Vulnerability in Public Transportation Networks

Aaron Aquino1, Noah Gamboa2, and Antariksh Mahajan3

1aquino@stanford.edu2ngamboa@stanford.edu

3am95@stanford.edu

I. INTRODUCTION

Transportation networks are indispensable to the livesof people in cities. In Europe, metros alone carry 31.6million passengers a day [12]. However, this meansthat the impact of disruptions in transport networks canhave particularly severe consequences. Following the2005 London bombings, which simultaneously targetedseveral stations in the London Underground, subwaypassenger ridership was down 30% on weekends and15% on weekdays even months later [3]. Given both theimportance and the fragility of public transit networks(PTNs), PTN vulnerability has become an important areaof research for city planning. In particular, using networkanalysis techniques to analyze PTN vulnerability canshed light on where new routes and stations can be addedto an existing system to enhance its resilience in theface of disruptions such as failing infrastructure, severeweather, or targeted attacks.

This project examines the PTN in Madrid, Spain.Madrid was the city of choice for several reasons. Itis heavily utilized, and thus important to many people:in 2013, almost 1.37 billion trips were taken on the PTN[4]. It is also interesting to study because it is complex,with five individual transit types comprised of two metro,two bus, and a train system. Vulnerability analysis onMadrid’s PTN is particularly relevant given the 11-Mtrain bombings in 2004, which killed more than 190people [14]. Finally, data on this PTN is available forall five transit types in a standardized format.

By analyzing an aggregate PTN graph that links allfive public transportation types, we aim to test thefollowing hypotheses:• Edges in the city center are the most critical. The

precise definition of criticality will be explored indetail in Section IV, but at a high level, a criticaledge is one whose disruption would cause the mostharm to the system.

• When going from a network containing one trans-port type to many, network diameter will increase,

but average travel time between stations will de-crease. For example, a network containing only thebus system will have a smaller diameter but longeraverage travel time between any two stations thana network containing all five transport types.

• Nodes with the smallest weighted clustering co-efficients occur at the geographical extremes ofthe network. A rigorous, mathematical definition ofweighted clustering coefficient is given in SectionIV, but at a high level, this means that peoplewho take public transport from the geographicalextremes of the network tend to have fewer optionsfor where they start their journeys.

Based on our investigation of these hypotheses, we aimto propose improvements to Madrid’s PTN through theaddition of new stops and routes.

II. LITERATURE REVIEW

Vulnerability has two key components: the probabilityof disruption and the consequences of disruption [6]. Toillustrate the difference between the two, a station couldhave high probability of disruption if its infrastructurewere aging, but limited consequences if it were used byvery few people. Jenelius et al. chose to define vulner-ability based solely on the consequences of disruption,since probability was difficult to measure using networkproperties. They then used the following definition ofvulnerability: “a node is vulnerable if loss (or substantialdegradation) of a small number of links significantlydiminishes the accessibility of the node, as measuredby a standard index of accessibility”. They applied thisdefinition to vulnerability in road networks, where theirstandard of accessibility was the cost of travel, whichincreased when parts of the road network were disrupted.They also identified two perspectives on this measure ofcost: “equal opportunity”, where all network edges wereconsidered equal, and “social efficiency”, where edgeswere weighted by how many people traveled on themper unit time.

Other studies used similar definitions of vulnerability,but different accessibility indices. Rodrıguez-Nunez andGarcıa-Palomares used the index of time instead of costto analyze the metro network in Madrid [9]. In otherwords, they measured the vulnerability of a station byanalyzing the expected increase in average travel timefrom that station to any other station when a randomnetwork link was disrupted. They used the “social effi-ciency” framework as defined by Jenelius et al., wherethe travel time between any two stations was weightedby how frequently people made the trip between thosetwo stations. They also defined the term “criticality” foran edge (as opposed to vulnerability, which is defined fora node) as the average increase in travel times across thenetwork when that edge is disrupted. Identical definitionswere used by Taylor in a study on vulnerability andtraffic incidents, showing that these are metrics usedconsistently in the literature [11].

