# Spectral Analysis of Delay Propagation in Air-Rail Intermodal Networks

Jean-Claude Lebegue, Daniel Delahaye

► To cite this version:

Jean-Claude Lebegue, Daniel Delahaye. Spectral Analysis of Delay Propagation in Air-Rail Intermodal Networks. SESAR Innovation Days, SESAR, Dec 2025, Bled, Slovenia. hal-05440815

HAL Id: hal-05440815

https://enac.hal.science/hal-05440815v1

Submitted on 5 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Spectral Analysis of Delay Propagation in Air-Rail Intermodal Networks

## A Case Study at Paris CDG Hub

Jean-Claude Lebègue

OPTIM

ENAC

Toulouse, France

jean-claude.lebegue@enac.fr

Daniel Delahaye

OPTIM

ENAC

Toulouse, France

daniel.delahaye@enac.fr

*Abstract*—Intermodal transportation networks integrating air and rail systems are critical for efficient and sustainable passenger mobility, yet their performance is often challenged by cascading disruptions originating from delays at key hubs. This paper presents a novel application of Graph Signal Processing (GSP) techniques to analyze delay propagation and characterize unexpected disruptions within a large-scale air-rail intermodal network. Using real flight data and simulated rail delay data for the Paris Charles de Gaulle (CDG) hub, we model the network of transportation legs as a graph where edges represent Pearson correlation coefficients of delay patterns. We introduce spectral metrics such as total variation and total energy derived from the graph Laplacian to quantify the spatial smoothness and irregularity of delay distributions. The spectral decomposition via the Graph Fourier Transform (GFT) enables identification of distinct delay patterns associated with both expected and irregular states of the network. Our results reveal that, while most days exhibit smooth and predictable delay propagation consistent with historical correlations, select days demonstrate significantly high total variation, indicating unexpected disruption dynamics. This methodological framework provides transport operators with actionable insights to detect, characterize, and potentially mitigate complex disruption scenarios in multimodal hubs.

*Keywords*—Air-rail intermodal network; Disruption management; Transportation delay analysis; Graph Signal Processing.

## I. INTRODUCTION

By 2050, the European Commission [1] foresees airports evolving into multimodal hubs closely integrated with other forms of transport. This vision requires a robust, connected infrastructure enabling air travel to blend seamlessly with rail and public transit, optimizing passengers' door-to-door journeys in terms of cost, time, and convenience. Airports, acting as pivotal nodes, support trade, tourism, and economic development, while High-Speed Rail (HSR) emerges as a sustainable, high-capacity alternative for short-and medium-distance journeys. The combination of air and HSR networks expands regional accessibility, with seamless multimodal connections offering clear benefits for travelers, operators, and local economies.

Disruptions within multimodal transportation networks can significantly degrade overall system performance, leading to

cascading effects that impact the coordination between interconnected transport modes. When a delay or disturbance occurs at one node—such as an airport or rail station—it can propagate through the network, affecting subsequent transfers and connections. This results in longer wait times, missed connections, and increased uncertainty for passengers, ultimately extending the door-to-door travel duration. The complexity of managing such disruptions is compounded by the diverse stakeholders involved and the varying operational constraints of different transport modes. Efficient coordination and timely information sharing among these actors are therefore crucial to mitigate delay propagation, optimize recovery strategies, and maintain a reliable and resilient multimodal travel experience for passengers.

Optimizing connectivity and resilience across these networks depends on efficient management of disruptions—a challenge rooted in effective data sharing. Transportation providers, including airports, airlines, rail, and public transport operators, often work in silos, hampering timely information exchange. As demonstrated by the SESAR-supported SIGN-AIR project [2], streamlining data-sharing agreements and implementing secure digital platforms is a prerequisite for integrated multimodal services. Real-time data exchange enables early detection of events, coordinated responses, dynamic rerouting, and single ticketing across travel modes. Such collaboration minimizes passenger inconvenience during disruptions, automates rebooking processes, and promotes sustainable mobility alternatives. By simplifying contractual frameworks and fostering standardized, secure information flows, innovations like SIGN-AIR pave the way for reliable, passenger-centric multimodal travel experiences.

