Scale-free Segregation in Transport Networks
aa r X i v : . [ phy s i c s . s o c - ph ] O c t Scale-free Segregation in Transport Networks
Ph. Blanchard, D. Volchenkov
Bielefeld-Bonn Stochastic Research Center (BiBoS) , Bielefeld University, Postfach 100131, 33501, Bielefeld, GermanyEmail: [email protected]
November 15, 2018
Abstract
Every route of a transport network approaching equilibrium can be represented by a vector of Euclideanspace which length quantifies its segregation from the rest of the graph. We have empirically observed thatthe distribution of lengths over the edge connectivity in many transport networks exhibits scaling invariancephenomenon. We give an example of the canal network of Veneice to demonstrate our result. The methodis applicable to any transport network.PACS: 89.65.Lm, 89.75.Fb, 05.40.Fb, 02.10.OxTransport networks are used to model the flow ofcommodity, information, viruses, opinions, or traf-fic. They typically represent the networks of roads,streets, pipes, aqueducts, power lines, or nearly anystructure which permits either vehicular movement orflow of some commodity, products, goods or service.The major aim of the analysis is to determine thestructure and properties of transport networks thatare important for the emergence of complex flow pat-terns of vehicles (or pedestrians) through the networksuch as the Braess paradox [1]. This counter-intuitivephenomenon occurs when adding more resources to atransportation network (say, a new road or a bridge)worsens the quality of traffic by creating longer delaysfor the drivers, rather than alleviate it. The Braessparadox has been observed in the street traffic of NewYork City and Stuttgart, [2].In the present Letter, we show that while approach-ing equilibrium, a transport network can be embed-ded into Euclidean space R N − , N being a numberof vertices. Then, every edge of the network is rep-resented by a vector which length quantifies its seg- regation from the rest of the graph. We have empir-ically observed that the distribution of lengths overthe edge connectivity in urban transport networks ex-hibits scaling invariance phenomenon. The relationbetween the connectivity of city spaces and their cen-trality known as intelligibility is a key determinant ofhuman behaviors in urban environments, [3].In most of researches devoted to the improving oftransport networks, a primary graph representationof urban networks is used in which streets and routesare considered as edges of a planar graph, while thetraffic end points and street junctions are treated asnodes. The usual city map based on Euclidean ge-ometry can be considered as an example of primarycity graphs.However, another graph representation can be use-ful if we are interested in describing the transport net-work at equilibrium. Given a connected undirectedgraph G ( V, E ), in which V is the set of nodes and E is the set of edges, we introduce the traffic function f : E → (0 , ∞ [ through every edge e ∈ E . It thenfollows from the Perron-Frobenius theorem [4] that1he linear equation f ( e ) = X e ′ ∼ e f ( e ′ ) exp ( − h ℓ ( e ′ ) ) , (1)where the sum is taken over all edges e ′ ∈ E whichhave a common node with e , has a unique positivesolution f ( e ) >
0, for every edge e ∈ E , for a fixedpositive constant h > metric length distances ℓ ( e ) >
0. This solution is nat-urally identified with the traffic equilibrium state ofthe transport network defined on G , in which thepermeability of edges depends upon their lengths.The parameter h is called the volume entropy of thegraph G , while the volume of G is defined as the sumVol( G ) = P e ∈ E ℓ ( e ) . The volume entropy h is de-fined to be the exponential growth of the balls in auniversal covering tree of G with the lifted metric,[5]-[8].The degree of a node i ∈ V is the number of itsneighbors in G , deg G ( i ) = k i . It has been shown in [8]that among all undirected connected graphs of nor-malized volume, Vol( G ) = 1, which are not cycles andfor which k i = 1 for all nodes, the minimal value ofthe volume entropy, min( h ) = P i ∈ V k i log ( k i − ℓ ( e ) = log (( k i −
1) ( k j − h ) , (2)where k i and k j are the degrees of the nodes linkedby e ∈ E . It is then obvious that substituting (2) andmin( h ) into (1) the operator exp ( − hℓ ( e ′ )) is given bya symmetric Markov transition operator, f ( e ) = X e ′ ∼ e f ( e ′ ) p ( k i −
1) ( k j − , (3)where i and j are the nodes linked by e ′ ∈ E , andthe sum in (3) is taken over all edges e ′ ∈ E whichshare a node with e . The symmetric operator (3)rather describes time reversible random walks overedges than over nodes. In other words, we are invitedto consider random walks described by the symmetricoperator defined on the dual graph G ⋆ .The Markov process (3) represents the conserva-tion of the traffic volume through the transport net-work, while other solutions of (1) (with h > min( h )) are related to the possible termination of travelsalong edges. If we denote the number of neighboredges the edge e ∈ E has in the dual graph G ⋆ as q e = deg G ⋆ ( e ), then the simple substitution showsthat w ( e ) = √ q e defines an eigenvector of the sym-metric Markov transition operator defined over theedges E with eigenvalue 1. This eigenvector is pos-itive and being properly normalized determines therelative traffic volume through e ∈ E at equilibrium.Eq.