Even with a given definition of vulnerability, thereare different ways of modeling PTNs using graphs.Sienkiewicz and Holyst identify two possible topologies[10]. The first is L-space, which is more representativeof PTNs in their real, physical form: nodes representstops and edges represent transitions between consecu-tive stops. The other is P-space, where nodes representstops but edges exist between stops that can be reachedfrom each other without transfers. In other words, ifa particular bus or train travels along a set of stopsS on a particular route, then the stops in S form acomplete subgraph of the broader network. These twotopologies show different types of relationships in thePTN; L-space models patterns in stops, whereas P-spacemodels patterns in routes. Von Ferber et al. identify othertopologies such as B-space (bipartite graph where bothroutes and stations are nodes) and C-space (only routesare nodes), but stated that they provided less insight thanL- and P-spaces, hence they are not within the scope ofthis paper [13].

Once a network topology is chosen, various propertiesof the network can be calculated. Latora and Marchioriintroduced the concept of network efficiency, a measureof how fast information flows across a network [7].When analyzing the efficiency of the Boston PTN, theyfound that efficiency changed significantly when lookingpurely at the subway versus looking at both bus andsubway, which provides a strong motivation for ourproject to account for all types of transit. Von Ferberet al. chose to calculate several properties of PTNs forfourteen different cities, including average node degree,maximal and mean shortest path length, betweenness

centrality, and the ratio of mean clustering coefficientto random graphs of the same size [13]. They then usedthese properties to identify interesting statistical patterns,such as the diversity in degree distributions of PTNs indifferent cities. However, they did not use these findingsto draw insights on how these PTNs could be improved,which motivated our approach of posing hypotheses andtesting them against the actual network.

Finally, these properties can be recalculated basedon various scenarios to analyze the network’s re-silience. Murray et al. identified three increasingly broadapproaches to scenario-based calculations: scenario-specific, which only accounts for particular disruptions;strategy-specific, which accounts for multiple disruptionsbased on a strategic attacker; and simulation, whichcreates random scenarios [8]. The choice of approachdepends on one’s objectives as well as computationalresources. This project focuses on the strategy-specificapproach, since this can help city planners to deal withworst-case scenarios where the most important links inthe network are disrupted.

III. MODEL

Data from Madrid’s governing transport body, theConsorcio Regional de Transportes de Madrid (CRTM),was used to construct our graphs. The data was publishedin General Transit Feed Specification (GTFS) format, astandardized format for specifying PTNs that includesdata such as stop locations, fares, routes, and time takenbetween stops. A full specification of GTFS can be foundin [1].

GTFS data from September 2017 was used fromfive different transit types: Metro (subway), EMT (bus),Metro Ligero (subway), Autobuses Urbanos (bus), andCercanıas (train). A node was created for every stop inevery network. A visualization of the physical locationsof Metro, EMT, and Autobuses Urbanos nodes is shownin Figure 1. The value of connecting different networkscan immediately be seen: many locations have stops fordifferent types of public transport located close to eachother, showing that there are multiple ways of gettingbetween destinations. Moreover, different networks havedifferent densities of stops inside and outside the heartof the city. For example, the blue nodes (EMT) arehighly dense within the heart of the city, while the yellownodes (Autobuses Urbanos) have clusters outside the citycenter.

Subsequently, edges were created using both the P-and L-space topologies (see Section II). This created fivedifferent disjoint subgraphs, one for each type of transit.

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Fig. 1: Physical locations of metro (red), EMT (blue),and Autobuses Urbanos (yellow) stops. Other networksare excluded for simplicity.

Fig. 2: Metro subgraph. Each color represents a metroline.

Figure 2 shows the subgraph for the metro system, withdifferent colors representing different metro lines.