This paper constitutes a contribution to the SIGN-AIR project's overarching goal of developing innovative solutions to characterize and enhance the resilience and predictability of multimodal transportation systems facing disruption challenges. Specifically, we leverage GSP methodologies to detect and quantify expected and unexpected patterns in delay distributions at a multimodal hub. By applying spectral decomposition techniques to a Hub-and-Spoke network representation, we analyze the structural signatures of delay propagation via

<figure><img src="image_2.png" /></figure>
eigenvalue and eigenvector analyses, providing novel insights into disruption dynamics and recovery mechanisms.

The paper is organized as follows: Section II reviews recent advances in delay and disruption management. In Section III, we introduce and detail the metrics employed to quantify unexpected delay patterns. Section IV presents the data sources and discusses the findings from the case study conducted at Paris Charles de Gaulle (CDG) airport. Finally, Section V summarizes the main contributions and outlines directions for future research.

## II. LITERATURE REVIEW

### A. Disruption management

Disruption management in air-rail and multimodal transport networks has become increasingly central to transport systems research, with recent literature providing a variety of frameworks and models that address operational complexities, user impacts, and systems resilience. Real-time disruption management has been reviewed extensively [3], highlighting the need for effective simulation-optimization and decision support tools as well as revealing a gap in studies addressing large-scale, cascading disruptions spanning multiple transport modes. Network robustness [4], [5] and resilience assessments focus on critical node identification, resilience metrics, and comparative studies of air, rail, and integrated networks, offering methodologies to evaluate and bolster multimodal network continuity under failure scenarios. From the perspective of traveler behavior [6], [7], disruption modeling leverages agent-based and simulation approaches to capture user delays, behavioral adaptation, and the propagation of delays throughout the interconnected system. In parallel, collaborative decision-making frameworks and integrated recovery models [8]–[10] underscore the operational benefits of tightly coordinated air-rail disruption response, emphasizing the importance of information sharing and joint contingency planning among stakeholders. Together, these complementary strands of the literature provide a solid basis for enhancing both analytical research and practical management of disruptions in complex, integrated air-rail networks.

### B. Delay analysis

The field of disruption management encompasses a breadth of topics, including network robustness [11], passenger response [12], freight reliability [13], and collaborative recovery [14]; this work focuses specifically on delay analysis within air-rail intermodal networks. Delay analysis has evolved as a critical domain in both passenger and freight research, leveraging sophisticated modeling and optimization approaches—such as Stochastic Petri Nets and Markov Chains—to identify inefficiencies and reduce total delay across integrated air-rail systems [15]. Recent hierarchical spatial and temporal models using mixed-integer and quadratic programming further minimize costs and emissions while enhancing overall network performances [16]. Empirical studies reveal that transfer and waiting times, along with schedule coordination and service frequency, are primary drivers of total

travel time, particularly in networks linking smaller cities to major hubs [17], [18]. The resilience of air-rail systems under disruption is increasingly supported by integrated spatiotemporal network recovery models and predictive analytics, including machine learning to forecast weather-induced delays and simulation to assess critical freight network segments vulnerable to operational disturbances [19]–[22]. Moreover, delays at intermodal hubs and switching points are shaped by hub resource allocation and dynamic path algorithms, directly affecting network reliability and efficiency [23], [24]. Collectively, these insights underscore the importance of advanced analytical methods, coordinated schedule management, and robust system design for mitigating delays and enhancing the reliability of air-rail intermodal networks.

### C. Graph signal processing techniques

GSP methods—particularly the GFT and Laplacian spectral decomposition—are emerging as powerful techniques for examining the connectivity, dynamics, and disruptions within complex transportation networks, including air-rail intermodal systems. Although research to date has not focused directly on air-rail networks, foundational studies show that GFT can decompose network-wide signals into interpretable frequency components, offering a principled way to capture patterns, anomalies, and structural bottlenecks using spectral projectors and directionality-aware transforms [25]–[27]. Laplacian spectral analysis further delivers metrics for robustness, path efficiency, and overall system resilience—properties impactful for multimodal network planning and disruption mitigation [27], [28]. Computational advances, such as signal-adapted decompositions and polynomial approximations, have enabled multi-scale analysis of large urban traffic networks and are readily transferable to intermodal transportation contexts [27], [29]. As Li et al. [30] highlight, applying GSP techniques in air traffic delay analysis provides rich operational insights and a framework for probabilistic classification of network events, suggesting these spectral methods are well-suited to future research aimed at optimizing air-rail intermodal systems through network-aware and disruption-resilient analysis.

