Robustness of the public transport network against attacks on its routes

During the last decades, there has been a growing focus on central aspects of urban infrastructure and its functioning within the framework of a nascent science of cities [21]. This surge in interest has been propelled by the utilization of spatial networks and the accessibility of public data alongside Geographical Information System (GIS) tools. Moreover, the perspective of considering cities as complex systems, as their overall properties are emergent and are not simple summations of their parts, is increasingly relevant [24]. Thus, many efforts seek to understand the social aspects underlying the dynamic change of cities driven by human mobility [29, 30], design innovative solutions for city services [28], and plan improvements that impact the mobility of inhabitants [27, 26, 19].

An essential infrastructure facilitating the daily movement of people within a city is the public transportation system, which comprises various routes or lines of different modes such as buses, trains, metros, and other public conveyance means. The Public Transport Network (PTN) integrates data from the public transportation system with spatial information about cities [17, 16, 25]. However, there is no singular method for representing the PTN solely based on information regarding a specific mode, such as buses. Instead, various representations exist to depict public transportation information within specific networks, including $\mathbb{L}$-space, $\mathbb{L^{\prime}}$-space, $\mathbb{C}$-space, and $\mathbb{P}$-space [14, 13]. Each of these representations is tailored to serve distinct analytical purposes or to study particular processes of interest in the transportation system, providing insights into different facets of transportation operations. In this study, our focus lies on examining the impact of route removal on the connectivity of the transportation system.

Each route, such as a bus line, consists of a sequential arrangement of stations spanning from an originating station to a terminal one. Passengers have the option to board or disembark at each station along the route. Our motivation in this study is to analyze scenarios wherein an entire bus line discontinues operation either due to a failure or attack, or based on a particular decision of the line’s company management or a strike by its employees, disrupting then the route as a whole.

In many interconnected complex systems across various domains such as biological, social, economic, and technological realms, public transportation networks can experience both intentional and unintentional failures, impacting individual system elements as well as the system as a whole [4]. Unintentional shortcomings typically manifest as minor random disruptions within the network, such as stations or lines being temporarily disabled. Sometimes, these failures are part of the dynamics of networks, making the study of cascading system crashes essential for understanding systems evolution, as discussed in [3, 2]. On the other hand, intentional failures, often termed attacks, aim to disrupt the entire network’s operation. These attacks commonly target nodes, links, or sets of nodes with the most significant potential to destabilize the network. The exploration of how diverse networks amplify failures at the systemic level, along with their stability and recovery, traverses numerous disciplines and has been investigated in networks beyond transportation, including biological, ecological, financial, and social networks [6]. This subject matter is closely related to the robustness and resilience of networks and systems [4], garnering considerable attention due to its significance in identifying and mitigating problems. Various solutions have been proposed, such as interventions to mitigate wildfires and extreme events [7], the development of early warning systems for different disruptive scenarios, the identification of critical and vulnerable elements within the system [6, 47, 48], and the proposal of improved networks to prevent future failures [49].

In the context of PTN, failures can potentially disrupt the daily lives of passengers and all city residents, leading to travel delays for passengers and impacting the city’s broader transportation system. Hence, it is crucial to understand the susceptibility of PTN to failures or attacks to anticipate critical collapses [5], plan future improvements in the city’s transportation infrastructure, and effectively respond to eventual collapses following a failure. The robustness of PTN has been examined across various cities, considering different attack strategies targeting nodes and links [38, 41, 42].

The central question guiding this research is to understand which type of failure or attack on the bus public transportation network yields the most significant impact on the city. It is worth noting that this approach differs from previous studies, which primarily focused on directed attacks targeting specific nodes or links within the network [17, 39, 37]. We operate under the assumption that each transport line, like a bus line, functions as a cohesive unit integrating a set of employees, a fixed organization, and logistical arrangements. As a result, the interruption extends from the starting station to the end station along the route. This paper compares and analyzes the effects of various strategies for selecting the removed line from the PTN based on network topological properties. Our objective is to evaluate which route attack strategy inflicts more significant damage to the network and quantify its impact. The proposed methodology aligns with the broader theory of bond (or edge) percolation. For instance, in the realm of PTN analysis, prior works such as [35, 36, 40] concentrate on attacking edges based on their centrality measures and delineate the effects of such attacks. However, when a route is attacked, instead of removing a single edge at each step from $\mathbb{L}$-space, all its set of consecutive edges are eliminated from the $\mathbb{L^{\prime}}$-space. Indeed, although there exist several works investigating the impact of removing straight rigid rods [45, 44] or $k^{2}$-mers [43] within the framework of regular lattices, the exploration of removing complex paths in real networks remains relatively unexplored.

