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 ⇠

slog ⇢

⇢

“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

P e

P e =X

e0 6=e

C(e0)(V (i)� V (j))2(Istem = N

Ii 6=stem = �1X

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 = KPiPj

daij

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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 ⇡ ⇡d20

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Scaling – Subways

n  We then obtain

n  Linear fit gives d0≈500meters

R ⇡ ⇠⇡d20⇢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

` =

rA

Ns

L = Ns` ⇠pANs

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Scaling – The railway case

L = Ns` ⇠pANs

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