Importance and exposure in road network vulnerability analysis. ACCEPTED for publication by TRANSPORTATION RESEARCH part A

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1 Iportance an exposure in roa network vulnerability analysis ACCEPTED for publication by TRANSPORTATION RESEARCH part A Erik Jenelius jenelius@infra.kth.se To Petersen top@infra.kth.se Lars-Göran Mattsson lg@infra.kth.se Affiliation: Division of Transport an Location Analysis (TLA), Departent of Transport an Econoics, KTH/Royal Institute of Technology, Stockhol Aress: Transport- och lokaliseringsanalys, KTH, Teknikringen 78 B, SE Stockhol, Sween Corresponing author: Erik Jenelius Tel: Fax: ABSTRACT The reliability an vulnerability of critical infrastructures have attracte a lot of attention recently. In orer to assess these issues quantitatively, operational easures are neee. Such easures can also be use as guiance to roa ainistrations in their prioritisation of aintenance an repair of roas, as well as for avoiing causing unnecessary isturbances in the planning of roawork. The concepts of link iportance an site exposure are introuce. In this paper, several link iportance inices an site exposure inices are erive, base on the increase in generalise travel cost when links are close. These easures are ivie into two groups: one reflecting an equal opportunities perspective, an the other a social efficiency perspective. The easures are calculate for the roa network of northern Sween. Results are collecte in a GIS for visualisation, an are presente per link an unicipality. In view of the recent great interest in coplex networks, soe topological easures of the roa network are also presente. KEYWORDS: travel reliability, vulnerability inex, contingency planning, roa network topology. 1

2 1. Introuction 1.1 Backgroun For at least a ecae, there has been a research interest in questions of vulnerability an reliability of transport networks. Events that have spurre this interest are the earthquake in Kobe in 1995 an the terrorist acts of Septeber 11, Issues of the vulnerability an reliability of large-scale systes have been stuie for a long tie before that in, e.g., ilitary logistics as well as coplex technical systes such as electronic circuits, inustrial prouction processes, etc. In networke systes, however, the issue has arisen quite recently, siultaneously with the increasing nuber of global connections in the econoy ( globalisation ), the rise of the Internet an the threats of coputer viruses, wors an other kins of attacks on coputers an servers. In retrospect, it has becoe ore an ore evient that interactions an interepenencies between ebers, parts an subsystes can be represente by graphs, an hence the theories of graphs can be applie for vulnerability an reliability analysis. The evelopent of ore powerful an available coputers has also ae it feasible to eploy the coputationally often heavy algoriths of iscrete an cobinatorial atheatics, which are associate with graphs. Given the iportance of the transport networks for journeys to work, for prouction logistics (notably the just-in-tie prouction philosophy), an for business travel, the reliability of transport networks is a key interest fro the point of view of transport syste users an hence planners at all levels, both in the public an private sectors. For business, the reliability of transports is actually, together with regularity an safety, one of the three ost value aspects of transport quality, an together with cost an tie the ost iportant aspect of transports altogether (SIKA, 2000a; b). For the purpose of roa anageent, prioritisation for roa aintenance an repair, contingency planning, an for the assessent of regional isparities, it woul thus be helpful with a general vulnerability inex attache to links an noes. In incient an contingency planning, an the planning of roa works, there shoul be an awareness about the ipacts of the reuce capacity on links. Increase efforts shoul be ae on links that are ore critical to the syste. Siilarly, increase awareness an efforts are calle for about the egree of vulnerability in ifferent places in the syste. In view of the strategic goals of the Sweish Roa Ainistration (SRA) of a positive regional evelopent an an accessible transport syste, relevant questions to ask are: Which regions are ost susceptible to isruptions in the transport syste? an: Which links are ost critical to the operation of the whole syste?. In orer to ake such istinctions, general easures of the criticality of links in the network, an the vulnerability of sites, unicipalities etc. nee to be evelope an teste. 1.2 Ai of the paper In the following sections, we first attept to efine what it is that we want to easure, an then we evelop such inices or easures. We argue that the concept of vulnerability can be ivie into two parts, one containing the probability of a hazarous event 2

