Collectively optimal routing for congested traffic limited by link capacity
aa r X i v : . [ phy s i c s . s o c - ph ] N ov Collectively optimal routing for congested traffic limited by link capacity
Bogdan Danila, ∗ Yudong Sun, and Kevin E. Bassler
1, 2, † Department of Physics, University of Houston, Houston TX 77204-5005 Texas Center for Superconductivity, University of Houston, Houston TX 77204-5002
We show that the capacity of a complex network that models a city street grid to support congestedtraffic can be optimized by using routes that collectively minimize the maximum ratio of betweennessto capacity in any link. Networks with a heterogeneous distribution of link capacities and with aheterogeneous transport load are considered. We find that overall traffic congestion and averagetravel times can be significantly reduced by a judicious use of slower, smaller capacity links.
PACS numbers: 89.75.Hc, 89.75.Da, 05.60.-k
I. INTRODUCTION
What are the best routes for us to use for driving hometonight in rush hour traffic? Choosing the best route isa task that many of us face everyday. Knowing the bestchoice is also important to urban planners who designcity transportation networks. To answer this, and otherimportant questions, a vast amount of research has beendevoted in recent years to the analysis and optimizationof transport on complex networks [1–24]. If we wouldcollectively choose to use the best routes, then the trafficcongestion and our commute times could be reduced.When traveling in a city, the best route to use is pre-sumably the one that takes the least time to go fromorigin to destination. If there is little or no traffic, thisroute is simply the so-called shortest-path route. Onthe complex network that describes the city’s streets,the shortest-path route is defined as the path for whichthe sum of the weights of each of the links along thepath is minimal. The weight of a link can be defined asthe time it takes to travel from one end of the link tothe other when traffic is light. Generally, these weightsare inversely related to the transport capacities of thelinks (streets that can handle a lot of traffic are trav-eled faster). On the other hand, when traffic is heavysome links may become congested, and then the quickestroute to the destination may be a longer one involvingsmaller capacity links. It is not obvious, however, exactlyat what level of congestion the use of alternative routesbecomes advantageous, or how much the improvementsin network transport capacity and average travel timeamount to. The optimal transport routes that we find isthe collective optimum that occurs if everyone uses thebest routes for collective results. As such, it describesan important limit of collective behavior. This contrastswith the goal of many traffic studies that seek to optimizeresults individually through a learning process within theframework of evolutionary game theory [4].In this paper, we present methods to help answer theseimportant questions and demonstrate the effectiveness ofour results within the framework of a simple model [18].that captures important characteristics of urban street networks. This model has been shown to produce planargeometric networks with an exponential distribution ofthe node distances from the origin and with other topo-logical characteristics similar to the actual street net-works of various cities. We use this model to demon-strate the effectiveness of our methods, rather than aparticular realistic example, in order to make a statisticalanalysis by taking ensemble averages over many examplecity networks. The networks generated using this modelhave only homogeneous links and are topologically moresimilar to the street networks of older cities rather thanthose of modern cities. To account for the possibility oflinks with different transport capacities, we will extendthe model to include two types of streets. Nevertheless,it remains a relatively simple model of street networks,and is, thus, ideally suited for the purpose of this paper,which is to develop methods to determine optimal routesfor congested flow. Note that our results are importantnot only for the study of vehicular traffic but also forother types of link capacity limited network transport,including information transport on the Internet.Unlike previous studies of optimal routing for con-gested flow [2, 11–15], which assumed that traffic is lim-ited by node congestion, here we assume that the ca-pacity of the network is limited by the amount of trafficeach link can support. This important and realistic dif-ference requires a nontrivial variation in our methods. Aswe have shown previously [12, 14, 15], when transport islimited by node congestion and all nodes have the sameprocessing capacity, the transport capacity of the net-work is maximized by using a set of routes that minimizethe maximum betweenness of the nodes. In [12] we intro-duced an algorithm for finding such a set of routes anddemonstrated that, for a number of commonly studiednetwork topologies, it finds routes for which the capac-ity, at least, scales optimally with system size. Here weshow that when transport is limited by heterogeneouslink capacities, the transport capacity of the network ismaximized by minimizing the maximum betweenness tocapacity ratio of links. We also consider the traffic opti-mization problem with uneven traffic demands betweenthe various pairs of origin and destination nodes. In par-ticular, we study the case of a rush hour traffic burst em-anating from a central location. To obtain our results, weuse a variant of our previous routing optimization algo-rithm. Furthermore, we prove a formula that allows thequick computation of the average of the sum along thepath of any link-related quantities and use this formulato compute average travel times.The rest of the paper is structured as follows. In Sec. IIwe present the model we use, including details about thenetwork structure, the transport process, and our meth-ods for determining the optimal routing. In Sec. III weprove a general formula for computing route averages oncomplex networks in terms of link betweennesses and ap-ply that formula to compute the average travel time. InSec. IV we present our results for network transport ca-pacity and average travel time. In Sec. V we presentour conclusions and suggest a few directions for possiblefuture research.