To analyze general cities and compute statistics of PTN, we will generate a synthetic PTN based on the model proposed in [17]. These synthetic networks emulate topological properties similar to those observed in large cities [16], incorporating a random component to allow for multiple realizations.

Moreover, we will conduct a similar analysis on the PTN network of a Latin American city, specifically focusing on public bus transportation. To do so, we will utilize open data from [32] to examine 489 bus lines within the Metropolitan Area of Buenos Aires (AMBA), Argentina. This region spans approximately 13,285 square kilometers [33] and serves around four million people daily [34].

The paper’s organization is structured as follows: Section 2 outlines the PTN methodology, details the route removal strategies for implementing attacks, discusses the metrics used to quantify the impacts of the attacks, and provides a description of the synthetic PTN employed. Section 3 presents the results of the attacks on the synthetic PTN. In Section 4, the same methodology is applied to the city of Buenos Aires. Section 5 covers the discussion and future research directions.

We systematically analyze multiple realizations of the synthetic network to explore the robustness of PTNs against attacks using the route removal described procedure. Figure 4 illustrates the procedure for a specific realization of the synthetic PTN, elucidating the analysis through a simple example. In the figure, the $\mathbb{L}$’-space of one realization of the synthetic model is depicted for $R=500$ and $S=45$ in the left panel ($\Phi=0$). In the top row of panels, snapshots display different instances when routes are randomly removed, while in the bottom row, routes are removed with priority based on betweenness in the $\mathbb{C}$-space. The same number of removed routes is visualized for both processes, indicated by $\Phi=0.07$, $0.26$, and $0.66$ in the left-to-right columns, respectively. Each of the first five largest components is shown in different colors, as indicated in the legend, while the rest of the components ($S_{i}$ with $i>5$) are displayed in light gray. Qualitatively, it is evident that targeted attacks following the betweenness measure are more effective in detaching nodes from the first giant component. The second component appears with a size comparable to the giant component. Intimately related to this, it is also noticeable that the network disassembly mechanism, in the case of betweenness, leads to detachments of entire network regions, illustrated by the successive emergence of different components of considerable size (bottom right panel).

Going further, we will systematically analyze the attack following the procedures described in the previous section: four centrality measures in the $\mathbb{C}$-space (degree, closeness, PageRank and betweenness), the maximal and minimal removal procedure described in the previous section, and random removal. Each procedure is computed for $10$ realizations of the synthetic network model. For random removal, the number of realizations is increased to $100$, ensuring the convergence of the procedure. In this manner, we were able to reconstruct the average values of $S_{1}/N$, $S_{2}/N$, and $\Sigma_{S}$ as a function of $\Phi$, as shown in Figure 5, across the different realizations.

For $S_{1}/N$, the maximal and minimal procedures serve as lower and upper bounds for a wide range of $\Phi$. The betweenness attack emerges as the most effective among the traditional centrality measures, leading to a substantial decrease in the giant component. Its effectiveness stems from its ability to break the network into several sizable components swiftly. In contrast, the maximal metric destructs the network similarly to betweenness (or even more effectively for small $\Phi$) based on the giant component, though without generating the emergence of a second component. When considering the entropy, the distinction becomes more evident. The maximal attack reveals its strategy: it removes routes to generate many isolated nodes, rapidly driving the entropy to high values. The disparity between random removal and classical metrics is notable. In the literature, for instance, as seen in [31], random attacks tend to be less detrimental when disassembling a network. The other three centrality measures in $\mathbb{C}$-space: degree, closeness and PageRank exhibit similar behavior in $S_{1}$, $S_{2}$, and $\Sigma_{S}$. One noticeable result is that the betweenness procedure leads to smaller values of $S_{1}$ (for $\Phi>0.6$) compared to the maximal method, which is defined to be the most effective one-step strategy.