(3) relates the equilibrium transport flows on thegraph G to the stationary distribution of randomwalks defined on its dual counterpart G ⋆ and empha-sizes that the degrees of nodes are a key determinantof the transport networks properties.The notion of traffic equilibrium had been intro-duced by J.G. Wardrop in [9] and then generalizedin [10] to a fundamental concept of network equilib-rium. Wardrop’s traffic equilibrium is strongly tiedto the human apprehension of space since it is re-quired that all travellers have enough knowledge ofthe transport network they use. Dual city graphs areextensively investigated within the concept of spacesyntax , a theory developed in the late 1970s, thatseeks to reveal the effect of spatial configurations onthe human perception of places and behavior in urbanenvironments, [11, 3]. Spatial perception that shapespeoples understanding of how a place is organized de-termines eventually the pattern of local movement,which is quantified by the space syntax measure be-ing nothing else, but an element of a transition prob-ability matrix of a Markov chain [12], with surprisingaccuracy [13]. Random walks embed connected undi-rected graphs into the Euclidean space R N − . Thisembedding can be used in order to compare nodeswith respect to the quality of paths they provide forrandom walkers and to construct the optimal coarse-graining representations.While analyzing a graph, whether it is primaryor dual, we assign the absolute scores to all nodesbased on their properties with respect to a trans-port process defined on that. Indeed, the nodes of G ( V, E ) can be weighted with respect to any mea-sure m = P i ∈ V m i i , specified by a set of posi-tive numbers m i >
0. The space ℓ ( m ) of square-assumable functions with respect to the measure m is the Hilbert space H ( V ). Among all measures which2an be defined on V , the set of normalized measures(or densities ), 1 = P i ∈ V π i i , are of essential inter-est since they express the conservation of a quantity,and therefore may be relevant to a physical process.The fundamental physical process defined on agraph is generated by the subset of its linear auto-morphisms which share the property of probabilityconservation; it can be naturally interpreted as ran-dom walks. The linear automorphisms of the graphare specified by the symmetric group S N includingall admissible permutations p ∈ S N taking i ∈ V to p ( i ) ∈ V and preserving all of its structure.Markov’s operators on Hilbert space form the nat-ural language of transport networks theory. Beingdefined on a connected aperiodic graph, the transi-tion matrix of random walks T ij is a real positivestochastic matrix, and therefore, in accordance to thePerron-Frobenius theorem [4], its maximal eigenvalueis 1, and it is simple. The left eigenvector πT = π associated with the eigenvalue 1 is interpreted as aunique equilibrium state π (the stationary distribu-tion of random walks). The Markov operator b T isself-adjoint with respect to the normalized measureassociated to the stationary distribution of randomwalks π , b T = 12 (cid:16) π / T π − / + π − / T ⊤ π / (cid:17) , (4)where T ⊤ is the transposed operator. In the theoryof random walks defined on graphs [14] and in spec-tral graph theory [15], basic properties of graphs arestudied in connection with the eigenvalues and eigen-vectors of self-adjoint operators defined on them.The orthonormal ordered set of real eigenvectors ψ i , i = 1 . . . N , of the symmetric operator b T defines abasis in H ( V ).In particular, the symmetric transition operator b T of the random walk defined on connected undirectedgraphs is c T ij = 1 / p k i k j , if i ∼ j , and zero other-wise. Its first eigenvector ψ belonging to the largesteigenvalue µ = 1, ψ b T = ψ , ψ ,i = π i , (5)describes the local property of nodes (connectivity),since the stationary distribution of random walks is π i = k i / M where 2 M = P i ∈ V k i . The remainingeigenvectors, { ψ s } Ns =2 , belonging to the eigenvalues1 > µ ≥ . . . µ N ≥ − global connect-edness of the graph. For example, the eigenvectorcorresponding to the second eigenvalue µ is used todefine the spectral bisection of graphs; it is called theFiedler vector if related to the Laplacian matrix of agraph [15].Markov’s symmetric transition operator b T definesa projection of any density σ ∈ H ( V ) on the eigen-vector ψ of the stationary distribution π , σ b T = ψ + σ ⊥ b T , σ ⊥ = σ − ψ , (6)in which σ ⊥ is the vector belonging to the orthogonalcomplement of ψ . In space syntax, we are interestedin a comparison between the densities with respectto random walks defined on the graph G . Since allcomponents ψ ,i >
0, it is convenient to rescale thedensity σ by dividing its components by the compo-nents of ψ , e σ i = σ i ψ ,i . (7)Thus, it is clear that any two rescaled densities e σ, e ρ ∈ H ( V ) differ with respect to random walksonly by their dynamical components, ( e σ − e ρ ) b T t =( e σ ⊥ − e ρ ⊥ ) b T t , for all t >