II. MODEL
In our network transport dynamics, particles are as-sumed to travel along the network according to staticrouting protocols. The number of nodes on the networkis denoted by N . We assume that each outgoing link ofa node has a separate “first in, first out” (FIFO) particlequeue. The capacity C ij of link ( i, j ) is defined as theaverage number of particles transported per time stepassuming an infinite number of particles in its queue.Transport on the network proceeds in discrete time stepsand is driven by inserting new particles at the nodes. Theaverage number of new particles inserted per time stepat node i with destination at node j is r ij and we denote r i = P Nj =1 r ij . Each new particle is inserted at the end ofthe appropriate queue at its node of origin, namely thequeue corresponding to the first link it has to traverseon its way to destination. We use a stochastic sequentialupdating of the positions of particles that erases the cor-relations between particle arrival times and ensures thatboth the arrival and the delivery of particles are Pois-son processes [14]. The load of the network is defined asthe average h r i over all nodes on the network of r i . Thetransport capacity of the network is the critical value ofthe load h r i c above which the average particle arrival fluxexceeds the processing capacity at some link [2]. Whenthis happens, the number of particles in the system willcontinue to grow and the transport becomes jammed .Note that this is not necessarily what happens in thereal world, where when jamming begins to occur at somelink traffic readjusts to avoid the link. Nevertheless, ourgoal here is to study the collective optimum that pre-sumably would eventually result from this readjustmentprocess. As we will see, when the optimal routing finallyjams it does so simultaneously at a large number of linksand there is no way to reroute traffic from a jammed link without causing jamming at other links.To account for an uneven traffic demand pattern be-tween the nodes while still using a single parameter tocharacterize the load of the network it is convenient towrite r ij = C h r i ρ ij , where h r i is the average number ofparticles per node generated in the course of a time stepand ρ ij are nonnegative demand weights ( ρ ii = 0). If theweights are normalized such that P Ni,j =1 ρ ij = N ( N − N is the number of nodes, we find r ij = h r i N − ρ ij . (1) Network Model
For simplicity, we consider networks with a binarydistribution of link capacities. Then, “streets” are lowcapacity links and “highways” are high capacity links.Each network realization consists of a network topology,generated using the algorithm introduced by Barth´elemyand Flammini [18], and a set of capacities, generated bychoosing any link to be a highway with probability P h .Our model neglects the correlations that one may expectin real world between the capacities of adjacent links.These correlations arise from the fact that the subset oflinks that are “highways” are typically planned to forma well connected subnetwork. Enforcing such correla-tions, however, would have substantially complicated themodel. In addition, we note that, on one hand, connectedsets of highways do form in our model and, on the otherhand, isolated high capacity links do exist in real streetnetworks.The network model we use [18] is based on the idea oflinks growing gradually towards population centers. Inits original form, the model does not account for any dif-ference in the capacities of the links. All links grow at thesame constant rate towards the centroid of their adjacentpopulation centers, and new population centers are gen-erated at constant average rate. When a link reaches thecentroid of its adjacent population centers it splits intoseparate links directed towards each center. At the end,the nodes of the network will be the population centersand the points where links have been split. This modelhas been shown to generate networks with characteris-tics similar to those of real-world street networks, partic-ularly when the distances of the population centers withrespect to the city center are exponentially distributed.These similar characteristics include the distribution ofthe node degrees, as well as size and shape distributionsof the areas delimited by streets. Using this model ratherthan a particular real-world example allows the calcula-tion of ensemble averages over thousands of model citynetworks.An ensemble of network realizations is characterizedby the number of centers generated per time step R C ,the rate of growth of the links d , the total number ofcenters N C , the ratio between the capacity of a highwayand that of a street C h /C s , and the probability P h fora link to be a highway. Note that the number of nodes N on these networks can only be controlled on average,through N C and the ratio R C /d . Without loss of gener-ality, the average distance of the population centers fromthe center of the city is assumed to be unity. All re-sults presented below have been computed for R C = 0 . d = 0 . C h /C s = 8. Due to the high value of C h /C s , shortest-path routing will typically forgo the useof lower capacity links unless they are strictly needed toachieve connectivity. Traffic Routing