Finally, in both the P- and L-space, edges were addedbetween stops that were within 100m of each other, bothacross subgraphs and within subgraphs. The distanceswere calculated using the standard WGS-84 ellipsoidconversion from latitude-longitude coordinates to dis-tance across the surface of the earth. It is important tonote that these added edges were conceptually differentin the two spaces. In the L-space, traversing an addededge represented walking between closely-located stopsas traveling another “stop”. On the other hand, in theP-space, traversing an added edge represented walkingbetween closely-located stops as a transfer betweenforms of transport. Figure 3 shows visualizations for bothspaces, where it can be seen that the P-space is far moredense, since stops on any given route form a completesubgraph instead of a chain. In total, the L-space graphhad 6,587 nodes and 20,540 edges, whereas the P-space

Fig. 3: L- (above) and P- (below) space graphs of thefull network. It is clear that the P-space graph is moredense, since stops along common routes form completesubgraphs.

graph had 6,587 nodes and 590,976 edges.

IV. IMPORTANT EQUATIONS

This section details important equations for calculatingcriticality, which is crucial for evaluating our first hy-pothesis. Along the way, we will detail how to calculateaverage travel times, which are important for evaluatingour first hypothesis.

First, we use travel times as our metric of accessibility.Given the shortest possible travel time between any twostations (found using Dijkstra’s algorithm as describedin Section V), we can calculate the average travel timefrom a station i to every other station. We use the“equal opportunity” approach to accessibility, meaningthat every trip is treated as equal, so the average traveltime from station i is:

Ti =1

n− 1

∑j

Tij ,

where Tij is the travel time from station i to station jand n is the total number of nodes in the network. Given

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Ti for all 1 ≤ i ≤ n, we can calculate the overall averagetravel time in the network:

T0 =1

n

∑i

Ti

Next, we analyze the impact of disruption of eachindividual link. However, it is necessary to note thatdisrupting a link can result in the graph becomingdisconnected, thus making travel times between certainnodes infinite. Thus we have to account for the nodeswhich become disconnected, and the nodes for which thetravel times simply increase (or stay the same). For thelatter case, let the set of stations which are still reachablefrom station i once edge a is disrupted be Ai,a. Then thenew average travel time from station i is given by:

Ti,a =1

|Ai,a|∑

j∈Ai,a

Tij

The unsatisfied demand, or number of nodes thatbecome disconnected from station i when edge a isdisrupted, is:

di,a = |V \Ai,a|

And the new overall average travel time in the networkis:

Ta =1

n

∑i

Ti,a

It is important to note that, since our network usedmany different transportation types, there were no caseswhere removing an edge disconnected the graph. How-ever, the above equation on unsatisfied demand is stillincluded for completeness.

We then measure the criticality of each link a by thechange in overall travel time when that link is disrupted:

Ca = Ta − T0

A final important equation is related to clusteringcoefficients. Since we are using weighted edges, we usea proposed generalization of the clustering coefficientformula to weighted graphs [2]:

ci =1

si(ki − 1)

∑j,h

wij + wih

2aijaihajh

si is the sum of the weights of node i’s outgoing edgesand ki is the degree of node i, while a denotes entriesin the adjacency matrix.

V. ALGORITHMS

A. Paths with shortest travel times

In order to compute the shortest trips between all pairsof nodes in our aggregate network, we implementeda version of Dijkstra’s algorithm that takes a singlesource node and then calculates the shortest paths fromthe source to every other node in the network. At ahigh level, Dijkstra’s initially assigns a tentative distancevalue of infinity to every destination node. It then iteratesover all previously unvisited nodes and for each one,computes a new tentative distance to that node basedoff its neighbors. If this new tentative distance is lessthan the previous one, then the value is updated. Whenthe algorithm finishes, each node now stores the lengthof the shortest path from the source to that node. Ourversion of Dijkstra’s also records the specific nodes ineach shortest path by keeping track of each destination’sprevious node.

The original implementation of Dijkstra’s runs inO(|V 2|) time, but we decided to implement a variationcreated by Fredman and Tarjan that utilizes a Fibonacciheap and has a runtime of O(|E| + |V | log |V |)[15]. This gain in efficiency comes from threespecial heap operations, add_with_priority(),decrease_priority() and extract_min(),which allow the algorithm to process nodes in order ofincreasing tentative distance.

To speed up our computations, we used multithreadingwhen running Dijkstra’s over all the nodes in our net-work. For a given node, we then calculated the averageover each of its shortest paths to determine the farnesscentrality, or mean travel time from that node.