3 an the other, which we call exposure, containing the consequences of the event in a certain place. Exposure is thus site-epenent. Siilarly, following Nicholson an Du (1994), we call the consequences for a collection of sites of a failing link or group of links the iportance of that link/group of links. As a easure of the consequences of failure we use the increase in generalise travel cost. In orer to reflect the two perspectives of equal opportunities an social efficiency, for the latter we weight this increase by travel ean. With the equal opportunities perspective, all roas are equally significant, regarless how often they are utilise. Each trip is equally iportant, which reflects the characteristic of roas as public goos : people shoul be given equal opportunities everywhere. In contrast, with the social efficiency perspective, the roas use the ost are consiere ore significant, because they serve ore people an businesses an thus generate ore positive social an econoic spin-offs. Which perspective to use in a specific situation is a atter of political jugeent. The easures are calculate for the roa network of northern Sween (Norrlan plus the county of Dalarna) 1, an are then aggregate over noes an/or links to yiel easures for, in turn, single noes an links, unicipalities, an the whole network. Norrlan (with Dalarna) is geographically the larger part of the country, but at the sae tie the ost sparsely populate part. The ajority of people, businesses an large cities are locate in the southern part of Sween. For exaple, the largest city in our stuy area, Ueå, is only nuber 11 in size in Sween with 109,000 inhabitants, an out of the 43 cities with a population of ore than 50,000 in all of Sween, only eight are locate there. The traffic, in general, is thus oerate an the network not liable to congestion. The paper is outline as follows: in the next section, the concepts an efinitions of vulnerability, reliability an criticality are iscusse. In section 3, our oel an easures are presente, together with soe of the forulas use. The presentation of the case stuy starts in section 4 with our assuptions, a short escription of the geographical context, after which the ata sources are iscusse. The etails of the calculations are presente in section 4.3. In view of the recent big interest in the properties of coplex networks, in section 4.4 we calculate soe topological easures of the roa network an copare it with other networks. We then present the ost interesting results fro the case stuy in section 4.5, in which the ost expose unicipalities to an averagecase (rano) an a worst-case scenario, as well as the ost iportant links, are ientifie. We then conclue (section 5) an iscuss the results an the caveats associate with our etho (section 6), an last we suggest soe options for further research in section 7. Definitions an all forulas are supplie in the Appenix. 2. Definitions of concepts 2.1 Vulnerability, reliability an risk 1 The Palt area, use by the SRA for regional policy analysis. 3

4 The concept of vulnerability oes not yet have a coonly accepte efinition, accoring to several authors (Einarsson an Rausan, 1998; Berica, 2002a; Holgren, Å., 2004; Taylor an D Este, 2004), an the eaning of the ter ay epen on the context. Holgren, Å. (2004) uses sensitivity to threats an hazars as a working efinition, however stressing that the negative events substantially reuce [the] ability [of the syste] to aintain its intene function (our ephasis). Other authors stress the infrequent occurrence of the events, such that vulnerability is a susceptibility for rare, big risks (Laurentius, 1994, cite in Berica, 2002a), or the suen, unpreicte occurrence (Sweish Departent of Defence, 1998). Abrahasson (1997) notes that in the vulnerability concept there is iplicitly the notion of little strokes fell great oaks, i.e., a relatively sall incient can, if it happens in an unfortunate (critical) place an tie, cause ajor aage or even the failure of the whole syste by chain reactions. If the incient is a blow of a hyrogen bob, we woul harly call the syste vulnerable if it fails. In case of oerate strains, networks are therefore, by construction, not vulnerable. If iniviual links, or contiguous areas of links fail, in ost cases there is still a way through the network this is the funaental iea of a network. Vulnerabilities appear when the network (or transport syste) is put uner pressure, when the capacity reaches its axiu, an a sall further stress coul cause a ajor aage by agnifying itself an cascae through the syste, possibly until it collapses. Berica (2002a) efines vulnerability as a susceptibility to incients that can result in consierable reuctions in roa network serviceability. Serviceability of a link/route/roa network, in turn, escribes the possibility to use that link/route/roa network uring a given perio. In the ters of Husal (2004), vulnerability is the nonoperability of the network uner certain circustances. Conversely, he states that reliability escribes the operability of the network uner varying strenuous conitions (i.e., the ability to continue to function). These efinitions are quite siilar to the one of Holgren above, given that the ter serviceability (non-operability) expresses the ability to aintain the intene function of the transport network (or transport syste). The eaning of both serviceability an operability is approxiately the sae as capacity (Chen et al., 2002) or perforance (Nicholson an Du, 1997), which all introuce the possibility of links an networks to be partially egrae (as in the case of congestion, accients, roa works etc.). With these efinitions of vulnerability, however, we still have the probles of escribing the perforance in ters that can be quantifie, an efining what a consierable reuction ( non-operability ) is, not to ention how to assess the susceptibility of the roa network. To begin with, it has to be ae clear what the scope of the vulnerability investigation is: single links, a part of the network, interactions between the network an iniviual links, or the whole transport syste incluing other oes as well, planning authorities etc. The use of the wor vulnerability in the context of the roa transport network (or syste) has soeties been confuse with the reliability of the sae syste. For exaple, Berica (2002a) proposes that vulnerability in the roa transport network can be seen as the copleent of reliability. Soe of the confusion with the concepts of vulnerability 4