To ensure the robustness of the observed results against parameter choices, we compute $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ for synthetic networks with different values of $R$ and $S$, as illustrated in Figure 6. The behavior across different values of $S$, denoted by the varying size of the circles, demonstrates minimal variations. $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ stabilize as functions of $R$ for $R\gtrsim 200$. Initially, we observe that the procedures can be ordered by decreasing value of $\mathcal{R}_{1}$ as follows: minimal; followed by the group of degree, PageRank, and closeness; then random; and finally, the pair of maximal and betweenness with the lowest values. Notably, the attacks by betweenness and maximal exhibit distinct behavior from the rest, with maximal being the most effective.

Regarding the second component, the order in $\mathcal{R}_{2}$ is as follows: betweenness has the largest value by a considerable margin, followed by the group of closeness, PageRank, maximal, and degree; then random; and finally, the minimal procedure. It is noticeable that the maximal procedure, which is the one with the best attack performance, together with betweenness, does not yield high values of $\mathcal{R}_{2}$, but quite the opposite. This emphasizes that while both attack strategies have the same impact, for the case of the maximal procedure, $S_{1}$ is not broken into large $S_{2}$. The bottom panel of Figure 6 illustrates both results in the $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ plane.

This section implements all defined attack procedures in a real PTN. In particular, we will work with the urban bus system of the Metropolitan Area of Buenos Aires (AMBA), the largest metropolitan region in Argentina (and one of the largest in Latin America).

We have obtained the information regarding the routes of the different $R=489$ bus lines operating in the AMBA using OpenStreetMaps [32]. By utilizing the DBSCAN algorithm implemented in [20], nearby stops within an $80$ meter radius were merged, aiming to consider nearby stops as the same node. This process resulted in obtaining the $\mathbb{L}$’-space, which consists of $N=6016$ nodes and $22698$ links, with $\Omega_{S}=1$. As for the $\mathbb{C}$-space, the network comprises $R=489$ nodes and $19201$ links.

The same analysis as in Section 3 is conducted on the AMBA bus network. In panel $[A]$ of Figure 7, $S_{1}/N$, $S_{2}/N$, and $\Sigma_{\mathcal{S}}$ are plotted for the different attack strategies.

One notable observation is the significant difference compared to the results for the synthetic networks, as all metrics except the minimal one are more effective at disrupting the network than the random strategy. This aligns with expectations from the literature and could indicate some shortcomings of the synthetic network model in mimicking real networks. In this real case, the minimal method is the least effective in damaging $S_{1}$, while the betweenness procedure emerges as the most effective for $\Phi>0.2$, surpassing the maximal strategy. In this case, both $S_{1}$ and $S_{2}$ exhibit considerable jumps, indicating a significant change when one route is attacked. It’s important to note that the only way $S_{2}$ can increase is by forming a new and larger second component from the giant component.

Observing the evolution of $S_{2}/N$ reveals that betweenness succeeds in disassembling the network by generating detachments of the giant component from important sets of connected nodes. Analogously, this behavior can be seen in the flow diagrams in panels $[B]$, $[C]$, $[D]$, and $[E]$, where the flow of nodes between the first five largest components is plotted for the minimal, degree, maximal, and betweenness strategies, respectively. For example, the minimal and degree attacks disassemble the network by removing nodes that are then isolated throughout the process. The same happens, but more quickly, for the maximal case. In contrast, the betweenness strategy quickly generates the detachment of secondary components that prevail throughout the attack process.

Interestingly, when observing the normalized entropy associated with the sizes of the different components throughout the process, the minimal and maximal strategies represent the lower and upper bounds. Although betweenness is more effective in breaking the network, the distribution of components along the process exhibits a lower entropy than for the maximal case. Additionally, we again observe a discrepancy with the synthetic network: the entropy for the random case is lower than that for the classical metrics. This difference highlights a key distinction between synthetic and real networks.

To quantitatively compare the overall robustness against the different strategies and how the network is disassembled during the process, we compute $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ for the actual network in the bottom panel of Figure 6. Thus, betweenness stands out as the most effective strategy due to its low $\mathcal{R}_{1}$, while it has the highest $\mathcal{R}_{2}$.

