Optimal intervention in traffic networks
Leonardo Cianfanelli, Giacomo Como, Asuman Ozdaglar, Francesca Parise
OOptimal intervention in traffic networks
Leonardo Cianfanelli , Giacomo Como , Asuman Ozdaglar , andFrancesca Parise Department of Automatic Control, Lund University, BOX 118, SE-22100, Lund, Sweden Department of Electrical Engineering and Computer Science, MIT Department of Electrical and Computer Engineering, Cornell [email protected], [email protected], [email protected], [email protected]://sites.google.com/view/leonardo-cianfanelli/, https://staff.polito.it/giacomo.como/,https://asu.mit.edu/, https://sites.coecis.cornell.edu/parise/
Abstract
We present an efficient algorithm to identify which edge should beimproved in a traffic network to minimize the total travel time. Ourmain result is to show that it is possible to approximate the variation oftotal travel time obtained by changing the congestion coefficient of anygiven edge, by performing only local computations, without the need ofrecomputing the entire equilibrium flow. To obtain such a result, we re-formulate our problem in terms of the effective resistance between twoadjacent nodes and suggest a new approach to approximate such effectiveresistance. We then study the optimality of the proposed procedure forrecurrent networks, and provide simulations over synthetic and real trans-portation networks.
Keywords : Wardrop equilibrium; transportation networks; effective resis-tance; network design problem.
Due to increasing populations living in urban areas, many cities are facing theproblem of traffic congestion, which leads to increasing levels of pollution andmassive waste of time and money (Schrank et al. [29]). The problem of mitigat-ing congestion has been tackled in the literature from two main perspectives.One approach is to indirectly influence the behaviour of the drivers, for in-stance by road tolling (e.g., in Brown and Marden [7], Fleischer et al. [16],Zhao and Kockelmann [39], Cole et al. [10]), information design (e.g., in Daset al. [12], Meigs et al. [25], Wu et al. [35, 36]) or lottery rewards (Yue et al.[38]), to minimize inefficiencies due to the autonomous uncoordinated decisionsof agents. A second approach is to intervene on the transportation network1 a r X i v : . [ c s . G T ] F e b irectly, by building new roads or enlarging existing ones. The corresponding network design problem (i.e., the problem of optimizing the intervention on atransportation network subject to some budget constraints, see e.g. LeBlanc[22]) is very challenging because of its bi-level nature (Farahani et al. [15]), i.e.,it involves a network intervention optimization problem given the flow distribu-tion for that particular network. We assume that each edge of the network isendowed with a delay function and the flow distributes according to a Wardropequilibrium, taking paths with minimum delay, defined as the sum of the delaysof edges along a path (see Beckmann et al. [3] and Wardrop [34]). Characteri-zation of Wardrop equilibrium is used to construct the lower level of the bilevelnetwork design problem.In this paper we study a special class of network design problem (NDP),where the planner can improve the delay function of a single edge. Our objec-tive is to strike a balance between a model that is simple enough to guaranteetractable analysis, yet rich enough to allow insights for more general classes ofNDPs. For this class of NDPs (with E denoting the number of edges), our firstmain result provides an analytical characterization of the cost variation corre-sponding to an intervention on a particular edge under a regularity assumption,which states that the edges that carry positive flow remain unchanged with anintervention. This assumption, tipically used in the traffic equilibrium literature(see Steinberg and Zangwill [31], Dafermos and Nagurney [11]) leads to charac-terization of Wardrop equilibria using a system of linear equations and enablesrepresenting edge interventions as rank-1 perturbations of the system. We showthat this assumption is satisfied provided that the total incoming flow to thenetwork is large enough, which may be of independent interest. We exploit thestructure of our characterization and linearity of delay functions to express thecost variation using the effective resistance of an edge (i.e., between the enpointsof the edge), defined with respect to a related resistor network. Computing theeffective resistance of a single edge requires the solution of a linear system witha matrix with dimension scaling with the size of the network (we indistinctlyrefer to the size of the network as the cardinality of the node and the edge sets,implicitly assuming that transportation networks are sparse in a such a waythat the average degree of the nodes is independent of the number of nodes, in-ducing then a proportionality between the number of nodes and edges). Hence,solving the NDP requires the solution of E of these problems. Since this can becomputationally untractable for large networks, our second main result proposesa method based on Rayleigh’s monotonicity laws to approximate the effectiveresistance of each edge with a number of iterations independent of the networksize, thus leading to a significant reduction of complexity. The key idea is thatthe effective resistance between two adjacent nodes i and j depends mainly onthe local structure of the network around the two nodes (i.e., the set of nodes N ≤ d that are at distance no greater than a small given constant d from at leastone of i and j ), and may therefore be approximated by performing only localcomputations. Since for networks with bounded degrees (as typical in trafficnetworks, think for instance of the bidimensional square grid) the size of N ≤ d continuous network design problems (e.g., Chiou [8], Li et al. [24], Wang et al. [32]), wherethe budget can be allocated continuously among the edges, and discrete formu-lations, in which the decision variables include which new roads to build (Gao etal. [19]), how many lanes to add to existing roads (Wang et al. [33]), or a mixof those two problems (Poorzahedy and Rouhani [26]), have been consideredin the literature, together with dynamical formulations (Fontaine and Minner[17]), and formulations where the optimum is achieved by removing, instead ofadding, edges, because of Braess paradox, as in Roughgarden [28] and Fotakis etal. [18]. For comprehensive surveys on the literature on NDP we refer to Yangand Bell [37] and Farahani et al. [15]. We stress that most of the literaturefocused on finding time polynomial algorithms to approximately solve NDPs intheir most general form. As noted above, we instead consider a problem thatcan be solved with a polynomial algorithm by simply enumerating all the candi-date edges and computing the cost corresponding to the intervention on each ofthose edges. Our main contribution is to define a simplified, more intuitive andtractable approach to solve such a design problem in quasilinear time insteadof polynomial, as well as providing intuition and a complete new formulation.For the future we aim at extending our techniques to more general cases, likethe multiple interventions case. Our work is related to Steinberg and Zangwill[31] and Dafermos and Nagurney [11], where the authors investigate the signof total travel time variation when a new path is added to a two-terminal net-work, under similar assumptions to ours, providing sufficient conditions underwhich Braess paradox arises. In our work we instead suggest an efficient al-gorithm to select the best edge to improve. As mentioned, a key step of ourapproach is to reformulate the NDP in terms of a resistance