5 an reliability ight be clarifie if the perspective of the observer is introuce. For exaple, Iers et al. (2004) regar reliability rather as a user-oriente quality of the transportation syste an not as a characteristic of the syste itself. They efine reliability as the egree of certainty with which a traveller is able to estiate his own travel tie, which epens on the probability istribution an stability of travel ties, on available inforation an on alternative travel options. Reliability is eficient when our expectations are not fulfille on a regular basis. On the other han, there are purely theoretical interpretations of reliability, like for exaple connectivity reliability an terinal reliability (the probability of an existing path between two noes). Reliability coul thus be viewe fro at least two angles. In the view of the iniviual traveller, it coul be seen as a binary ecision either the syste is operative or not. But in the aggregate view, regaring all travellers, there will be a portion of the fining the syste operative who will use it, while others will not, i.e., people have ifferent threshols for reliability. Lastly, there is also the theoretical probability of connectivity (regarless of travel tie). A nuber of objective reliability easures, focusing on ifferent aspects of roa transportation have been propose an use in recent years (see, e.g., Bell an Iia, 1997; Berica, 2002a; Chen et al., 2002; Nicholson, 2003). We alreay entione connectivity an terinal reliability; other easures are travel tie reliability an capacity reliability (Chen et al., 2002; Iers et al., 2004). However, as Berica (2002a) notes, these reliability easures are too uch concerne with probabilities, an she argues that the agnitue of the consequences shoul be ore ephasise. Vulnerability is relate to the concept of risk. Accoring to a generally accepte perception, risk inclues two coponents: probability an consequence. The consequences usually concern life, health an environent. If these issues are assigne onetary values, consequences can also be expresse as costs. The risk associate with a harful event X, R(X), then is a cobination of the probability of the event, Pr(X), an the cost, C(X). A coon way of operationalising the risk is to calculate the prouct of the two coponents, R(X) = Pr(X) C(X) (Einarsson an Rausan, 1998; Holgren, Å., 2004). In view of the above, we argue that vulnerability can be treate in the sae way as risk, an that the concept of vulnerability shoul be issociate into one coponent of probability, an one coponent of consequence. Sarewitz et al. (2003) also point to the isavantages of incluing the probability of failure in vulnerability stuies. Estiating the probabilities of extree events such as natural isasters an terrorist attacks is very ifficult. The probabilities are preicte fro historical ata, which iplies that the circustances aroun the event reain the sae at all ties, an that all causal connections are known. 2.2 Exposure, criticality an iportance For very rare events where the estiation of the probability is not feasible, the ter conitional vulnerability is often use (naely, given that a hazarous event occurs). We suggest the alternative ter exposure for conitional vulnerability. 5

6 D Este an Taylor (2003) are on the sae track when they argue that vulnerability shoul only be concerne with consequence. They state that a noe is vulnerable if loss (or substantial egraation) of a sall nuber of links significantly iinishes the accessibility of the noe, as easure by a stanar inex of accessibility. This efinition is ientical to our use of the ter exposure, which we prefer for the sake of clarity. The criticality of a certain coponent (link, noe, groups of links an/or noes) in the network involves both the probability of the coponent failing an the consequences of that failure for the syste as a whole. The ore critical the coponent, the ore severe is the aage to the syste when that coponent is lost. If the probability of an incient is high, the coponent (link etc.) is weak, an if the consequences are great, the coponent is iportant. If it is both weak an iportant, the coponent is critical (cf. Nicholson an Du, 1994). Following the line of arguent about conitional vulnerability an exposure above, it woul be natural to use an expression like conitional criticality for the iportance of a link or coponent. We get a ivision of the concepts as shown in Table 1. Table 1: Concepts in the analysis of vulnerability an criticality. Coposite concept, Concept A Concept B A B risk probability consequence vulnerability probability exposure, or conitional vulnerability criticality weakness (probability) iportance, or conitional criticality In the forulations in the beginning of section 2.1, vulnerability is efine in ters of reuce serviceability, operability (or even reliability ), an accessibility. These concepts all concern the function of the network, but reliability an accessibility approach the issue fro the ean sie, whereas serviceability/operability focus on the supply sie. Although these are all coplex concepts, we argue that a reasonable easure of the reuce serviceability/operability/ accessibility is the increase in generalise cost of travel (tie, istance, oney, etc.) for the users of the network. The cost of travel (weighte by ean or not) is surely a strong inicator of the function an the possibility to use the network, an it is one of the stanar inices of accessibility (for an iniviual) given in D Este an Taylor (2003). 3. Moel an easures 3.1 Measuring iportance Our approach is siilar to the one outline in Taylor an D Este (2004). We assue that the hazarous event is a link or a group of links, collectively calle an eleent, being copletely isrupte or close, which forces all travellers on those links to take other, less avantageous routes 2. The travellers are assue to behave accoring to the 2 The failure of a noe can be seen as being equivalent to the failure of the eleent consisting of all links connecte to the noe. 6

7 user equilibriu principle, i.e., they choose a route fro their origin to their estination that iniises their travel cost. This eans that the travel cost between each originestination (OD) pair is uniquely efine. At the ost basic level we look at an origin noe i, a estination noe j an an eleent e. In the following, the origins/estinations are calle ean noes. We enote the cost of travel fro ean noe i to ean noe j when eleent e has faile by unaage network. The quantity c (e), an let c represent the cost of the initial, (0) c ( k ) = c ( e) c (0) is the basis fro which easures of iportance an exposure are erive by aggregating over the appropriate sets of links an noes. Besie the increase in travel cost, there is another, ore severe, consequence that ust be taken into account. When an eleent e is close, the network ay be ivie into several isconnecte parts. The travel costs between noes in ifferent parts then becoe infinite. Although serious, inability to travel is harly infinitely worse than any finite increase in travel cost, an we woul want soe finite easure of the consequences of this event. For this purpose we introuce the concept of unsatisfie ean u (e), efine as u ( e) x = 0 if c if c ( e) ( e) =, <, where x is the travel ean fro ean noe i to ean noe j. The easure represents the nuber of trips fro i that are unable to reach j ue to the close eleent e. At aggregate levels, we generally easure the unsatisfie ean relative to the total ean. How to value unsatisfie ean in relation to increase travel cost is still an open question, however, or a atter of political ecisions. For siplicity, let us focus on the case where a single link is close. The closure will then result in either finite increases in travel cost (possibly zero) or positive unsatisfie ean. In the following, a link that causes finite increases in travel cost when close is calle a non-cut link, an the set of non-cut links is enote E nc while the set of all links is enote E, an the set of cut links is enote E c = E \ E nc (the set of E outsie E nc ). Measures base on increase travel cost ust be liite to the non-cut links, whereas easures base on unsatisfie ean are well-efine for all links. The iportance of a link can be calculate with respect to a single ean noe, a group of ean noes (such as those in a unicipality) or the whole network. Consier first a non-cut link k. In the case of the whole network, we aggregate over (k ) all OD pairs. Each OD pair is assigne a weight w that reflects its significance in relation to the other pairs. The iportance of link k with regar to the whole network is then c 7