B. Criticality

Given the size and density of the graphs, calculatingthe overall criticality was computationally infeasible.Even with multithreading, it took about an hour tocalculate the criticality of an edge, which means it wouldhave many months of non-stop computation to calculatethe criticality of every edge.

To bring the criticality calculations to within ourcomputational limits, we chose to sample a subset ofthe edges to calculate their criticality. In order to gainan understanding of criticality inside and outside the citycenter, edges were sampled separately from within andoutside the city center. We defined the city center asall points within 5km of Puerta del Sol, which was thecentral address of Madrid as defined by the Google MapsReverse Geocoding API [5].

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Station name Transit type Average traveltime (s)

Sol Cercanıas 1838Tribunal EMT 1854AlonsoMartınez

EMT 1855

Gran Vıa Metro 1859Callao EMT 1870

Fig. 4: Lowest 5 average travel times in the aggregategraph.

To calculate criticality, the average travel time Ti

from each node in the original graph were initiallystored. During the running of Dijkstra’s algorithm tocalculate these values, a map of each edge to a list ofsource-destination node pairs whose shortest path passedthrough that edge was also computed and stored. Next,for each edge sampled from the graph, the edge wasdeleted, and Dijkstra’s algorithm was reapplied to allthe source-destinations pairs whose old shortest pathspassed through that edge. The old average travel timeswere updated, and the new Ti,a was calculated for eachnode. Using the updated average travel times for eachnode, the criticality of the removed edge was found.

VI. RESULTS

A. Hypothesis 1: Edges in the city center are the mostcritical.

Our analysis of criticality began with travel times asour metric of accessibility. The GTFS feeds specifiedtravel times for each stop on each route, and thosetravel times were added to the graph as edge weights. Inaddition, a flat 3-minute weight was added to each edgethat was added for stops within 100m of each other.

As described in Section V, average travel times arecalculated based on the shortest possible travel timebetween any two stations. To find this path from eachnode to every other, we ran Dijkstra’s algorithm foreach node as described in Section VI. We found thefive stations with the shortest average travel time Ti,shown in Figure 4. As a quick sanity check, Sol isthe station in one of the busiest and most well-knownpublic squares in Madrid, so it is unsurprising that thisstation is the most central (where centrality is measuredby average travel time).

Station name Transit type Average traveltime (s)

Av. Berlın-Florencia

AutobusesUrbanos

92610

Bronce-Platino AutobusesUrbanos

89865

ConcejalV.Granizo-Victoria Kent

AutobusesUrbanos

87886

Av .Concordia-Marques de Hi-nojosa

AutobusesUrbanos

87666

Av. Juan Gris-Pirineos

AutobusesUrbanos

87367

Fig. 5: Highest 5 average travel times in the aggregategraph.

Conversely, we found the five stations with the longestaverage travel time Ti, shown in Figure 5. Anotherquick sanity check tells us that Av. Berlın-Florenciais on the distant eastern outskirts of Madrid, whichexplains its long average travel time.

Figure 6 shows a plot of criticality vs. distance of thesource node from the city center for a few edges. Onlya small sample of edges was included so as to make thisplot less noisy. It is clear that there is no obvious patternin the plot; there are small criticality values scatteredthroughout, while the peaks are inconsistent. However,we can see that the highest criticality values tend notto occur very near the city center or very far from it,but at the intermediate distances. This could suggestthat the city center is already sufficiently resilient todisruption or attack, since it already has many alternativeroutes within it. Thus, the truly critical edges havebecome those between the city center and the surround-ing districts, which are interfaces between residentialareas and the city center that are less well-served bypublic transportation. This implies that it might servepolicymakers well to focus more on creating alternativeroutes for intermediate-distance stops, since they couldcause severe delays in the case of disruption.

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Fig. 6: Criticality of node vs. distance from city center(m).

Network L-graph Diame-ter

P-graph Diame-ter

Full Network 78 9AutobusesUrbanos

92 11

Cercanıas 14 3EMT 55 5Metro 44 4Metro Ligero 27 2

Fig. 7: Graph diameters for the aggregate network andeach subnetwork.

B. Hypothesis 2: When going from a network contain-ing one transport type to many, network diameter willincrease, but average travel time between stations willdecrease.