problem. We dothis in two steps: first we reformulate the NDP in terms of random walks overa network (following a similar approach as in Rebeschini and Tatikonda [27],where however the problem of finding conditions under which a perturbationof the incoming flow to a small part of the network leads to local perturbationof Wardrop equilibrium), then following standard literature we reformulate therandom walk problem as an electrical one (see e.g. Doyle and Snell [13]). Tosummarize, the contribution of this paper is two-fold. From a methodologicalperspective, we provide a method to locally upper and lower bound the effectiveresistance between adjacent nodes, which may be of a separate interest beyondtraffic applications. From the network design perspective, we provide a newformulation of the design problem in terms of resistor networks, and we exploitour methodological result to approximate in an efficient manner a simplified ver-sion of the design problem where a single edge can be improved. For the future3e aim at extending our methods to the case of multiple interventions. Froma methodological perspective, it is worthwhile mentioning that the equivalencebetween Wardrop and electric flows has been already investigated in Klimm andWarode [21].The remainder of the paper is organized as follows. In Section 2 we definethe model and formulate the problem as a bi-level programming. In Section 3 wefirst translate the problem into a single-level program, and then rephrase it interms of resistor networks. In Section 4 we provide our method to approximateeffective resistance between neighbors. In Section 5 we analyze the asymptoticbehaviour of the bounds in the limit of infinite networks. In Section 6 we showsome simulations over relevant networks. Finally, in the conclusive section, wesummarize the work and discuss future research lines. G = ( N , E ) denotes a two-terminal strongly connected directed network withorigin o and destination d, where N and E are respectively the node and theedge sets. Let P denote the set of paths from o to d, and N, E and P denote thecardinality of N , E and P , respectively. We allow for multiple edges betweenthe same pair of nodes. We do not allow for selfloops, since they are not relevantin traffic networks. δ i , and I denote the unitary vector with 1 in position i and 0 in all the other positions, the column vector of all ones, and the identitymatrix, respectively, where the size of them may be deduced from the context. A T and v T denote the transpose of matrix A and vector v , respectively. Forsimplicity of notation we use A − ij instead of ( A − ) ij . The transportation network is modeled as a two-terminal strongly connecteddirected network G = ( N , E ). Let τ > ν = τ ( δ o − δ d ) ∈ R N denote the net inflow tothe network. An admissible path flow is a vector z ∈ R P satisfying the non-negativity and conservation of mass constraints z ≥ , z T = τ. (1)Let R ∈ R E × P denote the edge-path incidence matrix, with entries R lp = 1 ifthe edge l belongs to the path p or 0 otherwise. The path flow induces a uniqueedge flow f ∈ R E via f = Rz. (2)Let B ∈ R N × E denote the node-edge incidence matrix, with entries B nl = 1 ifthe node n is the tail of the edge l , − n is the head of l , or 0 otherwise. The4onstraints can be reformulated in terms of edge flows as f ≥ , Bf = ν. Every edge is endowed with a delay function, which is assumed affine, non-negative and strictly increasing, d l ( f l ) = a l f l + b l , a l > , b l ≥ , ∀ l ∈ E . The cost of a path p , under flow distribution f , is c p ( f ) = (cid:88) l ∈E R lp d l ( f l ) , (3)which is the sum of the delays of the edges belonging to that path. We alsodefine A ∈ R E × E and b ∈ R E as A := a . . . a . . . . . . a E , b := b b ... b E . Definition 2.1 (Affine routing game) . An affine routing game is a quadruple ( G , A, b, ν ) . A Wardrop equilibrium is a flow distribution such that no one has incentivein changing his path.
Definition 2.2 (Wardrop equilibrium) . A path flow z ∗ , with associated edgeflow f ∗ as defined in (2) , is a Wardrop equilibrium if for every path pz ∗ p > ⇒ c p ( f ∗ ) ≤ c q ( f ∗ ) , ∀ q ∈ P . It is shown in Beckmann et al. [3] that an edge flow f ∗ is a Wardropequilibrium of a routing game if and only if it solves the following minimizationproblem: minimize f (cid:88) l ∈E (cid:90) f l d l ( s ) ds subject to f ≥ , Bf = ν. (4)Since the delay functions are assumed strictly increasing, the objective functionis strictly convex and the Wardrop equilibrium f ∗ is unique.We now define the social cost, which is the total travel time at the equilibrium. Definition 2.3 (Social cost) . Let f ∗ be the unique Wardrop equilibrium of anaffine routing game ( G , A, b, ν ) . The social cost is C ( f ∗ ) = (cid:88) l ∈E f ∗ l d l ( f ∗ l ) . The social cost can be interpreted by a planner that aims at minimizing theoverall congestion as a measure of performance of the transportation network.5 .2 Problem formulation
We consider a problem in which a planner can intervene on the network withthe goal of minimizing the social cost. We propose as intervention to rescalethe slope of one edge l by a scaling parameter κ >
1, so that the slope of theedge l gets reduced from a l to ˜ a l = a l /κ . This intervention may correspondfor instance to adding a new lane in a street. In fact, every intervention on asingle edge may be seen as a rank-1 perturbation of the system and may behandled by our method (see § f ∗ ( l ) and C ( f ∗ ( l ))denote the Wardrop equilibrium when the slope of the edge l is rescaled, and thecorresponding social cost, respectively. Hence, the problem can be expressed asfollows. Problem 1.
Let ( G , A, b, ν ) be an affine routing game and κ > be the scalingparameter. Find edge l ∗ such that l ∗ ∈ argmin l ∈E C ( f ∗ ( l )) . We stress the fact that Problem 1 is bi-level, in the sense that the planneroptimizes the intervention over the edge set, but the cost function is a functionof the Wardrop equilibrium f ∗ , which in turn is the solution of the optimizationproblem (4).Problem 1 can be solved by a brute force approach, by enumerating all theedges and computing the corresponding equilibrium f ∗ ( l ) by solving the convexprogram (4) with d l ( f l ) = ˜ a l f l + b l instead of d l ( f l ). In this work we propose amethod that, given f ∗ before the intervention (which is assumed to be observableand therefore known) and other electrical quantities computed on a resistancenetwork related to the original unperturbed traffic network, provides an upperand lower bound to C ( f ∗ ( l )) with a computational complexity that does notscale with the size of the network. The main idea is that the effect of perturbingan edge may be well approximated by looking at a local portion of the network.Our method works under the assumption that the network is sparse in sucha way that the average degree of the nodes does not depend on the size ofthe network, and under the assumption that the set of the used edges doesnot change after the intervention. The first assumption is suitable for trafficnetworks, and the second one is standard in the literature on intervention intraffic networks (see Steinberg and Zangwill [31] and Dafermos and Nagurney[11]). We provide a more detailed discussion on this assumption in § In this section we provide two equivalent formulations for Problem 1. The firstformulation is based on the fact that modifying the slope of one edge is equiv-alent to introducing a rank-1 perturbation in the KKT conditions of (4) (for6n in depth discussion on KKT conditions we refer to Boyd and Vandenberghe[5]). The second formulation gives an interpretation using resistor networks.