8 Iportance net ( k) = i j i w ( c i j i ( k ) w c (0) ), k E nc. For c k E, ( ) = Iportance net k. If the weights w are all equal, all origins an estinations are equally iportant. This is the equal opportunities perspective that represents the role of the roa syste to provie access to all parts of the region (referre to as global iportance in the Appenix). If the travel ean x is use as weight, the severity of an increase in travel cost between two noes epens on the traffic between the. This easures the capability of the roa syste to provie socially an econoically efficient transports where the ean is the highest ( ean-weighte iportance in the Appenix). Corresponingly, the iportance pertaining to unsatisfie ean of any link k with regar to the whole network (relative to total ean) is Iportance uns net ( k) i = i j i u j i ( k ) x, k E. For nc k E, Iportance uns net ( ) = 0 k. 3.2 Measuring exposure Siilarly, exposure to a certain event can be calculate for a single ean noe, a group of ean noes or the whole network. When easuring the increase in travel (k ) cost for a unicipality, is aggregate over all origins i in the unicipality an c all estinations j in the entire network. One siple event is the failure of a single, ranoly chosen link. We then easure the expecte increase in travel cost, Exposure ran nc k E i V j i nc L i V ( k ) (0) w ( c c ) ( ) =, w j i where L nc is the nuber of non-cut links an V is the set of ean noes locate in unicipality. This average-case scenario represents the siplest possible oel of failures that have soe probability istribution across the region, for exaple those cause by extree weather. Another event is the failure of the ost iportant link for the unicipality. The exposure of unicipality is then the axiu value over all non-cut links, Exposure ax w i V j i i V ( k ) (0) ( c c ) ( ) = ax. nc k E w j i 8

9 This worst-case scenario woul correspon to an attack by an infore terrorist who wishes to cause as uch aage as possible to unicipality with only one strike. It is siilar to the situation stuie in Bell (2000) within a gae theory setting. As before, the weights w can be chosen to reflect either the equal opportunities perspective ( global exposure in the Appenix) or the social efficiency perspective ( eanweighte exposure ). The exposure pertaining to unsatisfie ean of each unicipality is the iportance, expresse as unsatisfie ean fro a noe in the unicipality to anywhere in the network, aggregate over all links using the average or the axiu aggregation rules. For the forulas, see the Appenix. 3.3 Inelastic ean For coputational reasons, the travel ean in our oel is inepenent of the link costs ( supply ), i.e., we assue the ean to be inelastic. Thus, we assue that the uration of the closure is long enough for a new user equilibriu to for, but short enough to not affect the travel ean significantly. In ters of the traitional four-step transport oelling process (trip generation, trip istribution, oe choice an route choice), only the route choice is affecte. Inelastic ean shoul be a reasonable assuption if the trip, as for the ajor part of our case, is a trip to work, which norally has to be pursue no atter what happens. With elastic ean, travellers woul alter their travel ecision, as well as their estination an/or oe choice, when facing increases in travel cost or other isturbances in the network. For an accurate assessent of the econoic consequences of link failures, it woul be ore satisfactory to use elastic ean in the calculations, like in Nicholson (2003). Berica an Eliasson (2004) also suggest an accessibility-base reliability easure, where changes in accessibility (consuer surplus) is eant to be calculate by the rule-of-a-half (see, e.g., Neuburger, 1971, p. 56 ff.). This approach presupposes an elastic, negatively sloping ean function, approaching zero when travel cost increases. Our approach coul be justifie by the fact that we are looking for general inices for the coparison of links an locations, an for the prioritisation of actions an easures by the roa authorities not priarily for use in cost-benefit calculations at this stage. An inconvenience with the inelastic approach, though, is that we are force to treat those links separately that copletely isconnect the network in two parts. 4. Case stuy: the roa network of northern Sween In the following case stuy, the travel tie between the ean noes is use as the generalise cost of travel, c. A perhaps ore realistic oel woul be to use a linear cobination of travel tie an istance. The rawback of this, however, is that a value of travel tie (VoT) woul have to be assue, which norally iffers fro person to person an fro tie to tie. In aition, a istance-epenent cost for the vehicle woul have to be assigne. In principle, however, there is nothing in our easures that prevents incluing istance epenence in the link costs. We assue that the travel ties are inepenent of the loa on the links. This approxiation shoul be reasonable for the network we stuy, since the region is sparsely populate an ost links have fairly low initial traffic loas: the eian loa is 450 ve- 9