We define diameter as the longest shortest path overthe network. We looked at each of the individual net-works and the aggregate of all 5 networks and evaluatedthe diameter for each, and the results are shown inFigure 7. Our hypothesis is that the diameter of thefull network would be longer than the diameter of theindividual networks. The results confirm this hypothesisfor some networks, but not others. Overall, we see anaveraging effect amongst the different networks, wherethe diameter of the aggregate network is somewherebetween the diameters of the original subnetworks.

We can understand this phenomenon by looking atthe original diameter for the Autobuses Urbanos (AU)network. Figure 7 shows that AU is by far the largestsubnetwork of the 5 with nearly an order of magnitude

Network L-graph MeanPath Length

P-graph MeanPath Length

Full Network 11.13 2.57AutobusesUrbanos

17.70 3.066

Cercanıas 4.97 1.75EMT 20.13 2.72Metro 14.55 2.16Metro Ligero 17.73 3.65

Fig. 8: Average path lengths for the aggregate networkand each subnetwork.

larger diameter than the rest of the subnetwork. Thismeans that adding it to the other networks will increasethe overall network diameter. To illustrate this phe-nomenon, consider the addition of the AU network (largediameter) to the Cercanıas network (small diameter).The original AU network has many bus stops that arenot very densely connected, hence its large originaldiameter. When combined with other Cercanıas stopsthat allow shortcuts between bus stops, the aggregatediameter decreases, but there are still many bus stops thatrequire many steps to reach. Conversely, from the pointof view of the Cercanıas network, adding the AU stopsincreases the number of stops that can be reached withoutproportionately increasing the number of edges betweenthem, thus the network diameter increases. Overall, whencombined with the other networks, the longest shortestpath shrinks by 15% and 18% for P- and L-space graphsrespectively, as compared to the AU network.

Another observation is that the P-graph diameters areall significantly smaller than their L-space counterparts.However, since we know that the former is far moredensely connected, this is an unsurprising result.

We now turn to average travel time. The results forthe calculations for the aggregate graph, as well as eachsubnetwork, are shown in Figure 8.

The first essential observation from Figure 8 is that,in the L-space graph, the average travel time for theaggregate graph is far smaller than that of almost allthe other networks (with the exception of Cercanıas).This lends support to the motivation behind our project,since it illustrates how combining different forms oftransportation has a significant impact on travel time,and thus a combined network is more likely to berepresentative of the real world.

Another example of the positive effects of combining

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Fig. 9: Heatmap of L-space graph for aggregate network,where nodes with shorter average path lengths are morered.

different transit networks can be seen when viewing thegraph as a heat map for which the hotter a node gets,the smaller its average path length becomes. To see this,simply compare the aggregate graph (Figure 9) with thatfor Autobuses Urbanos (Figure 10): it is clear that theformer contains more nodes with shorter average pathlengths.

Another interesting observation is the relative changesfor each graph from the L- to the P-space. For example,we see that out of all the subnetworks, EMT has thegreatest reduction in mean path length from L to P. Thissuggests that it has many long lines with few transfers.Meanwhile, in both spaces, the aggregate network hasa smaller mean path length than the EMT subnetwork.One question that this raises is: how do groups ofsubnetworks interact with each other? For instance, howmuch does the mean path length for the EMT graphdecrease when paired with each other network? Thiscan shed light on interactions between subnetworks, andpossibly give transportation planners some insight intothe synergies between different transportation types.

C. Hypothesis 3: Nodes with the smallest clusteringcoefficients occur at the geographical extremes of thenetwork.

The original justification for this hypothesis was thatinterchanges and intersections tend to occur close to each

Fig. 10: Heatmap of L-space graph for Autobuses Ur-banos, where nodes with shorter average path lengths aremore red.

other more in the city center than anywhere else, thusclustering coefficients should be higher in the city center.However, the results suggest that the answer may bemore nuanced.