Let us introduce the dual variables λ ∗ ( i,j ) associated to f ∗ ( i,j ) ≥ γ ∗ associ-ated to the constraint Bf = ν . The KKT conditions of (4) are: a ( i,j ) f ∗ ( i,j ) + b ( i,j ) + γ ∗ j − γ ∗ i − λ ∗ ( i,j ) = 0 ∀ ( i, j ) ∈ E , (cid:80) j :( j,i ) ∈E f ∗ ( j,i ) − (cid:80) j :( i,j ) ∈E f ∗ ( i,j ) + ν i = 0 ∀ i ∈ N ,λ ∗ ( i,j ) f ∗ ( i,j ) = 0 ∀ ( i, j ) ∈ E ,λ ∗ ( i,j ) ≥ ∀ ( i, j ) ∈ E ,f ∗ ( i,j ) ≥ ∀ ( i, j ) ∈ E . (5)The third condition is known as complementary slackness, and implies that allthe edges such that λ ∗ e > f ∗ e = 0. Let E + denote the set of such edges. Thus, the edges in E + and the last three conditionsof (5) can be removed, without affecting the solution of (5). With a slight abuseof notation, from now on let E denote E \E + . Thus, the KKT conditions become: (cid:40) a ( i,j ) f ∗ ( i,j ) + b ( i,j ) + γ ∗ j − γ ∗ i = 0 ∀ ( i, j ) ∈ E , (cid:80) j :( j,i ) ∈E f ∗ ( j,i ) − (cid:80) j :( i,j ) ∈E f ∗ ( i,j ) + ν i = 0 ∀ i ∈ N , (6)where the constraint f ∗ i,j ≥ f ∗ i,j ≥ i, j ) / ∈ E + . Without loss of generality, we order thenodes in such a way that the origin o and the destination d are the first and thelast node respectively. Observe from (6) that the optimal flows f ∗ ( i,j ) depend on γ ∗ only via the difference γ ∗ i − γ ∗ j , so that γ ∗ remains a solution if a constantvector is added to it. This is due to the fact that the matrix B is not full rank.Removing the last row of B is therefore equivalent to setting γ ∗ d = 0. Thus, wedefine x ∈ R N+E − and y ∈ R N+E − as x := (cid:20) fγ − (cid:21) , y := − (cid:20) bν − (cid:21) , where γ − and ν − denote respectively γ and ν where the last element of bothvectors is removed. Also, B − ∈ R (N − × E denotes the node-edge incidencematrix where the last row is removed. Finally, we define H ∈ R (N+E − × (N+E − as H := (cid:20) A − ( B − ) T − B − (cid:21) . With this notation and assuming γ ∗ d = 0, the KKT conditions (6) become: Hx ∗ = y. (7)7ecause we assumed γ ∗ d = 0, x ∗ is unique, and x ∗ = H − y. (8)The invertibility of H follows from the invertibility of A (the delays are strictlyincreasing) and from the invertibility of Q := B − A − ( B − ) T (see Horn and John-son [20]), which we prove in the proof of Theorem 1. Let A ( l ) and H ( l ) denotethe matrix A and H corresponding to the intervention on edge l . The opti-mal lagrangian multipliers γ ∗ have an useful interpretation, under the followingassumption. Assumption 1.
Let E + ( l ) be the set of edges e for which λ ∗ e ( l ) > in theWardrop equilibrium of ( G , A ( l ) , b, ν ) . We assume that E + ( l ) = E + for all l ∈ E . The intuition is that under Assumption 1 the KKT conditions (7) beforeand after the intervention on the edge l involve the same set of edges and differfor the value of a l only, allowing therefore to handle the intervention as rank-1perturbation in H . A detailed discussion on such assumption is given in § Proposition 1.
For any l ∈ E consider the modified game ( G , A ( l ) , b, ν ) ob-tained by changing the slope of edge l from a l to ˜ a l = a l /κ and construct thecorresponding primal and dual solution x ∗ ( l ) as in (8) . Then, C ( f ∗ ) − C ( f ∗ ( l )) = τ ( γ ∗ o − γ ∗ o ( l )) , where γ ∗ o and γ ∗ o ( l ) are the (E + 1) − th component of x ∗ and x ∗ ( l ) respectively.Proof. See the Appendix.Since τ is a given constant of the problem, Proposition 1 states that thegoal of the planner should be to select the edge l minimizing γ ∗ o ( l ), that is, theoptimal lagrangian multiplier of the origin after the intervention on the edge l . Observe that the brute force method requires the computation of the wholevector x ∗ ( l ) for every edge l . A natural question is whether it is possible toevaluate γ ∗ o ( l ) for every edge l without computing the whole vector x ∗ ( l ). Weprovide a positive answer under Assumption 1 in the next proposition, wherethe social cost variation is expressed in terms of the scaling parameter κ , theunperturbed equilibrium f ∗ , and selected elements of H − . Proposition 2.
Let ( G , A, b, ν ) be a routing game, κ > be the scaling pa-rameter and suppose that Assumption 1 holds. Then, the social cost variationcorresponding to the intervention on edge l is equal to C ( f ∗ ) − C ( f ∗ ( l )) = τ ( κ − a l H − ,l ) f ∗ l ( κ − a l H − ll − κ . (9) Proof.
See the Appendix.In the next section we provide an interpretation to the required elements of H − in terms of electrical quantities. Before doing that we discuss Assumption1 in detail. 8 n n d l l l l l l Figure 1: A directed network that is not series-parallel. On this network As-sumption 1 is not guaranteed to hold.
In the following we show that Assumption 1 is without loss of generality onseries-parallel networks, provided that the throughput is sufficiently high. First,we recall the definition of directed series-parallel networks, and then present theresult in Proposition 3.
Definition 3.1 (Directed series-parallel networks) . A two-terminal directed net-work G is series-parallel if and only if (i) it is a single edge from the origin tothe destination, or (ii) it is the result of connecting two directed series-parallelnetworks G and G in parallel, by merging o with o and d with d , or (iii)it is the result of connecting two directed series-parallel networks G and G inseries, by merging d with o . Proposition 3.
Let ( G , A, b, ν ) be a routing game. If G is a directed series-parallel network, it exists τ such that for every τ ≥ τ , E + = ∅ . Furthermore, if b = 0 , E + = ∅ for every τ > .Proof. See the Appendix.
Remark 1.
Proposition 3 immediately implies that Assumption 1 is without lossof generality on directed series-parallel networks provided that τ ≥ τ . However, τ depends on A and thus may change after the intervention. The next example shows that without the assumption of series-parallel net-works Proposition 3 may fail, even in case of linear delays.
Example 1.
Consider the network in Fig. 1, which is not series-parallel. Letus consider τ = 1 and linear delay functions, with a = a = a = a = a = 1 and a = 2 . By some computations, f ∗ = 611 , f ∗ = 511 , f ∗ = 711 , f ∗ = 411 , f ∗ = 111 , λ ∗ = 111 , f ∗ = λ ∗ = 0 . urning a from to / , we get: f ∗ = 613 , f ∗ = 713 , f ∗ = 113 , f ∗ = 513 , f ∗ = 813 , λ ∗ = 113 , f ∗ = λ ∗ = 0 . Thus, E + ( a = 2) = { l } and E + ( a = 1 /
2) = { l } . Since E + ( a = 2) (cid:54) = E + ( a = 1 / , Assumption 1 is violated, even in case of linear delays. In this section we explore the structure of H − to give an interpretation of itin terms of electrical quantities. To this end, by the well-known formula for the2 × H − = (cid:20) A − − KQ − K T − KQ − − Q − K T − Q − (cid:21) , (10)where K ∈ R E × (N − and Q ∈ R (N − × (N − are K := A − B T − , Q := B − A − B T − . From the definitions of B − and A , it follows K l : = δ Ti − δ Tj a l ∀ l = ( i, j ) ∈ E , (11)with the convention that δ d = 0 · (since we removed the destination), and Q ij = (cid:40) − (cid:80) l ∈{ ( i,j ) , ( j,i ) } a l if i (cid:54) = j (cid:80) l ∈ ∂i a l if i = j. ∀ i, j ∈ N \ d , where ∂i denotes the in and out neighborhood edges of i , that is ∂i := { l ∈ E : B il (cid:54) = 0 } . We remark that ∂i includes also edges pointing to the destination. The matrix Q allows for an interpretation in terms of electrical quantities. To this end, letus introduce the notion of resistor network and effective resistance between twonodes. Definition 3.2.