10 hicles per ay, sue over both irections, with a ean of 1,804 an a axiu of 35,742 (this latter on a two-lane otorway). One out of six (1,070) links have a calculate loa of zero. The travel tie of each link is obtaine by iviing the length by the free-flow spee fro the volue-elay function. The assuption that travel ties are inepenent of traffic loa allows us to use a sipler an faster algorith to fin the fastest routes between origins an estinations. The presence of network congestion woul necessitate an iterative network assignent proceure, since link costs change for each iteration, an therefore woul require consierably ore coputing tie. 4.1 Geographic context Northern Sween is generally ore sparsely populate than the southern part, which historically has to o with the lack of arable lan an the col cliate, an geologically with the Fennoscanian ountain chain along the borer to Norway in the north-west. The region also borers on Finlan in the north-east an the Botnian Sea an Gulf along the east coast. In the northernost an western parts, uch of the lan is arsh an ountains, unusable for cultivation. These are traitionally the grouns of the native, reineer-breeing Sai people, who also populates the tunras of Norway, Finlan an Russia. However, the areas along the coast of the Botnian Sea, as well as the counties of Dalarna an Gävleborg in the south an Jätlan in the i-west, have becoe ore populate ue to both the cliate an the access to arable lan. The population ensity of the unicipalities in the region is shown in Figure 1. Fro an econoic point of view, the attraction of the northernost parts of the region has historically ostly been its natural resources: valuable ores (iron, copper, gol an silver) have been foun there, as well as fur an pine forests, an the great rivers have been use for the construction of as for power generation an for transportation of tiber to the sawills on the coast. The scenic beauty in the ountains attracts tourists an outoor activities. Nowaays the tiber is transporte by roa an the ore by freight trains, but the iportance of the waterways for counication is still visible in that any roas an railways follow the river valleys, an that the inlan settleents are often locate on the shores of rivers, large lakes an lake systes. There are two ain roas going in the north-south irection: the E 4 European highway following the coast an the inlan national roa 45. Along the rivers, the European highways E 10, E 12 an E 14 an a few national roas are going in the north-west-to-southeast irection, starting in Norway an ening on the Botnian Sea coast. The cobination of heavy roa transports an the cliate, which in spring causes great Kiloeters People/sq k Figure 1: Population ensity of the unicipalities of northern Sween, as of Septeber 30, Data fro Statistics Sween an National Atlas of Sween. 10

11 ifficulties while the groun below the roa is thawing, akes the question of vulnerability relevant also fro an econoic prouction perspective. The thawing process starts fro above, an converts the groun into soething like a quagire, threatening to break up the roa surface. It is neither technically nor econoically feasible to construct roas that withstan thaw copletely. During a perio in spring that can last several onths, the bearing capacity of the roa is severely reuce, an restrictions have to be put on the volue an tie of ay for roa use. Of course, these restrictions hit the prouction possibilities in this area. 4.2 Network an travel ata The Sweish national travel ean oel syste SAMPERS (Beser an Algers, 2001) is use by authorities to forecast the effects of ifferent transportation policies, econoic evelopent an infrastructure investents. The SAMPERS oel syste inclues a network representation of the Sweish roa syste. For local an regional trips, Sween is ivie into approxiately 8,500 zones where all trips begin an en, each zone coprising approxiately 1,000 inhabitants. The centre of gravity of each zone is represente by a centroi noe that is attache to the roa network with two connector links, one in each irection. The SAMPERS oel is ivie into five regional suboels. In this stuy we use the roa network of the suboel Palt, which focuses on the six northernost counties of Sween: Dalarna, Gävleborg, Västernorrlan, Jätlan, Västerbotten an Norrbotten. This area also represents the ile an northern planning regions of SRA. In this part the network is represente at its ost etaile level, reflecting the situation in 2001, with 19,392 noes (incluing 1,208 centrois) an 42,956 irecte links (incluing 2,416 connectors). In the rest of the country the network is represente by the national roa network, which is coarser with 7,597 noes (incluing 191 centrois) an 17,796 irecte links (incluing 382 connectors). The network ata contains the length, volue-elay function (the relation between traffic loa an travel tie), type (fro European highway to tertiary county roa), nuber of lanes, an county of each link. The ata also contains the co-orinates of each noe. By iporting the ata into a GIS 3 an cobining it with other geospatial inforation, we have obtaine the unicipality an parish in which each link an noe is locate Kiloeters Vehicles/ay Figure 2: Traffic loa on the links in the siplifie roa network, fro a network assignent by SRA. 3 Geographic Inforation Syste 11