Figure 11 shows the geographic distribution ofweighted clustering coefficients (see Section IV) for theL-space graph, where darker colors represent higherclustering coefficients. It is evident that the darkest colorsare not clustered in the city center; rather they are spreadout throughout the city, including in its outskirts. Notethat in this case, the weights are travel times; therefore,darker colors represent node clusters where it takes along time to get between nodes in that cluster.

The existence of dark-colored nodes in the outskirts isunsurprising. Suburbs are likely to have fewer and lessfrequent public transport connections, so traveling timesare likely to be higher. However, the darker spots closerto the city center are more surprising. These suggest that,in spite of being close to the city center, there are certainpockets of nodes that are not very well-connected interms of short traveling times. This may reveal pockets ofpublic transport nodes that the government should lookto improve.

Next, Figure 12 shows a similar visualization for theP-space graph. Here, the results are less surprising. Wesee dark colors throughout the city including on the out-skirts, showing that there long transfer times throughoutthe city. However, we also see a collection of lighterpoints near the city center, showing that transfer times

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Fig. 11: Weighted clustering coefficients for L-spacegraph. Note that nodes with clustering coefficients of0 are excluded for clarity.

near the city center are shorter due to the better avail-ability of public transport. It is important to observe thatthis visualization compresses information about manythousands of nodes; a more detailed inspection of thismap could reveal more shades of differentiation betweenmore and less well-connected areas in terms of transfertime.

Finally, Figure 13 is the only one that generallyconfirms our hypothesis. This map shows unweightedclustering coefficients for the P-space, and the higheroccurrence of dark spots near the center show that thereis a higher availability of transfers near the city center,which is an expected result. However, we can also notethat there are pockets with slightly higher clusteringcoefficients even on the outskirts of the city (e.g., inthe southwest). This suggests the existence of suburbancenters, where transportation to the suburbs may con-gregate into hubs. This is an important observation forpolicy planning: if expanding public transportation to thesuburbs, transport planners may want to connect theirnew lines to these hubs, or to note that the areas aroundthe hubs are already well-served.

VII. SUMMARY AND CONCLUSIONS

This project aimed to propose improvements toMadrid’s PTN through the testing of the three hypothesesproposed in Section I. From hypothesis 1, we were ableto identify critical edges that, when removed, signifi-cantly impacted the performance of the PTN. Policymak-ers could consider adding in more parallel transportationmethods between the endpoints of those edges to in-

Fig. 12: Weighted clustering coefficients for P-spacegraph. Note that nodes with clustering coefficients of0 are excluded for clarity.

Fig. 13: Unweighted clustering coefficients for P-spacegraph. Note that nodes with clustering coefficients of 0are excluded for clarity.

crease resilience to attack. From hypothesis 2, we foundthat different subnetworks behave differently in the L-and P- spaces, which reveals characteristics about thenumber of stops versus transfers in those subnetworks.We also learned that exploring different combinations ofsubnetworks could reveal interesting synergies betweentransportation types. Finally, hypothesis 3 showed thatclustering coefficients can reveal pockets of PTNs thathave long travel times between them, and thus can beimproved through more frequent public transport service.

VIII. CHALLENGES AND FUTURE WORK

Future work in this domain would do well tonote that the C++ version of the SNAP libraryhas several bugs. In particular, there were several

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instances in the GetWeightedShortestPath()function where the syntax for retrieving values fromHashMaps was incorrect, resulting in hours of frustrat-ing and time-consuming debugging. However, the C++version does run significantly faster than the Pythonone, and additionally contains certain functions, such asGetWeightedFarnessCentr(), that the latter doesnot.

Another key challenge faced was lack of descriptivemethodology in the literature on creating PTN graphs.For instance, despite the fact that von Ferber et al.analyzed combinations of different transport types [13],they did not describe how edges were created acrossnetworks. Future work can continue experimenting withour 100m threshold for creating inter-network edges and3min weight for these edges to determine how thesevalues affect these results, since we lacked the time andcomputational power to do so. In addition, while a fewpapers separately calculated change in average traveltime and unsatisfied demand as we did in Section IV,none came up with a holistic measure to combine thosetwo metrics such that the impact of disrupting a linkcould be measured on a single scale. While this did notimpact our project due to the dense connectedness of ournetworks, it may have an impact on more sparse PTNs.