A resistor network is an undirected weighted network, wherethe weight matrix W represents a conductance matrix, i.e., W ij = W ji is theconductance between nodes i and j . Definition 3.3.
Let ∆ V = V i − V j be a difference of potential that is set betweennodes i and j on a resistor network. The effective resistance r ij between i and j is r ij = ∆ V i i , where i i denotes the total current flowing from node i under such potential.
10e define the resistor network G R obtained by making every directed edgeof the traffic network G undirected, with conductance matrix W ∈ R N × N W ij := (cid:40)(cid:80) l ∈{ ( i,j ) , ( j,i ) } a l if i (cid:54) = j i = j. Observe that W includes also the destination, is symmetric by construction,since G R is undirected, and the coefficients a l correspond to resistances. Finally,let D ∈ R N × N be the diagonal matrix of degrees in G R , i.e., D = diag( W ), and P = D − W ∈ R N × N the normalized adjacency matrix. The matrix Q may berelated to the truncated Laplacian of the resistor network G R . This is the keypoint to prove the next theorem. To this end, let us give the following definition. Definition 3.4.
Let V ∈ R N denote the potential over the nodes of G R whenthe boundary conditions V o = 1 and V d = 0 are imposed. V is an harmonicfunction (see Levin and Peres [23]), i.e., it satisfies V o = 1 , V d = 0 , V i = (cid:88) j ∈N P ij V j ∀ i ∈ N \ o , d . (12) Theorem 1.
Let ( G , A, b, ν ) be a routing game, κ > be the scaling parameter,and suppose Assumption 1 holds. The social cost variation corresponding to theintervention on edge l = ( i, j ) is ∆ C ( l ) := C ( f ∗ ) − C ( f ∗ ( l )) = ˜ τ f ∗ l ( V i − V j ) κ − + r ij a l , (13) where ˜ τ is a constant independent of l , and r ij is the effective resistance betweennodes i and j in G R .Proof. See the Appendix.Intuitively, Theorem 1 states the social cost variation after intervention onthe edge l = ( i, j ) depends: • proportionally on V i − V j , which may be interpreted as a gradient of relativeposition from the tail to the head of l , since V i may be seen as a measureof relative position with respect to the origin and the destination, with V o = 1 at the origin, V d = 0 at the destination, and every other node inbetween assuming an intermediate value that approaches 1 when the nodeis closer to the origin and far from the destination, and 0 in the oppositecase; • proportionally on the unpertured flow f ∗ l , which is a measure of impor-tance of the edge from the traffic perspective; • inversely on r ij /a l , which is a non-negative quantity, no greater than 1;this term is maximum when the edge l is a bottleneck and decreases asthe number of alternative paths from i to j increases.11n order to solve Problem 1 by the electrical formulation, we need to compute(13) for every edge l . The unperturbed equilibrium f ∗ is assumed to be ob-servable and therefore given, and the potential V can be derived by solving thelinear system (12). Observe that V has to be computed only once. However,the computation of r ij involves the solution of a linear system, and is neededfor every edge l = ( i, j ), so that the solution of Problem 1 by the electricalformulation requires to solve E linear systems (see Aldous and Fill [1]), whereasby formulation (9) we need to compute a row and the diagonal of the inverseof H , which still is computationally onerous when the network is large. In thenext section we propose a method to approximate the effective resistance be-tween a pair of neighbors that, under a suitable assumption on the sparsenessof the network, does not scale with the size of the network, allowing for a moreefficient solution to Problem 1. As seen in the previous section, Problem 1 may be rephrased in terms of elec-trical quantities over a resistor network. However, even in this formulation thecomplexity of the problem scales badly because it requires to solve E linearsystems whose size grows linearly with N. Since the computational bottleneckis represented by the effective resistance between every pair of adjacent nodesof the network, in the next subsection we propose a computationally cheapermethod to approximate this quantity. The main idea of our method is that,even though the effective resistance depends on the entire network, when i and j are adjacent nodes, r ij can be approximated by looking at a local portionof the network only. We then formulate an algorithm to approximately solveProblem 1 by exploiting the approximation of the effective resistance. Let us introduce the notion of cutting and shorting a network.
Definition 4.1 (Cutting at distance d ) . A resistor network G R is cut at distance d with respect to a pair of nodes ( i, j ) if every node at distance greater than d from both i and j is removed, and every edge having at least one end in the setof the removed nodes is removed. Let G U d ij and r U d ij denote such a network andthe effective resistance on it, respectively. Definition 4.2 (Shorting at distance d ) . An resistor network G R is shorted atdistance d with respect to a pair of nodes ( i, j ) if all the nodes at distance greaterthan d from both i or j are shorted together, i.e., an infinite conductance is addedbetween each pair of such nodes. Let G L d ij and r L d ij denote such a network andthe effective resistance on it, respectively. We refer to Fig. 2 for an example of these techniques applied to a regulargrid. We next prove that r U d ij and r L d ij are respectively an upper and a lower12
23 4 5 6789 10 11 12 13 141516171819 20 21 22 23 24 25 2627282930313233 34 35 36 37 3839404142434445 46 47 484950515253 545556 G U G L Figure 2: Square grid. Above: the yellow, orange and red nodes are at distance1, 2 and 3, respectively from the green nodes. Bottom left: cut at distance 1.Bottom right: shorted at distance 1. We stress that in the bottom right networkthe edges connecting yellow nodes with node s do not have unitary weights.bound for the effective resistance r ij for every pair of adjacent nodes. To thisend, let us introduce the Rayleigh’s monotonicity laws. Lemma 1 (Rayleigh’s monotonicity laws (Levin and Peres [23])) . If the resis-tances of one or more edges are increased, the effective resistance r ij betweenany two nodes i and j cannot decrease. If the resistances of one or more edgesare decreased, r ij cannot increase. Proposition 4.
Let G R be a resistor network, D max denote the maximal weighteddegree of the network, and r ij be the effective resistance between any two neigh-boring nodes i and j . Then, r U d ij ≥ r U d ij ≥ r ij ≥ r L d ij ≥ r L d ij , ∀ d > d ≥ . Moreover, /D max ≤ r L d ij ≤ r U d ij ≤ /W ij , ∀ d ≥ . (14)13 roof. Cutting a network at distance d is equivalent to setting to infinity the re-sistance of all the edges that have one node at distance greater than d . Shortinga network at distance d is equivalent to setting to zero the resistance betweenany pair of nodes at distance greater than d . Then, by Rayleigh’s monotonicitylaws, r U d ij ≥ r ij ≥ r L d ij . Similar arguments may be used to show that, if d < d , r U d ij ≥ r U d ij and r L d ij ≤ r L d ij . The right inequality of (14) follows from noticingthat, by Rayleigh’s monotonicity laws, the effective resistance computed in thenetwork with only nodes i and j , which is equal to 1 /W ij , is an upper boundfor r U ij . The left inequality follows from noticing that the effective resistanceon the network in which every node except j is shorted with i , which results ina network with only two nodes and a conductance between i and j not greaterthan D max (hence, resistance no less than 1 /D max ) is a lower bound of r L ij .Proposition 4 states that cutting and shorting a network provides upper andlower bound for the effective resistance. Moreover, the tightness of the boundsis a monotone function of the distance d . Based on the method for approximating the effective resistance, we here proposean algorithm to approximately solve Problem. Our approach is detailed inAlgorithm 1.
