transport and routing on coupled spatial networks
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Shall we take the train or the car?(. . . route choice on coupled spatial networks)
Richard G. Morris1 and Marc Barthelemy2
1University of Warwick, Gibbet Hill Road, Coventry, U. K.
2Institut de Physique Théorique, CEA-DSM, Saclay, France.
December 5th, 2013
Aside: subjects I am interested in. . .
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Approaches to complex networks and coupled /interdependent networks
1. Everyone knows Networks 101, but what about Networks 102?
2. Coupled networks are a trendy but ill-defined sub-class ofcomplex networks.
3. Hard (read: impossible) to characterise a generic type ofbehaviour associated to such a broad class of systems i.e.:3.1 percolation-like.3.2 cascading sandpile-like.
4. Better to be ‘problem-led’: defining a model in order to answerspecific questions about well defined / motivated systems.
Routing in 2D: fast-but-sparse vs. slow-but-dense
1. Start with generic questions: how robust are currenttransport/routing infrastructures?1.1 Centralised power generation → distributed renewable
generation.1.2 Bandwidth changes in packet routing (e.g., internet or other
ICT).1.3 Technology changes in transport systems (e.g., rail/road).
2. Consider specific characterisation of such systems: namely,when two different modes are available: ‘fast but sparse’(long-range) networks and ‘slow but dense’ (short-range)networks.
Time is of the essence
Need two (as yet unspecified) networks that are at the same timedifferent, but also connected. . . imagine they share some (not all)nodes.
Consider a (static) route assignment problem, based on travel time.
Use weighted-shortest-paths: for a spatial graph G (V , E ) whereV = {xi} ∀ xi ∈ R2, 1 ≤ i ≤ n, the weight of undirected edge(xi , xj) is given by:
wij = |xi − xj | /v , (1)
where v is the ‘speed’ of each network (i.e., weights ∝ time).
Topology: a planar subdivision
Take each network to be a Delaunay triangulation DT (V ):
. . . where V = {Xi} such that the Xi are i.i.d random variables inR2, distributed uniformly within a disk of radius r .
Connecting the two networks
. . . construct the two triangulations DT (1) and DT (2) such thatV (2) ⊆ V (1) (. . . where β = |V (2)|/|V (1)| ≤ 1):
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V = V (1),E = E (1) ∪ E (2).
Sources and sinks. . . sources and sinks are now characterised by an’origin-destination’ matrix Tij (of size |V (1)|2).
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Start with a (directed) star-graph centred on x∗:E = {(xi , x∗)} ∀ xi 6= x∗ and rewire using the followingalgorithm:
I for each e ∈ E :I with probability p:I replace (xi , x∗) with (xi , yi ),I where yi is chosen uniformly at random from V = V (1).
System characterisation: coupling
. . . the ’coupling’ is now a consequence of system parameters:I p,I α = v (1)/v (2),I β = n(2)/n(1),
where α, β ≤ 1.
Define:
λ =∑i 6=j
Tijσcoupled
ijσij
, (2)
where σij is the total number of weighted shortest paths betweenxi and xj , and σcoupled
ij is the number of weighted shortest pathsthat use both networks.
[Note: T is normalised, i.e.,∑
ij Tij = 1.]
System characterisation: efficiency & utility
Average route distance:
τ̄ =∑i 6=j
Tijwij . (3)
Gini coefficient of betweeness-centrality:
b (e) =∑i 6=j
Tijσij (e)
σij(4)
G =1
2b̄|E |2∑
p,q∈E|b(p)− b(q)|. (5)
Results
〈. . .〉 represents an average over the ensemble defined by values ofα, β and p.
Fix β and systematically vary α and p.
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à p=0.2
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Results: Gini-coefficient at low p
Heatmap of normalised betweeness-centrality: 1 0I almost monocentric origin-destination matrix.
α = 0.9 α = 0.1
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Gini-coefficient unchanged by increased coupling.
Results: Gini-coefficient at high p
Heatmap of normalised betweeness-centrality: 1 0I almost random origin-destination matrix.
α = 0.9 α = 0.1
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Gini-coefficient increased by increased coupling.
Results: system utility
Define system utility as F = 〈τ̄〉+ µ〈G〉.
. . . where λ∗ is defined such that F (λ∗) is minima.
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Wrap-up
SummaryI Simple toy model—analysed by simulation—exhibiting
unexpected behaviour.I Two regimes emerge p > p∗ and p ≤ p∗: Optimisation of the
system relies on the routing behaviour.Outlook
I Interacting or coupled networks play a prominent role inmodern life.
I Understanding and classifying the behaviour of such systemsis important.
I Familiarity with statistics and quantitative analysis isimportant but ’off-the-shelf’ physics models are oftenunhelpful.
R. G. Morris and M. Barthelemy, Phys. Rev. Lett. 109 (2012)
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