## III. METHODOLOGY

In this section, we present the graph delay model use to analyze transportation delay at a hub and define the total variation (TV) and the total energy (TE) of a signal.

Let us consider a graph $G$ defined by a set of nodes $n_i \in \mathcal{N}$ with $N = |\mathcal{N}|$ and edges $e_{ij} \in \mathcal{E}$. An edge connects two nodes $e_{ij} = (n_i, n_j)$. The weighted adjacency matrix is the matrix $A = [a_{ij}]$ where $a_{ij}$ is the weight of the edge $e_{ij}$. The degree matrix $D = [d_{ii}]$ is a diagonal matrix where the element $d_{ii}$ is the strength of node $n_i$. The strength $d_{ii}$ is defined as follows:

$$d_{ii} = \sum_{j=1}^{N} a_{ij} \qquad (1)$$

Given the adjacency matrix $A$ and the degree matrix $D$, the Laplacian matrix $L$ of the graph $G$ is defined by $L = D - A$.

<figure><img src="image_3.png" /></figure>
If $L$ is a real symmetric matrix, then it possesses a full set of orthonormal eigenvectors. Let us note the Laplacian eigenvectors by $\nu_i \in \mathbb{R}^{N \times 1}$ for $i \in \{1, \dots, N\}$. The associated eigenvalues $\lambda_i$ are the solutions of $L\nu_i = \lambda_i\nu_i$. The eigenvalues can be sorted so that $\lambda_1 \le \dots \le \lambda_N$.

The sum of each row of the Laplacian is equal to zero. Thus, $\lambda_1 = 0$ is always an eigenvalue and its multiplicity equals the number of connected components of the graph $G$. When the graph is fully connected, there is only one connected component therefore $\lambda_1 = 0 < \lambda_2 \le \dots \le \lambda_N$.

The set of eigenvectors $\nu_1, \dots, \nu_N$ form the graph Fourier basis for any graph signal $x \in \mathbb{R}^{N \times 1}$ on the nodes of the graph. For example, consider an airport network and the delay signal. Then, $x$ has as many components as there are nodes in the network, and each component represents the delay experienced by an airport over a time period. We will call $x$ a nodal graph signal or a graph signal with the understanding that it is supported on the nodes of the graph. The graph Fourier transform of a graph signal $x$ is given by $(\alpha_1, \dots, \alpha_N)$, where each $\alpha_i$ is computed as follows:

$$ \alpha_i = x^T \nu_i \quad (2) $$

The graph Fourier transform (also known as spectral decomposition) quantifies the “contribution” of each eigenvector in the graph Fourier basis to $x$. The TE of a signal $x$ is defined as follows:

$$ TE = \|x\|^2 = \sum_{i=1}^{N} \alpha_i^2 \quad (3) $$

The energy spectrum of a signal, represented as $(\alpha_1^2, \dots, \alpha_N^2)$, measures how energy is distributed among various graph eigenvectors. The term $\alpha_i^2$ is also known as the energy contribution of the i-th mode to the graph signal $x$.

The TV of a signal $x$ is defined by the following equation:

$$ TV = x^T L x = \frac{1}{2} \sum_{i \neq j} a_{ij} (x_i - x_j)^2 \quad (4) $$

A signal with a lower TV is considered smoother compared to another signal with a higher TV.