12 The travel ean is easure in nuber of vehicles on an annual aily average. The vehicles inclue personal cars for private an copany use, personal cars in coercial traffic, an heavy trucks with an without trailer. The users only inclue Sweish travellers within the confines of Sween, not trips abroa or transit trips. We have also available the results fro an assignent of traffic on the roa network perfore by SRA using SAMPERS. The oel inclues possible congestion effects, an generalise cost is base on a linear cobination of travel tie an istance. More precisely, the travel tie is value at 136 SEK (about 19 USD) per hour an the istance at 1.30 SEK (0.18 USD) per kiloetre. Figure 2 shows the traffic loa on each link in the network accoring to the assignent. The network here is a siplifie version of the original network, which will be introuce in the next section. 4.3 Calculations Siplifying the network In orer to reuce the tie consuption of the travel tie calculations, we first siplify the network as uch as possible without changing its iportant characteristics, even uring link closures. The first step is to ake the network unirecte. Analysis of the network ata shows that for each link there is another link of the sae length an travel tie going in the opposite irection. It is therefore possible to substitute the two irecte links by one unirecte link, which is assigne the length, travel tie an the su of the traffic loas of the ol links. As the secon step we reove all the centrois an connectors, since these are not part of the actual roa network. Instea, we ark the noes closest to the centrois (in travel tie) within the etaile part of the network as ean noes. Thus, for each centroi within this region (1,208 centrois) we ark its only neighbour as a ean noe. For each centroi outsie the region (191 centrois) we perfor a shortest path search to fin the closest noe within the region. During the process, we keep a recor of which ean noe each centroi is appe on. Since all ean has been ove insie the stuy region, we can now reove the coarse parts of the network. There are two kins of noes in the network that can be reove without affecting the travel ties an istances between the ean noes. The first kin is the ea-en noes with only one neighbour. No shortest paths between the ean noes will pass through these noes, unless they are ean noes theselves. The secon kin is the joint noes with two links connecte to the. Unless ean noes, they can be replace with a irect link between the two neighbours. The length an travel tie of the new link is set to the su of the lengths an travel ties of the ol links, respectively, while the traffic loa is preserve. Using an iterative proceure, all noes of these two kins are reove. The last step is to reove a few noes that are not connecte to the rest of the network. After this we arrive at the final network, with N = 4,470 noes, out of which N = 1,136 are ean noes, an L = 6,362 unirecte links, out of which L c = 168 are cut links. 12

13 4.3.2 Calculating ean an travel ties Fro the ata we know the ean between every pair of centrois in the original network. For the siplifie network, however, we nee to know the ean between every pair of ean noes. To obtain the latter we use the any-to-one ap between centrois an ean noes that was aintaine uring the siplification process. For every centroi h, let (h) be the ean noe that h is appe on. Then, for every pair of centrois h i an h j, one of the following actions is chosen: If at least one of the centrois is locate insie the stuy area, the ean fro h i to h j is ae to the ean fro (h i ) to (h j ). If both centrois are locate outsie the region, or if (h i ) = (h j ), the ean is not ae. The ean is store in a N N atrix X = (x ). The original ean ata is alost syetrical with respect to start an en noes, an we ake the ean atrix copletely syetrical by aing its transpose an iviing by two, so that x = x ji for all i, j. The travel ties between all ean noes are calculate using a cobination of Dkstra s shortest paths algorith (Dkstra, 1959) an a sei-ynaic upating algorith (Buriol et al., 2004). The algorith of Dkstra (1959) is ipleente with approxiate buckets, which is highly efficient for this particular kin of network (Zhan an Noon, 1998). Starting fro ean noe i, we perfor a shortest paths search to all other noes using the algorith of Dkstra (1959) with the travel ties of the links as weights. For each link k in the network, we then take one of two possible actions: If k is not in the shortest paths tree roote at i, the travel ties are unaffecte by the reoval of k an no recalculations are necessary. If k is in the shortest paths tree, we reove k by assigning an infinite travel tie to it an upate the shortest paths tree using the algorith of Buriol et al. (2004). This reuces the coputational buren significantly copare to fining the shortest paths fro scratch after each link reoval. The link is then replace an the shortest paths are restore. Since the shortest paths tree contains N 1 links, the shortest paths will be upate N 1 ties. This proceure is then repeate for each ean noe, giving a total of N original shortest paths searches an N (N 1) upates. The coputation of reoving an replacing all 6,362 links an storing the results takes about inutes with this algorith on a Xeon 1.8 GHz processor. The progras are ipleente in C++ an are escribe in Jenelius (2004). 4.4 Topological properties In recent years, the properties of coplex networks have becoe a topic for a great eal of research. One interesting question has been whether networks fro ifferent fiels have any characteristics in coon. In the literature, there are particularly three easures that have gaine a lot of attention: the average geoesic istance, the clustering coefficient an the egree istribution (Albert an Barabási, 2002; Newan, 2003). A stuy of these easures for roa networks is calle for by Waters (2004). The geoesic istance between two noes is the iniu nuber of links that have to be traverse in orer to get fro one noe to the other. The average geoesic istance between every pair of noes, here enote by l, is a easure of how copact or scattere the network structure is. 13