Algorithm 1:Input:
The resistor network G R = ( N , E R , W ), the rescale parameter κ and the distance d ≥ Output:
The optimal edge l ∗ d for the intervention.Compute V by solving the sparse linear system V o = 1 , V d = 0 , V i = (cid:88) j ∈N P ij V j ∀ i ∈ N \ o , d; for each l = ( i, j ) ∈ E do Construct G U d ij and G L d ij ;Compute r U d ij and r L d ij on G U d ij and G L d ij . end Select l ∗ d such that l ∗ d ∈ argmax l =( i,j ) ∈E ∆ C d ( l ) := f ∗ l ( V i − V j ) κ − + r Udij + r Ldij a l . Note that the performance of Algorithm 1 depends on the choice of theparameter d . Specifically, the higher d is the better is the approximation of the14ffective resistance and the closer is the output of Algorithm 1 to the achievingthe minimum of Problem 1. Theorem 2.
Let ∆ C ( l ) be the cost variation corresponding to intervention onedge l = ( i, j ) ∈ E as given in Theorem 1, ∆ C d ( l ) be the cost variation estimatedby Algorithm 1 for a given distance d ≥ , and (cid:15) ijd := r U d ij − r L d ij a l . Then, (cid:12)(cid:12)(cid:12)(cid:12) ∆ C ( l ) − ∆ C d ( l )∆ C ( l ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) ijd (cid:16) κ − + r Udij + r Ldij a l (cid:17) ≤ (cid:15) ijd (cid:16) κ − + D max · a l (cid:17) Furthermore, ∆ C ( l ) ≥ ˜ τ f ∗ l ( V i − V j ) κ − + r Udij a l . (15) Proof.
See the Appendix.In the next section we provide conditions for (cid:15) ijd to go to zero for largedistance d in the limit of infinite networks. In the rest of this section we showthat the tightness of the bounds (and therefore (cid:15) ijd ), and their computationalcomplexity (for a fixed d ) depend only on the local structure around the edge l , and do not scale with the size of the network, under a suitable assumption. Assumption 2.
Let G R be a resistor network, l = ( i, j ) an arbitrary edge of thenetwork, and N ≤ d denote the set of nodes that are at distance no greater than d from at least one of i and j . We assume that the network is sparse in such away that the cardinality of N ≤ d does not depend on N for any d . Assumption 2 is suitable for transportation networks, because of physicalconstraints not allowing for the degree of the nodes to grow unlimitedly (thinkfor instance of a square grid, where the degree of the nodes is 4 no matter whatthe size of the network is). Notice also that, under Assumption 2, N and E areproportional. Hence, from now on we refer indistinctly to N or E to denote thesize of the network.
Proposition 5.
Let G R = ( N , E R , W ) be a resistor network, ( i, j ) be a pair ofneighbors, d ≥ . Then, the time complexity of the bounds and the tightness ofthe bounds are functions of the structure of G R within distance d + 1 from i and j only. Furthermore, under Assumption 2 they do not depend on the size of thenetwork.Proof. See the Appendix. 15 emark 2.
Proposition 5 states that under Assumption 2 the time complexityto approximate a single effective resistance does not scale with the size of thenetwork for every distance d . Therefore, all the effective resistance may be ap-proximated in linear time. V is computed via a diagonally dominant, symmetricand positive definite linear systems. The design of fast algorithms to solve thisclass of problem is an active field of research in the last years. To the best ofour knowledge, the best algorithm has been provided by Cohen et al. [9] and hascomplexity O ( M log k N log 1 /(cid:15) ) , where (cid:15) is the tolerance error, k is a constant,and M is the number of nonzero elements in the matrix of the linear system.Since in our case M scales with E , and since E scales with N under Assumption2, Algorithm 1 is quasilinear in N . In this section we provide a characterization of the tightness of the boundsof the effective resistance between neighbors in terms of random walks over theresistor networks G R , G U d ij and G L d ij . We then use this characterization to providea sufficient condition on the network under which the approximation error ofthe bounds vanishes asymptotically as the distance d grows. To this end, weintroduce the following notation. Let • T S and T + S denote the hitting time, (i.e., the first time t ≥ S ), and return time (i.e., the first time t > S ), respectively. • N d denote the set of the nodes that are at distance d from either i or j and at distance greater or equal than d from the other node (we omit i and j for simplicity of notation). • p k ( X ), p U d k ( X ) and p L d k ( X ), denote the probability that the event X oc-curs, given a random walk that starts in k at time 0 and evolves over theresistor networks G R , G U d ij and G L d ij , respectively.The next proposition provides a characterization for the distance between theupper and lower bound on r ij in terms of probabilities of random walks over G R , G U d ij and G L d ij . Proposition 6.
Let G R = ( N , E R , W ) be a resistor network. Based on therandom walk on the resistor network, r U d ij − r L d ij ≤ D ii ( W ij ) p i ( T N d < T j ) (cid:124) (cid:123)(cid:122) (cid:125) Term 1 · max g ∈ N d (cid:0) p U d g ( T i < T j ) − p L d g ( T i < T j ) (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) Term 2 . (16) Proof.
See the Appendix.In the next subsection we use this result to study the asymptotic behaviourof the error term (cid:15) ijd = ( r U d ij − r L d ij ) /a l . In § § → § → (cid:57) → (cid:57) § (cid:15) ijd → (cid:57)
0, see Levin andPeres [23]) if Term 2 → In this section we show that a sufficient condition under which the distancebetween the upper and the lower bound vanishes as the distance d goes toinfinity is that the network is recurrent. We start by introducing this class ofnetworks. Definition 5.1 (Recurrent random walk) . A random walk is recurrent if, forevery starting point, it visits its starting node infinitely often with probabilityone (Levin and Peres [23]).
Definition 5.2 (Recurrent network) . An infinite network G = ( N , E , W ) isrecurrent if the random walk on the network is recurrent. The next theorem states that the distance between the upper and the lowerbound on recurrent networks vanishes as d goes to infinity, provided that thedegree of every node is finite. Theorem 3.
Let G R be an infinite recurrent resistor network, and let the max-imal weighted degree D max be finite. Then, for every edge l = ( i, j ) , lim d → + ∞ ( r U d ij − r L d ij ) = 0 . Proof.
It is proved in Levin and Peres [23, Proposition 21.3] that a graph isrecurrent if and only if lim d + →∞ p i ( T N d < T j ) = 0 ∀ i, j ∈ N . (17)Observe that, to hit any node at distance d + 1, the random walk starting from i has to hit at least a node at distance d . Hence, the sequence (cid:8) p i ( T N d < T j ) (cid:9) ∞ d =1 is non-increasing in d and the limit is well defined. Then, from (16), (17), from17he fact that 0 ≤ p U d g ( T i < T j ) − p L d g ( T i < T j ) ≤ g , and fromthe assumptions D max < + ∞ and W ij > i and j are adjacentnodes), it followslim d → + ∞ r U d ij − r L d ij ≤ D max ( W ij ) lim d → + ∞ p i ( T N d < T j ) = 0 , which completes the proof. Remark 3.