Consider two nodes $n_i$ and $n_j$ that are connected by an edge $e_{ij} = (n_i, n_j) \in \mathcal{E}$. We observe $k$ samples of the paired data $\{(x_{n_i}^1, x_{n_j}^1), \dots, (x_{n_i}^k, x_{n_j}^k)\}$, where $(x_{n_i}^l, x_{n_j}^l)$ represents the l-th observation of the pair of random nodal signals on nodes $n_i$ and $n_j$, for $l = \{1, \dots, k\}$. In the previous airport network example, $x_{n_i}^l$ represents the delay experienced by airport $n_i$ during time period $l$. Let a pair of observations $(x_{n_i}^l, x_{n_j}^l)$ be drawn from the bivariate Gaussian distribution defined by:

$$ \begin{pmatrix} X_{n_i}^l \\ X_{n_j}^l \end{pmatrix} \sim \mathcal{N} \left[ \begin{pmatrix} \mu \\ \mu \end{pmatrix}, \sigma^2 \begin{pmatrix} 1 & \rho_{X_{n_i}^l, X_{n_j}^l} \\ \rho_{X_{n_i}^l, X_{n_j}^l} & 1 \end{pmatrix} \right] \quad (5) $$

In Eq. (5), $\rho_{X_{n_i}^l, X_{n_j}^l}$ is the Pearson correlation for random variables $X_{n_i}^l$ and $X_{n_j}^l$. We now define the elements $a_{ij}$ of the adjacency $A$ more formally by using a weight mapping. Let

<figure><img src="image_4.png" /><figcaption>Figure 1. Flight Origin-Destination (OD) pairs passing through CDG airport in June 2018.</figcaption></figure>

$w: E \to [0, 1]$ be defined as $e_{i,j} \mapsto w(e_{i,j}) = a_{ij} = r_{x_{n_i}, x_{n_j}}^k$. This weight mapping associates each edge $e_{i,j}$ with the projected sample Pearson correlation coefficient $r_{x_{n_i}, x_{n_j}}^k$ for a signal between nodes $n_i$ and $n_j$, computed from $k$ previous data observations $\{(x_{n_i}^1, x_{n_j}^1), \dots, (x_{n_i}^k, x_{n_j}^k)\}$:

$$ \rho_{x_{n_i}, x_{n_j}}^k = \frac{\sum_{l=1}^k (x_{n_i}^l - \hat{\mu}_i)(x_{n_j}^l - \hat{\mu}_j)}{\sqrt{\sum_{l=1}^k (x_{n_i}^l - \hat{\mu}_i)^2} \sqrt{\sum_{l=1}^k (x_{n_j}^l - \hat{\mu}_j)^2}} \quad (6) $$

$$ r_{x_{n_i}, x_{n_j}}^k = \max \left( 0, \rho_{x_{n_i}, x_{n_j}}^k \right) \quad (7) $$

In Eq. (7), we denote the in-sample means of the node signals, $x_{n_i}$ and $x_{n_j}$, as $\hat{\mu}_i$ and $\hat{\mu}_j$, respectively. Additionally, $r_{x_{n_i}, x_{n_j}}^k$ serves as an estimate (specifically, a statistic) for the population Pearson correlation coefficient $\rho_{X_{n_i}^l, X_{n_j}^l}$.

In our context, the Pearson correlation serves as a quantitative indicator of how delays on two transportation elements are statistically related. A high positive value suggests that delays tend to occur simultaneously or increase together, meaning a disruption on one leg is likely to propagate to the other, either due to shared resources, scheduling dependencies, or operational connectivity. By mapping these correlations, the analysis identifies key connections where delay propagation is most likely, helping to understand and manage the spread of disruptions across a multimodal network.

## IV. RESULTS

This section presents the main results of the methodology presented in the previous section applied to the Paris CDG hub.

### A. Data processing

The primary objective of this study is to analyze delays and their propagation at the hub level. Accordingly, the chosen signal represents the delay in transportation legs (both train

<figure><img src="image_5.png" /></figure>
<figure><img src="image_6.png" /><figcaption>Figure 2. Train OD pairs passing through CDG station in June 2018.</figcaption></figure>

and flight connections) servicing the CDG airport. CDG is a multimodal hub, served by flights and trains, which constitute the two main transportation modes analyzed in this work. To construct the multimodal delay network at CDG, we use data from EUROCONTROL and Société Nationale des Chemins de Fer (SNCF). The multimodal network is modeled as a bi-layer network composed of a high-speed rail layer and a flight layer, based on June 2018 data.