14 The clustering coefficient C is use to easure to what extent the noes are foring sall, tightly connecte groups, which is coon in, e.g., social networks. The clustering coefficient C i of a noe i is efine as C i Li = ki ( ki 1) / 2 0 if k i else, > 1, where k i is the egree, i.e., the nuber of neighbours, of i an L i is the nuber of links between the neighbours of i. The enoinator is the axiu possible nuber of links between the neighbours of i. C is the average of C i over all noes. Figure 3 gives a siple exaple of how to calculate the clustering coefficient of a noe. The egree istribution P(k) is the fraction of noes with egree k for k = 0, 1, 2,. The egree istributions of any stuie networks have been foun to follow powerlaws, P(k) k γ, for soe positive paraeter γ usually ranging between 2 an 3. The functional for of P(k) has been foun to have a great ipact on the robustness of the network structure against rano noe failures as well as hostile noe attacks (Albert an Barabási, 2002). In Table 2 are shown the clustering Figure 3: Exaple of how to calculate the clustering coefficient of a noe i. The neighbours of i have a total of four links between each other. The axiu possible nuber of links woul be six, so the clustering coefficient of i is 4/ i coefficient, the average geoesic istance an soe other topological easures of the initial network in its unirecte for, with an without the centrois, as well as the final siplifie network. N an L enote the nuber of noes an links, respectively. The clustering coefficient is shown relative to what it woul be on average if the links were rawn copletely at rano, which is C ran = 2L/(N(N 1)). k enotes the average egree of the noes, while is the average travel istance (in kiloetres) an t is the average travel tie (in inutes) between every pair of noes. For coparison, Table 2 also inclues soe other infrastructure networks of siilar size. The first is the high voltage power transission network of the Noric countries (Sween, Norway, Finlan an parts of Denark), which has been stuie in Holgren, Å. (2004). The secon is the high voltage power transission network of the western Unite States (Watts an Strogatz, 1998). In both these networks, the links represent power lines an the noes represent generators an substations. The thir network is the physical structure of the Internet (Newan, 2003), where the noes represent oains an the links are connections between the oains. The final section of the table contains a few roa networks that have been use to test algoriths for shortest paths an 14

15 traffic assignent. The first an thir of these are use in Zhan an Noon (1998) an the secon in Holgren, J. (2004). Table 2: Topological easures of the initial an the siplifie roa network, as well as other unirecte networks. Roa network version N L k C/C ran l [k] t [in] Initial w centrois 26,989 30, Initial w/o centrois 25,590 28, Siplifie 4,470 6, Infrastructure network N L k C/C ran l Noric power gri a 4,789 5, W. USA power gri b 4,941 6, Internet c 10,697 31, Other roa network N L k Georgia, USA (coarse) 2,878 4, Chicago Regional, USA e 12,982 19, Louisiana, USA (fine) 35,793 49, a b c Holgren, Å. (2004). Watts an Strogatz (1998). Newan (2003). Zhan an Noon (1998). e Holgren, J. (2004). It can be seen that the average topological istance l is far bigger in the roa network than in the power gris an the Internet. This inicates that the constraint that geographical istance puts on which noes are connecte is the strongest for the roa network. It is siply not econoically profitable to buil irect links between istant locations, an such links are ore likely to be establishe by, e.g., flight or train. For the Internet on the other han, the cost of istance is uch lower an any noes have irect connections between each other. Meanwhile, the clustering coefficient C of the siplifie roa network is siilar to that of the power gri of Western USA but significantly lower than for the Internet. Thus, given that we can reach two locations (noes) irectly fro where we are, it is less likely in the roa network than in the Internet that there exists a irect link between these two locations. It ay also be note that the average egree is siilar in all roa networks, aroun or slightly below three. Table 3 shows the egree istribution of the roa network before an after the siplification process in absolute nubers. Without the centrois, the axiu egree is six, which is a ore restrictive boun than in ost other networks ue to the liite space available at roa intersections. Note that in the siplifie for, all noes with egree one an two are ean noes. For the two power gris the egree istributions ecay exponentially, with axiu egrees of 14 an 19, respectively (Holgren, Å., 2004). The egree istribution of the Internet ecays accoring to a power-law, which eans that soe noes have very high egrees (Newan, 2003). Table 3: The egree istribution of the initial an the siplifie roa network. For each value k = 0, 1, 2,, the nuber of noes with egree k is shown. Roa network version k = Total Initial w centrois 1 2,816 15,870 7,126 1, ,989 15

16 Roa network version k = Total Initial w/o centrois 1 1,484 16,627 6, ,590 Siplifie , , Results We now present soe of the results fro the case stuy. The iportance of each iniviual link with respect to the whole network is calculate using the three variants introuce in section 3.1: global, ean-weighte an unsatisfie ean-relate iportance. For each unicipality, the global, ean-weighte an unsatisfie eanrelate exposure is calculate with respect to the two scenarios escribe in section 3.2: First, the average-case scenario, in which we calculate the expecte effects of closing a ranoly selecte link; secon, the worst-case scenario, in which the ost iportant link is close. All forulas can be foun in the Appenix Link iportance for the whole network The global iportance of each link for the whole network is easure by the increase in travel tie per OD pair when the link is close (see Forula 1 in the Appenix). It can be seen fro Figure 4 that the ost iportant links are sections of the E 4 European highway that stretches along the entire coast. It is clear that this roa constitutes an iportant backbone for fast access across the network. This is to be expecte since the roa allows travel at high spee an since any ean noes are locate close to the coast. We can also see that the ost iportant link, locate in the ile part of the E 4 in the rocky High Coast region, causes an average increase in travel tie by ore than ten inutes per OD pair when close. Minutes/OD pair Veh inutes/veh Kiloeters Figure 4: Global link iportance for the whole network (Forula 1) Kiloeters Figure 5: Dean-weighte link iportance for the whole network (Forula 2). 16