Theorem 3 implies that lim d → + ∞ (cid:15) ijd = 0 on recurrent networksfor every neighboring nodes i and j . Hence, by Theorem 2, the cost variationcorresponding to intervention on edge l = ( i, j ) can be estimated with vanishingerror. Observe that not only the error term (cid:15) ijd , but also the relative error (cid:15) ijd /r ij , vanishes asymptotically, since r ij ≥ /D max . Recurrence is a sufficient condition to guarantee lim d → + ∞ (cid:15) ijd = 0, but isnot necessary, as discussed in the next subsection. We here provide examples of infinite networks for all of the cases in Table 1.Observe that, for every edge l = ( i, j ) ∈ E R , the network cut at distance d from l and the network shorted at distance d from l differ for a node only. Let s denotesuch node, which is the result of shorting all the nodes at distance greater than d from both i and j in a unique node. Intuitively speaking, our conjecture isthat Term 2 in (16) is small when the network has many short paths. In fact, inthis case, adding the node s does not affect too much the probability, startingfrom any node in N d , of hitting i before j , thus making Term 2 small. Thisintuition can be made more clear by the next examples. Consider an infinite unweighted bidimensional grid as in Fig. 3. This networkis relevant for the NDP since many transportation networks are very similarto grids. This network is recurrent (Levin and Peres [23]), hence Theorem 3guarantees that Term 1 and thus (cid:15) ijd go to 0 for large d . Our conjecture,confirmed by numerical simulations, is that, for every node g ∈ N d ,lim d → + ∞ p U d g ( T i < T j ) = 1 / , lim d → + ∞ p L d g ( T i < T j ) = 1 / . Hence, this is recurrent network for which also Term 2 vanishes asymptotically.
Consider an infinite unweighted tridimensional grid. This network is not recur-rent (Levin and Peres [23]), therefore Term 1 does not go to 0 and we cannot18
23 4 5 6789 10 11 12 13 141516171819 20 21 22 23 24 25 26272829303132
Figure 3: Bidimensional square grid, cut at distance d = 3. The red nodesbelong to N d . As d grows, p g ( T i < T j ) approaches 1 / g ∈ N d , becausethere are many short paths.conclude that (cid:15) ijd → g ∈ N d ,lim d → + ∞ p U d g ( T i < T j ) = 1 / , lim d → + ∞ p L d g ( T i < T j ) = 1 / . Hence, this is a non-recurrent network for which Term 2 (and therefore (cid:15) ijd )vanishes as the distance grows.
Consider an infinite unweighted ring as in Fig. 4. Consider nodes c and e as inFig. 4. Then, p U d c ( T i < T j ) = 1 , p U d e ( T i < T j ) = 0 . for each d (even d → + ∞ ), whereas, p L d c ( T i < T j ) = d d + 1 −−−−−→ d → + ∞ , p L d e ( T i < T j ) = d + 12 d + 1 −−−−−→ d → + ∞ , since this case is equivalent to the gambler’s ruin problem (Levin and Peres [23]).Hence, Term 2 does not vanish for the ring. This is due to the fact that, on thering, all the paths from c to j not passing in i include the node s . Still, Term1, and therefore (cid:15) ijd , vanish asymptotically by Theorem 3, since this network isrecurrent. We finally propose an infinite network for which (cid:15) ijd does not converge asymp-totically. This network is not relevant for traffic applications, since it admits one19 L ij i ja bc es G U ij i ja bc e Figure 4: Left: shorted ring at distance d = 2. Right: cut ring at distance d = 2. i j Figure 5: The double tree is an infinite non-recurrent network. On this networklim d →∞ (cid:15) ijd = 1 / i and j respectively, linked byan edge l = ( i, j ), as in Fig. 5, and it is assumed unweighted. It can be shownthat on the double tree network the probability that the random walk, startingfrom i , returns on i is equal to the same quantity for a biased random walkover an infinite line (for more details we refer to the Supplementary Materials).Since the biased random walk on a line is not recurrent (see Levin and Peres[23]), this equivalence shows that the double tree network is non-recurrent, andTerm 1 →
0. Moreover, we show in the Supplementary Materials thatlim d → + ∞ r U d ij − r L d ij = 13 , thus implying that Term 2 (cid:57) Infinite regular grids are useful to test the performance of the bounds. Indeed,despite having an infinite number of nodes, the effective resistance between20
Distance B ound t i gh t ne ss Figure 6: Average relative error of the bounds on Oldenburg network as afunction of distance d .adjacent nodes can be computed exploiting the symmetric structure of the grid.We focus on the square grid, but similar arguments can be applied to any regularinfinite grid. Lemma 2 (Bartis [2]) . Let G R be an infinite square grid with unitary resis-tances. Then, the effective resistance between two neighboring nodes is / . In Table 2 the performances of the upper and lower bounds are shown.Numerical simulations show that for every edge l = ( i, j ), r U d ij − r ij r ij = r ij − r L d ij r ij = O (1 /d ) . We underline that the relative errors of the bounds are symmetric only in thesquare grid, but they scale similarly in all the regular bidimensional grids. Ob-Table 2: Table of upper and lower bound in infinite square grid. d = 1 d = 2 d = 3 d = 4 d = 5( r U d ij − r ij ) /r ij / . . . . r ij − r L d ij ) /r ij / . . . . d = 5, the upper and thelower bounds give a good estimation of the true effective resistance. In this section we present the performances of the cutting and shorting tech-niques on the traffic network of the city of Oldenburg (Brinkhoff [6]). Thenetwork is composed of 6105 nodes and 7035 edges, and its diameter is 104.21he network is assumed to be unweighted, with a l = 1 for every edge l . Theaverage relative error of the bounds, i.e., AT d := 1 E (cid:88) ( i,j ) ∈E r U d ij − r L d ij r ij is shown in Table 3 and Fig. 6. Even for this network, the error of the boundsTable 3: Table of the average relative error of the bounds at distance d .d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 d=9 d=10 AT d In this work we study a discrete network design problem, where a single edge canbe improved. We reformulate the problem in terms of electrical quantities, inparticular in terms of the effective resistance between the two nodes at the end ofthe edge. We then provide a method to approximate such effective resistance byperforming only local computations. Both the tightness and the computationalcomplexity of our bounds do not depend on the size of the network, but on thelocal structure only. Based on the electrical formulation and our approximationmethod for the effective resistance we propose an efficient algorithm to solve theoriginal design problem.An interesting direction for the future is a deeper analysis on tightness of thebounds on effective resistance as a function of the network and the distance d ,since so far we have a result for the asymptotic behaviour only. Future researchlines also include extending to the case of multiple interventions, dealing withdifferent interventions (e.g building new roads), and the relaxation of someassumptions like the single origin and destination, and the assumption that theset of used edges is not affected by the intervention. Acknowledgments.