We also faced difficulties in working with the dataitself. We continually uncovered subtleties and new de-tails as we worked with the data; for instance, we foundthat the GTFS specification had multiple stop IDs foridentical stops, which led to duplicate nodes in our graphearly on. It is also difficult to check the correctness of ourgraphs, given their huge size and complexity. However,cross-checking small random samples of the graph withthe true PTN network maps was effective in helping usensure the correctness of our networks.

Finally, computational power was a key resourceconstraint. In order to calculate the criticality of eachedge, we had to recalculate all the shortest paths inthe graph and all the average travel times. This meantthat it was only possible to calculate a small numberof criticality values. Unfortunately, we did not anticipatethis constraint, and thus could not gain access to bettercomputational resources in time. Any work that seeksto build on this project should make sure to securecomputational resources that can feasibly calculate thevalues we could not.

One important area for future research is the inves-tigation of different combinations of subnetworks. Asmentioned in Section VI, looking at how the one networkbehaves when paired with various other networks could

shed light on pairwise interactions between subnetworks.This could help policymakers improve PTNs acrosssubnetworks; for instance, if they find that AU and Metrocomplement each other, they could reinforce that synergythrough greater AU stop availability near Metro stops.

Overall, this work demonstrates that graphical tech-niques can be a useful tool in analyzing PTN perfor-mance and resilience. Moreover, looking at PTNs inaggregate can yield significantly more practical insightsthan examining individual subnetworks. While compu-tational power is a power when looking at graphs ofsuch complexity, such analysis has the potential to be apowerful tool in informing policymakers’ decisions onhow to develop PTNs.

VIII. REFERENCES

[1] Google Transit APIs. General Transit Feed Speci-fication Reference, October 2017.

[2] A. Barrat, M. Barthelemy, R. Pastor-Satorras, andA. Vespignani. The architecture of complexweighted networks. Proc Natl Acad Sci U S A,101(11):3747–3752, March 2004.

[3] A. Chen, C. Yang, S. Kongsomsaksakul, andM. Lee. Network-based Accessibility Measuresfor Vulnerability Analysis of Degradable Trans-portation Networks. Network Spatial Economics,December 2007.

[4] Consorcio Regional de Transportes de Madrid. De-manda de transporte pblico colectivo. Technicalreport, 2013.

[5] Google. Reverse geocoding, 2017. [Online; ac-cessed 6-December-2017].

[6] E. Jenelius, T. Peterson, and L.G. Mattsson. Impor-tance and exposure in road network vulnerabilityanalysis. Transportation Research Part A: Policyand Practice, August 2006.

[7] V. Latora and M. Marchiori. Efficient Behavior ofSmall-World Networks. Physical Review Letters,November 2001.

[8] Alan T. Murray, Timothy C. Matisziw, and Tony H.Grubesic. A Methodological Overview of Net-work Vulnerability Analysis. Growth and Change,39(4):573–592, December 2008.

[9] E. Rodrguez-Nez and J.C. Garca-Palomares. Mea-suring the vulnerability of public transport net-works. Journal of Transport Geography, 35:50–63,February 2014.

[10] J. Sienkiewicz and J. A. Hołyst. Statistical analysisof 22 public transport networks in Poland. PhysicalReview E, 72(4):046127, October 2005.

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[11] M.A.P. Taylor. Critical Transport Infrastructure inUrban Areas: Impacts of Traffic Incidents AssessedUsing Accessibility-Based Network VulnerabilityAnalysis. Growth and Change, November 2008.

[12] Union Internationale des Transports Publics. WorldMetro Figures. Technical report, October 2010.

[13] C. von Ferber, T. Holovatch, Yu. Holovatch, andV. Palchykov. Public transport networks: empiricalanalysis and modeling. The European PhysicalJournal B, 68:261–275, 2009.

[14] Wikipedia. 2004 madrid train bombings —Wikipedia, the free encyclopedia, 2017. [Online;accessed 15-November-2017].

[15] Wikipedia. Dijkstra’s algorithm — Wikipedia, thefree encyclopedia, 2017. [Online; accessed 14-November-2017].

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