The transport capacity of a network depends on therouting protocol used and on the processing capacities ofthe various links. To demonstrate the importance of ourresults, we consider two different types of routing. One isthe “natural” shortest path (SP) routing computed withlink weights in inverse proportion to their respective ca-pacities. The other one is the “optimal” routing (OR) forcongested traffic resulting from the application of a rout-ing optimization algorithm which finds the set of routesthat maximizes the transport capacity of the network.Note that the problem of finding the absolute best setof optimal routes is mathematically related to the prob-lem of finding the minimal sparsity vertex separator [13],which has been shown to be an NP-hard problem [25]. InNP-hard problems the number of flops necessary for thecomputation of an exact solution can increase with thenumber of nodes N faster than any polynomial. Thus, asfor other problems that are known to be NP-hard [3, 26–29], it is useful to have an algorithm that finds, at least,nearly optimal routes and runs in only polynomial time.The following algorithm does just that. Its running timeis O ( N log N ) [ O ( N log N ) for one iteration and re-quiring O ( N ) iterations].Routing on a network is characterized by the set ofprobabilities P ( t ) ij for a particle with destination t cur-rently at node i to be forwarded to node j . The between-ness b ( s,t ) i of a node i with respect to a source node s anda destination node t is defined as the sum of the prob-abilities of all paths between s and t that pass through i . Node betweenness can be computed in terms of theprobabilities P ( t ) ij [2, 30]. The betweenness of link ( i, j )going from i to j is defined by [26] b ( s,t ) ij = b ( s,t ) i P ( t ) ij . (2)The average number of particles crossing link ( i, j ) in the course of a time step is given by w ij = N X s,t =1 b ( s,t ) ij r st . (3)Using Eq. 1, the flow of particles through link ( i, j ) be-comes w ij = h r i B ij N − , (4)with the total weighted betweenness of link ( i, j ) givenby B ij = N X s,t =1 b ( s,t ) ij ρ st . (5)Avoiding jamming requires w ij ≤ C ij for every link( i, j ). Consequently, maximum transport capacity isachieved when the highest betweenness-to-capacity ra-tio on the network ( B/C ) max is minimized and is givenby h r i c = ( N − / ( B/C ) max . (6)The minimization of ( B/C ) max can be achieved by vari-ous methods including, but not limited to, simulated an-nealing and extremal optimization. The results presentedhere were obtained using a variant of extremal optimiza-tion [31, 32]. Specifically, we modified an earlier algo-rithm that has been shown to converge to near-optimalrouting in the case of both geometric and small-worldnetworks if traffic is limited by the processing capacitiesof the nodes [12, 14, 15]. The original algorithm aimsat minimizing the maximum betweenness of any node onthe network. The variant employed here minimizes themaximum betweenness-to-capacity ratio of any link onthe network. It is, therefore, capable of maximizing thetransport capacity of networks with traffic limited by ar-bitrary link capacities. The algorithm works as follows:1. Assign an initial weight equal to one to every linkand compute the shortest paths between all pairsof nodes and the betweenness of every link.2. Find the link that has the highest betweenness-to-capacity ratio ( B/C ) max and increase its weight byone.3. Recompute the shortest paths and the between-nesses. Go back to step 2.Note that this algorithm, like many other transportoptimization algorithms, is essentially a shortest path al-gorithm with variable weights that are tuned in order toreduce congestion. The variable weights allow traffic tobe routed around the locations in the network that aremost likely to jam. T a vg r (a) P h =0.0(b) P h =0.3(c) P h =0.7 FIG. 1: (Color online) Average travel time (in time steps)vs. network load (in particles per time step per node) for atypical network realization with N = 200 nodes. Results areshown for shortest-path routing, SP (blue dashed lines), andfor optimal routing, OR (red solid lines), at three differenthigh capacity link densities, (a) P h = 0, (b) P h = 0 .
3, and (c) P h = 0 .