This research was carried on within the framework of the MIUR-funded
Progettodi Eccellenza of the
Dipartimento di Scienze Matematiche G.L. Lagrange , Po-litecnico di Torino, CUP: E11G18000350001. It received partial support fromthe MIUR Research Project PRIN 2017 “Advanced Network Control of Fu-ture Smart Grids” (http://vectors.dieti.unina.it), and by the
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Preliminaries on connection between Green’s function, ran-dom walks and effective resistance
Let G R = ( N , E R , W ) denote a connected resistor network, D = diag W ∈ R N × N denote the degree matrix, and P = D − W ∈ R N × N the associated ran-dom walk on the network. We define as k P ∈ R (N − × (N − the matrix wherethe row and the column referring to the node k are deleted. k P can be thoughtof as the transition matrix of a killed random walk obtained by creating a ceme-tery in the node k . We then define the Green’s function k G ∈ R (N − × (N − as k G := ∞ (cid:88) t =0 ( k P ) t = ( I − k P ) − , (18)where the last inequality follows from the connectedness of G R and from the factthat k P is substochastic and irreducible. Hence, it has spectral radius ρ < I − k P ) − = (cid:80) ∞ t =0 ( k P ) t (see Horn and Johnson [20]). Since (( k P ) t ) ij is the probability that the killed random walk starting from i is in j after t steps, k G ij indicates the expected number of times that the killed random walkvisits j starting from i before hitting k (Ellens and Spieksma [14]). While theseresults hold for any network, it is known (see Ellens and Spiesksma [14]) thatthe Green’s function of the random walk on a resistor network can be relatedto electrical quantities. In particular, with the convention that k G ik = k G ki = k G kk = 0 ∀ i ∈ N , (19)it is known that for any node k , k G ii − k G ji D ii + k G jj − k G ij D jj = j G ii D ii = 1 D ii p i ( T j < T + i ) = r ij , (20)where p i ( T j < T + i ) is as defined in Section 5, and r ij is the effective resistanceas defined in § Proof of Proposition 1
From (6), for all the used edges, γ ∗ i − γ ∗ j = a ( i,j ) f ∗ ( i,j ) + b ( i,j ) ∀ ( i, j ) ∈ E . So, by (3), any used path p = (o , n , n , · · · , n p , d), namely a path containingonly used edges, has cost at the equilibrium c p ( f ∗ ) = a (o ,n ) f ∗ (o ,n ) + b (o ,n ) + · · · + a ( n p , d) f ∗ ( n p , d) + b ( n p , d) = ( γ ∗ o − γ ∗ n ) + ( γ ∗ n − γ ∗ n ) + · · · + ( γ ∗ n p − γ ∗ d ) = γ ∗ o − γ ∗ d = γ ∗ o , (21)26here the last equivalence follows from the assumption γ ∗ d = 0. Hence, all theused paths at the equilibrium have the same cost γ ∗ o . Then, the social cost is C ( f ∗ ) = (cid:88) l ∈E f ∗ l d l ( f ∗ l ) = (cid:88) l ∈E d l ( f ∗ l ) (cid:88) p ∈P R lp z ∗ p = (cid:88) p ∈P z ∗ p (cid:88) l ∈E R lp d l ( f ∗ l ) = (cid:88) p ∈P z ∗ p c p ( f ∗ ) = γ ∗ o (cid:88) p ∈P z ∗ p = γ ∗ o τ, where the second equivalence follows from (2), the forth from (3), the fifthfrom (21) and the last one from the mass constraint (1). The proof follows byapplying similar arguments for C ( f ∗ ( l )). Proof of Proposition 2
Let x ∗ ( l ) = H ( l ) − y be the equilibrium after the intervention on edge l . Noticethat H ( l ) = H − a l (cid:16) − κ (cid:17) δ l δ Tl , Since only the element A ll is perturbed, H ( l ) is a rank-1 perturbation of H .Hence, its inverse can be computed by Sherman-Morrison formula (Shermanand Morrison [30]): H ( l ) − = H − − − a l (cid:0) − κ (cid:1) H − δ l δ Tl H − − a l (cid:0) − κ (cid:1) δ Tl H − δ l . By right multiplying by y , and recalling that x ∗ = H − y , it follows x ∗ − x ∗ ( l ) = − a l (cid:0) − κ (cid:1) H − δ l δ Tl x ∗ − a l (cid:0) − κ (cid:1) δ Tl H − δ l . Observe that γ ∗ o ( l ) is x ∗ E +1 ( l ) by construction. Then, by selecting the (E − γ ∗ o − γ ∗ o ( l ) = − a l (cid:0) − κ (cid:1) H − ,l ) x ∗ l − a l (cid:0) − κ (cid:1) H − ll = ( κ − a l H − ,l ) x ∗ l ( κ − a l H − ll − κ . The statement follows from Proposition 1 and from noticing that x ∗ l = f ∗ l byconstruction. Proof of Theorem 1
By letting ˜ W ∈ R (N − × (N − and ˜ D ∈ R (N − × (N − denote the restrictionover N \ d, i.e., all the nodes except the destination, of W and D respectively,it is straightforward to check that Q = ˜ D − ˜ W .
27e notice that, d P , as defined in §
7, is d P = ˜ D − ˜ W . d P is substochastic, since the rows referring to nodes pointing to the destinationsum to less than one. We now prove that Q is invertible. Indeed, Q − = ( ˜ D − ˜ W ) − = ( ˜ D ( I − d P )) − = ( I − d P ) − ˜ D − = ∞ (cid:88) t =0 ( d P ) t ˜ D − = d G ˜ D − , (22)where the penultimate equivalence follows from strongly connectedness of G (and therefore, connectedness of G R ) and (18).Let l = ( i, j ) ∈ E an arbitrary edge. From (10) it follows: H − ll = 1 a l − K l, : Q − K T : ,l . By (11), H − ll = (cid:40) a l − a l ( Q − ii + Q − jj − Q − ij − Q − ji ) if j (cid:54) = d a l − a l ( Q − ii ) if j = d . (23)We now construct ˆ Q − ∈ R N × N and d ˆ G ∈ R N × N by adding a zero column anda zero row to Q − and d G and ˆ K ∈ R E × N by adding a zero column to K corresponding to the destination. Thus, (23) can be written as H − ll = 1 a l − a l ( ˆ Q − ii + ˆ Q − jj − ˆ Q − ij − ˆ Q − ji ) ∀ l ∈ E , and, by ˆ Q − = d ˆ GD − (which follows from (22)), H − ll = 1 a l − a l (cid:16) d ˆ G ii − d ˆ G ji D ii + d ˆ G jj − d ˆ G ij D jj (cid:17) . Finally, by noticing that the definition of d ˆ G is coherent with (19), and by (20),we get H − ll = 1 a l − r ij a l . (24)Using the same notation with ˆ Q − and ˆ K − to handle also edges l pointing tothe destination, from (10), (11), (22) and from symmetry of ˆ Q − , it follows: H − ,l = − ( ˆ Q − ˆ K T ) l = − ˆ Q − i − ˆ Q − j a l = − ˆ Q − i − ˆ Q − j a l = − d ˆ G i o − d ˆ G j o a l D oo , (25)since the first node is the origin by construction.We now prove that V i = d ˆ G i od ˆ G oo ∀ i ∈ N . (26)28o this end, notice that both the potential and the Green’s function are har-monic functions (see Levin and Peres [23]) satisfying same boundary conditionson the origin and the destination, i.e., V o = 1 = d ˆ G ood ˆ G oo , V d = 0 = d ˆ G dod ˆ G oo , since d ˆ G do = 0 by construction, which implies (26) by Levin and Peres [23,Proposition 9.1]. Plugging (26) into (25), we get H E+1 ,l = − d ˆ G oo a l D oo ( V i − V j ) . (27)The statement follows from plugging (27) and (24) in (9) with the assignment˜ τ = τ · d ˆ G oo /D oo = τ · d G oo /D oo . Proof of Proposition 3