The flight layer is built using the European flight schedule provided by EUROCONTROL. This dataset includes departure and arrival airports (see Fig. 1), scheduled and actual off-block and arrival times. Departure delay is defined as the difference between the actual off-block time and the scheduled time at the departure airport. Similarly, arrival delay is the difference between actual and scheduled arrival times at the destination airport. Total delay is calculated as the sum of departure and arrival delays. We compute the daily delay signal for each leg by aggregating total delays of connections operating on that leg within the same day. To illustrate this concept, let us consider a specific flight leg from $n_o$ to $n_d$. In this example, three flights, designated as $f_1$, $f_2$, and $f_3$, have operated on a specific day, $D$, at distinct departure times. These flights have experienced varying delays, indicated by the total delays $d_1$, $d_2$, and $d_3$. Subsequently, the aggregated total delay for day $D$ on that leg, denoted by the pair $(n_o, n_d)$, is equivalent to the sum of the individual delays, $d_1 + d_2 + d_3$.

Due to data limitations, we cannot construct the daily rail network for June 2018. Consequently, we assess rail delay data using an alternative methodology, described as follows. First, we identified train OD pairs passing through the CDG train station using General Transit Feed Specification (GTFS) data from July 2025, obtained from the SNCF open data platform. This dataset provides static train schedules for the entire month. We began by determining the area ID corresponding to the CDG TGV station, which allowed us to pinpoint all relevant stops within this area. Subsequently, we filtered all trips whose routes included at least one of these stops, thereby

<table><caption>TABLE I. ASSOCIATION BETWEEN MISSING STATIONS USED AS OD PAIR AND THEIR CLOSEST STATIONS INCLUDED IN THE REGULARITY DATA.</caption><thead><tr><td>Missing station</td><td>Associated station</td></tr></thead><tbody><tr><td>Lille Flandres</td><td>Lille</td></tr><tr><td>Aéroport Charles de Gaulle 2 TGV</td><td>Marne la Vallee</td></tr><tr><td>Bruxelles-Midi</td><td>Lille</td></tr><tr><td>Amsterdam Centraal</td><td>Lille</td></tr><tr><td>Montpellier Saint-Roch</td><td>Montpellier</td></tr><tr><td>Lyon Perrache</td><td>Lyon</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Figure 3. Estimated number of trains per day, frequencies of delayed departures and arrivals, and corresponding average delays per OD pairs.</figcaption></figure>

isolating those that pass through CDG station. Each OD pair transiting via CDG is split into two new OD pairs: one connecting the former origin to CDG station and the other connecting CDG station to the former destination. The train OD pairs included in this study are shown in Fig. 2.

Next, we assess train delays using monthly regularity data from SNCF. This data provides statistics on scheduled trains, cancellations, and delays at departure and arrival for each leg. Many OD pair stations were not present in the regularity data; for these, we associate the closest available station from the dataset. The association table is provided in Table I.

For example, using Table I, the OD pair (Montpellier Saint-Roch, Bruxelles-Midi) becomes the associated OD pair (Montpellier, Lille). The average number of trains operating in June 2018 for each OD pair is then estimated using the corresponding associated OD pair, divided by the number of days in the month. In cases where the associated OD pair is missing from the regularity data, we estimate train counts by averaging the number of departures at the origin and arrivals at the destination stations.

The same methodology is applied to assess the daily frequency of delayed trains at departure and arrival, along with their average delays. The results are illustrated in Fig. 3.

The average number of trains per day per OD pair ranges from 4 to 10, with an overall mean of 9. The proportion of delayed trains varies from 1% to 54% at departure and from 6% to 37% at arrival. Departure delays for delayed trains range from 12 to 33 minutes, averaging 15.6 minutes, while arrival

<figure><img src="image_8.png" /></figure>
<figure><img src="image_9.png" /><figcaption>Figure 4. Total daily delay (minutes) in June 2018 for flights and trains.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 5. Graph showing high-correlated (correlation greater than 0.8) OD pairs passing through CDG, where blue and orange nodes represent flight and train legs, respectively, and links indicate delay correlations between legs.</figcaption></figure>

delays range from 21 to 51 minutes, averaging 30.1 minutes. These delays are consistent with SNCF's definitions, which classify a train as on time if:

* The arrival delay at the terminus is less than 5 minutes for journeys under 1 hour 30 minutes;

* The arrival delay is less than 10 minutes for journeys between 1 hour 30 minutes and 3 hours;

* The arrival delay is less than 15 minutes for journeys longer than 3 hours.