17 The ean-weighte iportance of each link is the increase in travel tie per trip when the link is close (Forula 2). The ost iportant links are generally short roas within cities an o not show well on the ap in Figure 5. In particular, any of the links are parts of the E 4 going through Gävle an Ueå, two of the three largest unicipalities in the area. These roa sections therefore accooate both regional an local traffic. The ost iportant link, locate in the city of Gävle to the south, causes an average increase in travel tie per trip by ore than one inute when close. However, these results shoul be treate with soe caution since the network ata oes not inclue all the sall streets in the cities. The effects of closing a city street ay therefore appear ore severe than they woul be in reality Kiloeters Unsatisfie veh/total Figure 6: Unsatisfie ean-relate link iportance for the whole network (Forula 3). Siilarly, the links that cause the largest relative aount of unsatisfie ean when close (Forula 3) are generally short roas in or near cities. In particular, Figure 6 shows that any of the links are locate east of the E 4, close to the coast where there is little roo for alternative links. Others are locate in the sparsely populate ountain areas an are often sections of roas that continue into Norway. Still others lie on the southern bounary an can be consiere a bounary effect, since they woul not ivie the network if the stuy area were larger. When the ost iportant link, again locate in Gävle, is close, nearly one in every forty vehicles is unable to reach its estination. Since this easure also is sensitive to inaccuracies in the network ata, the results shoul not be taken too efinitely Municipality exposure Average-case scenario Global unicipality exposure is easure by the average increase in travel tie per origin in the unicipality an estination in the whole network. In the average-case scenario a ranoly chosen non-cut link is close (Forula 4). As shown in Figure 7, the ost expose unicipalities are locate in the northernost part of the region. Here the roa network an the population are particularly sparse. In any network, however, peripheral noes will suffer the ost when a rano link is reove, since they rely on a saller nuber of links in their proxiity for their connection to the rest of the network. As a result of this the southernost region is affecte as well, even though the network is enser here. This is an exaple of a borer effect that shoul be carefully consiere, an because of such effects it ight be necessary to inclue a larger network than the one uner stuy. 17

18 Minutes/OD pair Veh inutes/veh Kiloeters Figure 7: Global unicipality exposure to the closure of a ranoly chosen link (Forula 4) Kiloeters Figure 8: Dean-weighte unicipality exposure to the closure of a ranoly chosen link (Forula 5). The ean-weighte exposure of a unicipality is the increase in travel tie per trip beginning in the unicipality an ening anywhere in the network (Forula 5). Since ost ean is to nearby estinations, the effects of closing a link shoul be ore local here than in the global perspective, an a few links shoul be responsible for a large part of the exposure of each unicipality. Figure 8 shows that there appear to be soe inlan regions where the exposure of the unicipalities is higher than elsewhere. One reason for this is that the local roa networks in these areas are sparser, with fewer goo alternatives, than elsewhere. The ost expose unicipalities are also aong the ost sparsely populate. One thing to note is that in the northernost unicipalities the trips appear to be very little affecte by a rano link closure. This is not the full picture, however, as we will see when we turn to the easure of relative unsatisfie ean. Figure 9 shows that the northernost unicipality of Kiruna is aong the ost expose ones regaring the relative aount of unsatisfie ean (Forula 6). Inee, several links in this region will cut off ean noes fro the rest of the network when close. For the non-cut links, however, the finite increases in travel tie per trip are sall, as seen above. The conclusion is that unsatisfie ean shoul be the first easure to stuy, since being unable to reach your estination is a ore serious scenario. Fro the figure we also notice that several of the ost ensely populate coastal unicipalities are highly expose, while the least expose ones are locate in the centre of the region, away fro the bounaries. This coul also be expecte, if the topography oes not place restrictions on roa construction: in general, internal noes have a higher connectivity (higher egree). 18

19 Unsatisfie veh/total Kiloeters Figure 9: Unsatisfie ean-relate unicipality exposure to the closure of a ranoly chosen link (Forula 6). Worst-case scenario Figure 10 shows that the unicipalities in the northern parts of the area are globally expose to the closure of the ost iportant link (Forula 7). Soewhat surprisingly, however, the south-east corner, where both the population an the roa network is relatively ense, is the ost expose part of the area. It is in fact a short section of the E 4 European highway with no nearby alternative links that is the ost iportant one for all the unicipalities in this region. This is another inication of the great iportance of this roa. The qualitative situation for the ean-weighte unicipality exposure (Forula 8) shown in Figure 11 is siilar to that in the average-case scenario (Figure 8), which strengthens the notion that a few iportant links are responsible for ost of the travel tie increase for each unicipality. The ost iportant link is always locate within the sae unicipality. Again, the view that the northernost region an a few other unicipalities appear to withstan the event well ust be revise when unsatisfie ean is consiere. The ost expose unicipality is Arjeplog to the northwest, where travellers will experience an average increase in travel tie by alost one an a half hour when a certain link is close. Finally, for the unsatisfie ean-relate unicipality exposure (Forula 9), the general picture shown in Figure 12 is also siilar to the average-case scenario (Figure 9). This confirs that only one or a few links account for the ajority of unsatisfie ean in each unicipality. In the north-western unicipalities of Dorotea an Sorsele, over 80 per cent of the trips will not reach their estinations if a certain link in the sae unicipality is close. 19