minimum.
- large: EMST
- small: star network
Optimization and growth
Fabrikant et al, 2002

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§ Growing Networks+optimization
Optimization and growth
Gastner and Newman, 2006

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n Centers (homes, businesses, …) need to be
connected to the road network
n When a new center appears: how does the road
grow to connect to it ?
A simple model for the road/streets network
n We assume that the existing network creates a
‘potential’ V(x)
n Two main parameters: “freedom” and
“wealth” (number of connections)
P(x) ⇠ e V (x)

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A (very) simple model
n Algorithm
q (0) Generate initial seed of a few centers connected by
roads
q (1) Generate a center in the plane with proba P(x)
q (2) Grow the n (n depends on the wealth) roads from the
center to the existing network
q (3) back to (1)until N centers
MB and Flammini 2008, 2009; Courtat et al, 2010

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A (very) simple model
MB and Flammini 2008, 2009

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Illustration: presence of an obstacle
MB and Flammini 2008, 2009

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A simple model: Problem of the area distribution
q Empirically: the density decreases with the distance to the
center (Clark 1951) => Generate centers with exponential
distribution
Surprisingly good agreement !
MB and Flammini 2008, 2009

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n The new centers are not uniformly distributed:
economical factors
Co-evolution of the network and centers
q Choice of location (for a new home, business,…):
depends on many factors.
q We can focus on two factors: rent and transportation
costs
n Rent price increases with density
n Centrality
q Very simplified model, but gives some hints about
possible more complex and realistic models
P(x) ⇠ e V (x)V (x) = Y CR(x) CT (x)
/ ⇢(x) g(x)

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Co-evolution of the network and centers
n Competition renting price- centrality
Most important:rent
Most important:centrality

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A simple model for the road/streets network
A large variety of patterns (Courtat et al, 2010)

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A simple model
n Local optimization seems to reproduce
important features of the road network
n Points to the possible existence of a
common principle for transportation networks
n Simple economical ingredients lead to interesting
patterns

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Cost-benefit analysis
of growth

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Railway growth model
n Add a new link of length which maximizes
where
n Crossover from a ‘star-network’ ( small) to a minimun
spanning tree ( large)
n Emergence of hierarchical networks
Tij = K
PiPj
da
ij

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Railway growth model
n Crossover from a ‘star-network’ ( small) to a minimun
spanning tree ( large)
n Emergence of hierarchical networks
n Most empirical networks display: where is
obtained for Benefit≈Cost
⇤

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Spatial network growth model
n Most networks in developed countries are in the regime
where the average detour index is minimum (due to the
largest variety of link length)
⇠ ⇤

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Scaling for
transportation systems
How are network quantities
related to socio-economical
factors ?

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Scaling
n Network properties
- Total length L
- Number Ns of stations
- Ridership R (per year)
n Socio-economical quantities
- GDP G (or GMP for urban areas)
- Population P
- Area A
n Difference subway-railway
- subway: urban area scale
- Railway: country scale

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Scaling – general framework
n Iterative growth: add a link e such that
is maximum
n In the `steady-state’ regime: operating costs are
balanced by benefits
Z(e) = B(e) C(e)
Z =
X
e
Z(e) ⇡ 0

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Scaling – Subways
n Benefit: R (total ridership per unit time); f ticket price
n Costs: per unit length (and time) for lines; and per unit
time and per station.
n Estimate of R ? For a given station i, we have
where the “coverage” is
Zsub = Rf ✏LL ✏sNs
Ri = ⇠iCi⇢
Ci ⇡ ⇡d2
0

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Scaling – Subways
n We then obtain
n Linear fit gives d0≈500meters
R ⇡ ⇠⇡d2
0⇢Ns

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Scaling – Subways
n Estimate of d0:
where the average inter-station distance is
n Interstation
distance constant !
(138 cities)
2d0 ⇡ `1
`1 =
L
Ns
`1 ⇡ 1.2km

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Scaling – Subways
Ns /
G
✏s
n Relation with the economics of the city
where G is the GMP (Gross Metropolitan Product)
n Large fluctuations…

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n The railway connects cities distributed in the country
n The intercity distance is
where A is the area of the country.
n The total length is
Scaling – The railway case
` =
r
A
Ns
L = Ns` ⇠
p
ANs

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Scaling – The railway case
L = Ns` ⇠
p
ANs
n A power law fit gives an exponent ≈0.5

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n For railways we write
n T is the total distance travelled is the relevant quantity
(not R)
n fL ticket price per unit distance
n In the steady-state regime and assuming
Scaling – The railway case
Ztrain ' TfL ✏LL
T ⇠ R`
R ⇠
✏LNs
fL

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Scaling – The railway case
R ⇠
✏LNs
fL
n Large fluctuations…

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n Relation with the economics of the city
where G is the GMP (for railways Cost(lines)>>Cost(stations))
n There is some
dispersion. Importance
of local specifics.
Scaling – Railways
L /
G
✏L

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n A simple framework allows to relate the properties of the
networks (R, L, Ns) to socio-economical quantities such as
G, P, A.
n These indicators allow to understand the main properties.
Fluctuations are present and might be understood,
elaborating on this simple framework
n Fundamental difference subway-railway
- The interstation distance is imposed by human
constraints in the subway case
- Railways: the network has to adapt to the spatial
distribution of cities
Scaling – Railways

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Discussion
n Few models of (realistic) planar graphs
n Even for the evolution of spatial networks
n Interesting direction: socio-economical indicators and
network properties…

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Thank you for your attention.
(Former and current) Students and Postdocs:
Giulia Carra (PhD student)
Riccardo Gallotti (Postdoc)
Thomas Louail (Postdoc)
Remi Louf (PhD student)
R. Morris (Postdoc)
Collaborators:
A. Arenas
M. Batty A. Bazzani H. Berestycki
G. Bianconi P. Bordin M. Breuillé S. Dobson
M. Fosgerau M. Gribaudi J. Le Gallo J. Gleeson
P. Jensen M. Kivela M. Lenormand Y. Moreno
I. Mulalic JP. Nadal V. Nicosia V. Latora
J. Perret S. Porta MA. Porter JJ. Ramasco
S. Rambaldi C. Roth M. San Miguel S. Shay
E. Strano MP. Viana
Mathematicians, computer scientists (27%)!
Geographers, urbanists, GIS experts, historian (27%)!
Economists (13%)!
Physicists (33%)!
!
http://www.quanturb.com
marc.barthelemy@cea.fr