A sufficient condition under which E + = ∅ is that the first E components of x ∗ = H − b , corresponding to equilibrium edge flows, are nonnegative. Indeed,since (4) is strictly convex, if the flows corresponding to x ∗ = H − b satisfy theconstraint f ∗ ≥ , then f ∗ is feasible and is the unique Wardrop equilibrium,with λ ∗ = 0 because of the complementary slackness. Hence, we look for con-ditions satisfying x ∗ l ≥ l ∈ { , · · · , E } . Consider an arbitrary edge l = ( i, j ). From (5) and (10), it follows: x ∗ l = − b l a l + [ KQ − K T ] l : b + [ KQ − ] l : ( ν − ) . With same arguments as in Proof of Theorem 1, we replace Q − with ˆ Q − and K with ˆ K to take into account edges pointing to the destination. Since ν − = τ δ o , x ∗ l = − b l a l + [ ˆ K ˆ Q − ˆ K T ] l : b + τ [ ˆ K ˆ Q − ] l o = − b l a l + [ ˆ K ˆ Q − ˆ K T ] l : b + τ d ˆ G i o − d ˆ G j o a l D oo where the last equivalence follows from (25) and from the fact that the origin isthe first node by construction. If d ˆ G i o − d ˆ G j o >
0, then, for any τ ≥ τ l with τ l = b l a l − [ K ˆ Q − K T ] l : b d ˆ G i o − d ˆ G j o a l D oo it holds x ∗ l ≥
0, which in turn implies that if τ ≥ τ := { τ l } L l =1 , then E + = ∅ .Moreover, if the delays are linear, d ˆ G i o − d ˆ G j o > x ∗ l ≥ E + = ∅ τ , because b = 0. By (26) and Ohm’s law (see [1]), d ˆ G i o − d ˆ G j o is proportional to V i − V j = i l a l , where i l denotes the current flowing throughedge l . Then, it suffices to show that i l >
0. To this end, we observe that,by definition, if the network is series-parallel, it is a single edge (o , d) or itcan obtained by connecting in series or in parallel two series-parallel networks.Thus, a series-parallel network can be reduced to a single edge by recursivelyi) merging two edges l and l connected in series into a single edge l , with a = a + a (recall that the coefficients a e correspond to resistances on theresistor network), and ii) merging two edges l and l connected in parallel intoa single edge l , with a = a a / ( a + a ). Moreover, observe that in both cases i > i > i >
0. Indeed, in case i) i = i = i , and incase ii) i = i a /a and i = i + i . Obviously, when the network is reducedto a single edge, the flow on the unique edge is positive because τ >
0. Then,by applying those arguments recursively, for every edge l = ( i, j ) ∈ E , i l > ⇒ V i − V j > , implying that if τ ≥ τ then x ∗ l ≥ E + = ∅ . Proof of Theorem 2
Using the definitions, | ∆ C ( l ) − ∆ C d ( l ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ τ f ∗ l ( V i − V j ) κ − + r ij a l − ˜ τ f ∗ l ( V i − V j ) κ − + r Udij + r Ldij a l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ τ f ∗ l ( V i − V j ) κ − + r ij a l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Udij + r Ldij − r ij a l κ − + r Udij + r Ldij a l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , Note also that | r U d ij + r L d ij − r ij | a l ≤ | r U d ij − r ij | + | r ij − r L d ij | a l = r U d ij − r ij + r ij − r L d ij a l = r U d ij − r L d ij a l = (cid:15) ijd . Putting those two together, and using (13), we get (cid:12)(cid:12)(cid:12)(cid:12) ∆ C ( l ) − ∆ C d ( l )∆ C ( l ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) ijd (cid:16) κ − + r Udij + r Ldij a l (cid:17) ≤ (cid:15) ijd (cid:16) κ − + D max · a l (cid:17) , where the last inequality follows from (14). Finally, (15) follows from r U d ij ≥ r ij . Proof of Proposition 5
The cut and shorted networks are obtained by finding the neighbors withindistance d and d + 1 from ( i, j ), respectively. The neighbors of a node i can30e found by checking the non-zero elements of W ( i, :). The neighbors withindistance d can be found by iterating such operation d times. Hence, the time forbuilding the cut and the shorted network depends on the local structure, which,under Assumption 2, does not depend on the size of the network. Since thebounds of the effective resistance are computed on these subnetwork, their timecomplexity and tightness depends on local structure, which, under Assumption2, is independent of the size of the network. Proof of Proposition 6
We introduce the following notation: • The index U d and L d indicate that the random walk takes place over G U d ij and G L d ij , respectively. So, for instance, k G U d ij denotes the expected numberof times that the random walk on the network G U d ij , starting from i , hits j before hitting k . • p i ( T u < T S ), with u ∈ S , denotes the probability that the random walkstarting from i hits the node u ∈ S before hitting any other node in S .By applying (20) to the effective resistance of edge l = ( i, j ) in the shorted andthe cut network, it follows r U d ij = j G U d ii D ii , r L d ij = j G L d ii D ii , where we recall that j G U d ii and j G L d ii are the expected number of visits on i ,before hitting j , starting from i , of the random walk defined on G U d ij and G L d ij respectively. The visits on i before hitting j can be divided in two disjoint sets:the visits before hitting j and before visiting any node in N d , and the visitsbefore hitting j but after at least a node in N d has been visited. Let G
We prove that the double tree network is not recurrent by showing that p i ( T i 3, respectively. Hence, the double tree is equivalent to a biasedrandom walk on a line as in Fig. 7, which is not recurrent (see Levin and Peres[23]).Since in the actual network and in the cut network there are no paths between i and j except the edge ( i, j ) (see Fig. 9 (a) and (b)), r ij = r U d ij = 1 . Computing r L d ij is more involved. First, referring to Fig. 8, we note that, becauseof the symmetry of the network, the effective resistance between i and j in theshorted network (c), which is r L d ij , is equivalent to the effective resistance in(d). Indeed, if we set potential V i = 1 and V j = 0, because of symmetry everyyellow node has potential 1 / 2. Thus, adding infinite conductance between allof them, i.e., shorting them, does not affect the current in the network (thisprocedure is also known in literature as gluing , see Levin and Peres [23]), andtherefore the effective resistance. The network (d) is series-parallel, so that theeffective resistance can be computed iteratively. Specifically, we refer to Fig. 9to explain the recursion that leads to r L d ij . From top to bottom, it is easy to seethat the first network has effective resistance between the two blue nodes equalto 3. The second network is the parallel composition of two of these, in serieswith two single edges. This procedure is iteratively repeated d − d = 2), leading to a network that, composed in parallel with34 (0) = 3 i j r (1) = 2 + r (0)2 i j r L ij = r (1) + 1Figure 9: The network in Fig. 8(d) is series-parallel. Then, it can be obtained byrecursively making parallel and series compositions of series-parallel networks.a copy of itself and with a single edge, is G L d ij . Hence, r L d ij is the result of thefollowing recursion. r (0) = 3 ,r ( n ) = 2 + r ( n − , d > n ≥ ,r L d ij = (1 + r ( d − ) − , which has solution (cid:40) r ( n ) = (2 d +2 − / d , d > n ≥ ,r L d ij = d +1 − d +1 +2 d − −−−−−→ d → + ∞ ..