0. Therefore, we can de-fine the distance k . . . k T between any two densitiesestablished by random walks by k σ − ρ k T = X t ≥ De σ ⊥ − e ρ ⊥ (cid:12)(cid:12)(cid:12) b T t (cid:12)(cid:12)(cid:12) e σ ⊥ − e ρ ⊥ E , (8)or, using the spectral representation of b T , k σ − ρ k T = N X s =2 h e σ ⊥ − e ρ ⊥ | ψ s ih ψ s | e σ ⊥ − e ρ ⊥ i − µ s , (9)where we have used Diracs bra-ket notations espe-cially convenient for working with inner products andrank-one operators in Hilbert space.If we introduce a new inner product for densities σ, ρ ∈ H ( V ) by( σ, ρ ) T = X t ≥ N X s =2 h e σ ⊥ | ψ s ih ψ s | e ρ ⊥ i − µ s , (10)3hen (9) is nothing else but k σ − ρ k T = k σ k T + k ρ k T − σ, ρ ) T , where k σ k T = N X s =2 h e σ ⊥ | ψ s ih ψ s | e σ ⊥ i − µ s (11)is the square of the norm of σ ∈ H ( V ) with respectto random walks defined on the graph G .We finish the description of the Euclidean spacestructure of G induced by random walks by mention-ing that given two densities σ, ρ ∈ H ( V ) , the anglebetween them can be introduced in the standard way,cos ∠ ( ρ, σ ) = ( σ, ρ ) T k σ k T k ρ k T . (12)The cosine of an angle calculated in accordance to(12) has the structure of Pearson’s coefficient of lin-ear correlations. The notion of angle between anytwo nodes of the graph arises naturally as soon as webecome interested in the strength and direction of alinear relationship between the flows of random walksmoving through them. If the cosine of an angle (12)is 1 (zero angles), there is an increasing linear rela-tionship between the flows of random walks throughboth nodes. Otherwise, if it is close to -1 ( π angle),there is a decreasing linear relationship. The correla-tion is 0 ( π/ i , which equals1 at i ∈ V and zero otherwise, acquires the norm k i k T associated to random walks. Its square, k i k T = 1 π i N X s =2 ψ s,i − µ s , (13)expresses the access time to a target node in randomwalk theory [14] quantifying the expected number ofsteps required for a random walker to reach the node i ∈ V starting from an arbitrary node chosen ran-domly among all other nodes with respect to the sta-tionary distribution π .The notion of spatial segregation acquires a statis-tical interpretation with respect to random walks bymeans of (13). In urban spatial networks encoded bytheir dual graphs, the access times (13) strongly varyfrom one open space to another and could be verylarge for statistically segregated spaces. It is remark-able that the norm a canal of Venice acquires withrespect to random walks scales with its connectivity(see Fig. 1).The Euclidean distance between i and j inducedby random walks, k i − j k T = N X s =2 − µ s (cid:18) ψ s,i √ π i − ψ s,j √ π j (cid:19) , (14)is the commute time in theory of random walks beingequal to the expected number of steps required for arandom walker starting at i ∈ V to visit j ∈ V andthen to return back to i , [14].Indeed, the structure of vector space R N − inducedby random walks cannot be represented visually, how-ever if we choose a node of the graph as a point ofreference, we can draw the 2-dimensional projection4igure 1: The scatter plot of the connectivity vs. thenorm a node in the dual graph representation of 96 Vene-tian canals acquires with respect to random walks. Threedata points characterized by the shortest access times rep-resent the main water routes of Venice: the Lagoon ofVenice, the Giudecca canal, and the Grand canal. Fourdata points of the worst accessibility are for the canal sub-network of Venetian Ghetto. The slope of the regressionline equals 2.07. of Euclidean space by arranging other nodes at thedistances and under the angles they are with respectto the chosen reference node. The 2-dimensional pro-jection of the Euclidean space of Venetian canals setup by random walks drawn for the the Grand Canal ofVenice (the point (0 , The 2-dimensional projection of space syntaxof Venetian canals built from the perspective of the Grandcanal of Venice chosen as the origin. The labels of thehorizontal axes display the expected number of randomwalk steps. The labels of the vertical axes show the degreeof nodes (radiuses of the disks). the traffic volume is conserved. It is evident fromFig. 2 that disks of smaller radiuses demonstrate aclear tendency to be located far away from the originbeing characterized by the excessively long commutetimes with the reference point (the Grand canal ofVenice), while the large disks which stand in Fig. 2for the main water routes are settled in the closestproximity to the origin that intends an immediateaccess to them.Probably, the most important conclusion of spacesyntax theory is that the adequate level of the pos-itive relationship between the connectivity of cityspaces and their integration property (vs. segre-gation) called intelligibility encourages peoples way-finding abilities [3]. Intelligibility of Venetian canalnetwork reveals itself quantitatively in the scaling ofthe norms of nodes with connectivity shown in Fig. 1and qualitatively in the tendency of smaller disks tobe located on the outskirts of the Venetian space syn-tax displayed in Fig 2.The support from the Volkswagen Foundation(Germany) in the framework of the project ”
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