In this study, most trip durations exceed 1 hour 30 minutes, so delayed trains have at least a 10-minute arrival delay.

Based on the estimated averages, we model the number of departures and arrivals with delays and their respective delays using an exponential distribution — a model validated in the literature for train delays [31], [32]. Let $N_d$ denote the random variable modeling the number of trains departing with a delay from the departure station of a given leg. We assume that $N_d$ follows an exponential distribution $Exp(\frac{1}{\beta})$, where $\beta$ is the scale parameter. The probability density function of this exponential distribution is defined as follows:

$$f(x, \beta) = \begin{cases} \frac{1}{\beta}e^{-\frac{x}{\beta}} & x \ge 0, \\ 0 & x < 0. \end{cases} \quad (8)$$

The use of an exponential distribution to model delayed train legs is supported by prior studies reporting that train delay times often exhibit memoryless characteristics consistent with this distribution, particularly in the context of minor to moderate delays. This assumption provides analytical tractability and aligns with empirical evidence in rail delay literature, allowing for a straightforward representation of delay duration distributions at intermodal hubs. Nonetheless, we recognize that real-world delay distributions may deviate from pure exponential form due to factors such as varying passenger load, infrastructure constraints, or extreme disruption events.

The period from May to June 2018 experienced multiple strike disruptions [33], which we account for by introducing a

random variable uniformly distributed over $U([0, 1])$. During June 2018, seven strike days severely affected high-speed train operations, with only 90% of trains running. On these days, delays were modeled to be twice the usual magnitude.

Figure Fig. 4 depicts the simulated daily delays generated using the mean values from Fig. 3 as input parameters for the exponential and strike disruption models. Delays vary between 5,000 and 15,000 minutes per day, averaging approximately to 8,000 minutes. The seven peaks correspond to strike-related disruptions. Compared to flight delays at CDG, the maximum rail delays are lower than the minimum flight delays, likely due to the smaller number of trains servicing CDG relative to flights.

## B. Case Study: Paris Charles de Gaulle Hub

This study focuses on analyzing delay patterns at the CDG hub using actual flight data from June 2018 alongside simulated train data for the same period. Rather than analyzing correlations between airports as done in previous works [30], we investigate delay propagation between transportation legs at the Paris CDG intermodal hub. Here, the graph signal $x$ corresponds to the total delay on each leg, defined as the sum of departure and arrival delays.

We model this problem using a graph delay network, where nodes represent legs connected to the CDG hub and edges encode the Pearson correlation coefficients of delays between those legs, computed with Eq. (7). The resulting correlation graph, shown in Fig. 5, contains over 700 nodes; for clarity, only edges with correlations greater than 0.8 are displayed.

Both flight and train legs are present in the graph, with flight legs accounting for 95% and train legs for 5%. The graph reveals numerous connected components of varying size; most components have fewer than three nodes, while a few form complex, large clusters. This correlation graph serves as a valuable tool for identifying legs where delays historically propagate, enabling targeted mitigation strategies

<figure><img src="image_11.png" /></figure>
<figure><img src="image_12.png" /><figcaption>Figure 6. Total variation of the delay signal at CDG hub as a function of daily total delay (minutes).</figcaption></figure>

during disruptions. Notably, some correlations exist between flight and train legs, albeit limited, potentially reflecting the current integration level between these two transport modes at CDG.

Our analysis demonstrates that the correlation graph effectively captures delay propagation among different legs at CDG, facilitating identification of potentially impacted OD pairs during disruptive events. This methodology enables detection of expected disruption patterns at the hub. To further investigate unexpected delay distributions on the intermodal network, we apply spectral measures such as TV and TE.