30 100 300 1000
The average travel time can be computed from linkbetweennesses assuming the arrival and the delivery ofparticles at every node are Poisson processes. We willnow prove a general formula for calculating averages overthe entire set of routes considered on a network and thenapply this formula to calculate the travel time. Let Q ij be any quantity associated with the links of a network.To calculate the average over all particle routes of thesum of Q ij along the route it is convenient to write thebetweenness in terms of the probabilities for completeroutes. Let π n ( s, t ) be the ordered set of all nodes alongthe n -th distinct route between s and t (including s butexcluding t ) and p n ( s, t ) be the probability for a particleto be routed along that route. The number of distinctroutes between s and t is N ( s, t ). Then, b ( s,t ) ij = N ( s,t ) X n =1 p n ( s, t ) X k ∈ π n ( s,t ) δ ik δ j,next ( k | n,s,t ) (7)where next ( k | n, s, t ) is the successor of node k in π n ( s, t ) ∪ { t } . Let us now compute the quantity (Σ Q ) ( s,t ) avg defined by (Σ Q ) ( s,t ) avg = N X i,j =1 Q ij b ( s,t ) ij . (8)By substituting Eq. 7 into Eq. 8, we find(Σ Q ) ( s,t ) avg = N ( s,t ) X n =1 p n ( s, t ) X k ∈ π n ( s,t ) Q k,next ( k | n,s,t ) . (9)The inner sum on the right-hand side of Eq. 9 is the sumof Q ij along the n -th route and (Σ Q ) ( s,t ) avg is its averageover all routes between s and t .The network-wide average of (Σ Q ) ( s,t ) avg weighted by theelements of the traffic demand matrix r st is then(Σ Q ) avg = 1 N ( N − N X i,j =1 Q ij B ij , (10)where B ij are defined in Eq. 5 and we used the fact thatthe sum of all ρ st equals N ( N − i, j ) is taken to be T ij = 1 + h q ij i C ij , (11)where h q ij i is the average length of the queue associatedwith this link and C ij is the link capacity. The 1 /C ij term accounts for the travel time in the limit of low trafficand is consistent with the idea that higher capacity linksare covered faster. The additional travel time due to linkcongestion is accounted for by the second term. Note thatthis is a very simple model of the traffic flow along thelink, which we use here for the purpose of illustrating ourmethod. More complex (and more accurate) nonlinearempirical formulas could be used to connect traffic flowand link capacity to the average travel time along a link.Such formulas could also be used in conjunction withEq. 10 to calculate network-wide route averages.Since both the arrival and the delivery of particles atevery queue are well approximated by Poisson processes,the average queue length is given by [33] h q ij i = w ij C ij − w ij . (12)By using Eqs. 11 and 12 in 10 we obtain the averagetravel time as a function of load T avg = 1 N N X i,j =1 B ij ( N − C ij − h r i B ij . (13)Additional time delays associated with traveling alongthe links may also be included in the calculation of T avg by using Eq. 10. IV. RESULTSAverage Travel Time
Figure 1 shows a comparison between the SP and ORaverage travel times as functions of network load r for onenetwork with uniform traffic demand at three differentvalues of P h . The total travel time along a link is taken tobe the waiting time in its queue plus an additional delayequal to the inverse of the link’s capacity. The waitingtime in queues accounts for congestion, while the inverseof a link’s capacity accounts for its travel time in the limitof low traffic. Note that routing is explicitly optimizedto maximize the transport capacity and not to minimizethe average travel time. The results shown are typical fornetworks with an average number of nodes h N i ≈ N C = 100.The load above which OR outperforms SP is in this caseabout 80% of the critical load under SP routing. Thismay be regarded as the threshold for congestion whenSP routing is used. Optimization increases the transportcapacity of this network by a factor of about 1.7 if allroutes have the same capacity, while for values of P h between 0.2 and 0.8 the factor of improvement increasesto more than 2. Note also the decrease of the averagetravel time at low loads with increasing P h regardless ofthe type of routing. This is due to the increased averagecapacity of the links. P h < B / C > FIG. 3: (Color online) Ensemble averages of the networkaverage and network maximum betweenness-to-capacity ra-tios, h ( B/C ) avg i and h ( B/C ) max i respectively, as functionsof P h for networks with an average of 250 nodes. Resultsare for (cid:10) ( B/C ) SPavg (cid:11) (black solid), (cid:10) ( B/C ) ORavg (cid:11) (red dotted), (cid:10) ( B/C ) SPmax (cid:11) (green dashed), and (cid:10) ( B/C ) ORmax (cid:11) (blue dot-dashed).
