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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DT (2)

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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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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.

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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)