The strategies applied to the AMBA PTN can be classified in the $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ plane, where betweenness is always located in the bottom right corner and minimal in the opposite one. The results of the real PTN (shown in circles) can be compared with the cases of synthetic modeled PTN (represented by star symbols with error bars denoting the standard deviation obtained from $10$ realizations). Although they do not perfectly coincide, there is a similar behavior between the strategies applied to the real PTN and the modeled one.

To conclude the analysis of the real network and better understand the disassembly process of the network, we decided to study the betweenness removal strategy for different values of $\Phi$. In particular, we wish to examine what happens when there are jumps in $S_{2}/N$ in Figure 7[A]. For this purpose, in Figure 8, we plot the $\mathbb{L}$’-space of the AMBA for 3 different values of $\Phi$. In the upper panel, we mark the line to be removed in the next step, while in the lower panel, we observe the effect produced by such removal. The different components are plotted in different colors.

The first observation is that removing the path with the highest betweenness does not necessarily result in the appearance of a new second giant component. For instance, at $\Phi=0.07$, $S_{2}$ remains constant, while a new $S_{3}$ emerges. However, what appears to be more significant is that in all three cases, the components detaching from the lattice belong to the periphery of the lattice. A similar effect is observed in the example shown in Figure 4. This suggests that the routes with higher betweenness in the $\mathbb{C}$-space serve as bridges between the core of the network and its expansions into the periphery.

We found that the results obtained from the model were not significantly different from those of the real network. However, to understand the differences in $S_{1}$ and $S_{2}$ between them, we can propose certain improvements to the model. One major simplification of the synthetic model is assuming all routes ($R$) have the same number of stations ($S$). However, this assumption does not hold in the real network, where routes have a varied number of stations. Therefore, by considering the same synthetic model but with a distribution of stations on the routes forced to be similar to the real one, we obtain results that are much more similar to those of the real network. For example, in cases of $S_{1}(\Phi)$, we can observe how the procedures given by the metrics in the $\mathbb{C}$-space (degree, closeness, and PageRank) are more harmful than the random ones in these synthetic models with ”realistic” routes. This indicates that improvements can be proposed to the synthetic model to mimic the topological and robustness properties against attacks more accurately, which goes beyond the objectives of this work.

In this work, we focused on studying the robustness and sensitivity of public transport networks (PTNs) to route attacks. This novel approach allows for obtaining different results than node or link attacks on the network. We defined various network attack procedures based on centrality measures for $\mathbb{C}$-space, where nodes represent routes. As metrics of damage, we used measures based on the sizes of the components formed in the networks where stations are nodes (both $\mathbb{L}$-space and $\mathbb{L}^{\prime}$-space). Additionally, we introduced two new route attack procedures, maximal and minimal, based on the most and least harmful tactics in the network damage at the next step. All of this was studied in both synthetic PTNs, which allowed for statistical analysis, and a real network of considerable size and well-established structure in Buenos Aires. We observed how different centrality measures or strategies generated distinct outcomes. Specifically, we illustrated that degree, closeness, and PageRank metrics show similar behaviors while betweenness operates differently. The latter procedure progressively breaks down the network into several components of comparable sizes. We found that the minimal strategy inflicts the most minor damage to the network. However, following the betweenness method can either surpass or be on par with the maximal strategy in terms of damage, albeit with markedly different behavior in the second component. Finally, we propose improvements to synthetic networks by observing the differences in the behavior between the random strategy and the other three centrality measures across synthetic and real networks. These enhancements in properties may contribute to producing new synthetic models of metropolitan areas for future studies. We also believe that the analysis developed in this work can aid in enhancing and designing PTNs that are efficient and resilient to difficulties. Furthermore, we believe that continuing in this direction would make it feasible to assess the accessibility and efficiency of a PTN by utilizing actual data on residents’ mobility patterns.

We acknowledge the support of PIP 2021-2023 11220200102701CO from CONICET for their financial assistance.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

TC collected and analyzed the data. TC, IC and LE interpret the results and wrote the manuscript. EL and IC conceived the study. All authors read and approved the final manuscript.