Network Transport Capacity
In Fig. 2 results are presented for the ensemble av-erages of (
B/C ) avg and ( B/C ) max for networks with h N i ≈
250 which were generated for N C = 120. All er-ror bars represent 2 σ estimates. The subscripts avg and max denote the average and respectively maximum overall links of a network with a given link capacity configura-tion. Averaging over an ensemble of network realizationsis denoted by angular brackets. This is an average overan ensemble of network topologies and over an ensembleof link capacity configurations for each network topol-ogy. The averages shown in Fig. 2 were computed for 100network topologies and 30 link capacity configurations.Each ensemble of network realizations is characterized bythe average number of nodes h N i and by the probability P h for a link to be a highway.Figure 2(a) shows a log-log plot of h ( B/C ) avg i and h ( B/C ) max i for both SP and OR at P h = 0 .
3. Since thelines are nearly straight, the ensemble averages of thesequantities scale with average network size as a power law.Similar power law dependence has been observed in otherstudies [13, 14] in the case of both the maximum and theaverage node betweenness and is discussed in [13]. Notethat any decrease in h ( B/C ) max i can only be obtained atthe expense of an increase in h ( B/C ) avg i since avoidingcongestion along the shortest path means taking longerroutes that contribute to the betweenness of more links.With OR, the slopes of h ( B/C ) avg i and h ( B/C ) max i areessentially the same. Thus, the capacity of the routeswe find, at least, scales optimally with system size. Thisbehavior is similar to that of the node betweenness whentransport is limited by node processing capacity. How-ever, the finite size effects are stronger in the current caseand the error bars under estimate the true error, sincethey are calculated assuming that the values of ( B/C ) avg and ( B/C ) max for the various network realizations arenormally distributed while our simulations show they arenot. Figure 2(b) shows the exponents of the power lawscaling of the four quantities as functions of P h . Per-haps somewhat surprisingly, note that the exponents for (cid:10) ( B/C ) ORavg (cid:11) and (cid:10) ( B/C ) ORmax (cid:11) are essentially equal overthe whole range of P h which, again, argues in favor ofthe optimality of routing. Note also that the exponentfor (cid:10) ( B/C ) SPmax (cid:11) exhibits a dip around P h = 0 .
9. Thisis an interesting feature, indicating that the SP routingworks unusually well when there is a small but nonzeroconcentration of low capacity links.Figure 3 shows h ( B/C ) i versus P h for h N i ≈
250 gen-erated for N C = 120. The averages shown in this figurewere also computed for 100 network topologies and 30link capacity configurations. The error bars represent2 σ estimates. Note that h ( B/C ) avg i varies monotoni-cally between P h = 0 and P h = 1 corresponding to allstreets and all highways, respectively, while h ( B/C ) max i exhibits a midrange maximum in the case of SP routingthat corresponds to a dip of the network transport ca-pacity. This type of behavior is due to the fact that SProuting forgoes the use of low capacity links as long asthey are not strictly needed to achieve connectivity (justas we tend to do in real life), which increases congestionon highways. On the other hand, by optimally using alllinks, it is possible to avoid this phenomenon. Uneven Traffic Demand
A comparison between the SP and OR average traveltimes for one network with uneven traffic demand is pre-sented in Fig. 4. Specifically, we look at a “rush hour traf-fic burst” with particles originating from the innermost10% of the nodes that we use to model a “downtown”.Their destinations are chosen uniformly from among allnodes with the only constraint being that the node oforigin must be different from the destination. This isthe same network that was used for Fig. 1 and again theresults are representative for networks with an averageof about 200 nodes. Note the lower maximum capacityof the network, which is due to the uneven distributionof traffic. Nevertheless, a judicious use of the low ca-pacity links again results in a significant increase of thetransport capacity. The factor by which transport capac-ity is increased may be even higher than in the case ofuniform traffic demand. This result is particularly impor-tant since traffic congestion usually develops is situationsof uneven traffic demand. T a vg r (a) P h =0.0(b) P h =0.3(c) P h =0.7 FIG. 4: (Color online) Average travel time (in time steps)vs. network load (in particles per time step per node) for a“rush hour traffic burst” with particles originating from theinnermost 10% of the nodes on a typical network realizationwith N = 200 nodes. Results are shown for shortest-pathrouting, SP (blue dashed lines), and for optimal routing, OR(red solid lines), at three different high capacity link densities,(a) P h = 0, (b) P h = 0 .
3, and (c) P h = 0 . V. CONCLUSIONS