Fig. 6 presents the TV of the 30 days of June 2018 depending on the total delay per day. We remind the reader that, in this work, TV quantifies the irregularity in the distribution of delays across the network: a low TV corresponds to smooth and expected delay patterns, whereas a high TV indicates unusual or unexpected disruptions causing significant variability between connected legs. We also plot the 99% prediction interval computed based on a quadratic fit curve. As we can see, all the values except one are included in the prediction interval. The exception is 2018-06-01. This date presents an unexpected high level of TV. According to Fig. 4, this days shows unusual high delays on the rail network corresponding to strike simulation but classical delays on the air network. Moreover, the delays on the rail network of that day are very similar to 2018-06-02. The flight delays are a bit different but the this latter day does not present such a high TV. To better understand the difference between those two situations a deeper analysis is necessary.

$$TV = \sum_{i} \alpha_{i}^{2} \lambda_{i} \qquad (9)$$

Building on [30], the TV can be expressed via the graph signal's spectral decomposition (Eq. (9)), where $\alpha_i$ and $\lambda_i$ are the mode energies and Laplacian eigenvalues, respectively. The eigenvalues of the graph Laplacian represent the frequencies of the corresponding modes, with lower eigenvalues indicating smoother, more gradual variations in delays across the network, and higher eigenvalues reflecting more abrupt, localized

<figure><img src="image_13.png" /><figcaption>Figure 7. Graph Fourier Transform of delay signals on June 1 and 2, 2018, at the CDG hub, showing the energy (min²) distribution across modes, alongside the corresponding Laplacian eigenvalues arranged in ascending order; here, a mode refers to an eigenvector of the graph Laplacian representing a distinct frequency component of the delay signal on the network.</figcaption></figure>

changes. The energy associated with each mode quantifies how much that particular frequency component contributes to the overall delay signal. Modes with higher energy thus highlight dominant patterns influencing the propagation of delays within the network.

The choice of spectral methods is motivated by their unique ability to capture both global and localized patterns of delay propagation in complex, multilayer air-rail networks. Unlike conventional clustering approaches, which primarily group nodes based on similarity in delay profiles, spectral analysis enables the decomposition of delay signals into frequency components, allowing for the quantification of smooth (expected) and abrupt (unexpected) delay variations through total variation metrics. This approach provides a principled framework to distinguish systemic disruptions from isolated anomalies, as demonstrated in recent studies on air traffic and transportation networks [30]. Furthermore, spectral methods offer interpretable, mathematically grounded metrics that reveal the underlying structure and resilience of the network, supporting operational insights that go beyond cluster membership alone. While clustering can identify groups with similar delay characteristics, the spectral approach delivers a richer, multi-scale understanding of delay dynamics, justifying its selection for this analysis.

Analyzing the GFT of June 1 and 2 delay signals (shown in Fig. 7) clarifies the unexpected TV behavior. The first mode, associated with eigenvalue zero, is omitted from the plot due to its dominant energy contribution and null effect on TV. Overall, the spectral energy is higher for June 1, especially in three mode ranges: below 100, between 350–400, and 650–690, with energy peaks exceeding $10^5$ min². Energy in modes below 100 contributes less to TV due to their lower eigenvalues.

We present in Table II the top five modes contributing most

<figure><img src="image_14.png" /></figure>
<table><caption>TABLE II. RANKING OF MODES—EIGENVECTORS OF THE GRAPH LAPLACIAN REPRESENTING DISTINCT FREQUENCY COMPONENTS OF THE DELAY SIGNAL—CONTRIBUTING MOST TO TE ON JUNE 1 AND 2, 2018.</caption><thead><tr><th rowspan="2">Rank</th><th colspan="2">2018-06-01</th><th colspan="2">2018-06-02</th></tr><tr><th>Mode</th><th>Percentage</th><th>Mode</th><th>Percentage</th></tr></thead><tbody><tr><td>1</td><td>0</td><td>16.28</td><td>0</td><td>23.47</td></tr><tr><td>2</td><td>357</td><td>2.42</td><td>682</td><td>2.45</td></tr><tr><td>3</td><td>365</td><td>1.98</td><td>95</td><td>2.23</td></tr><tr><td>4</td><td>371</td><td>1.78</td><td>163</td><td>1.63</td></tr><tr><td>5</td><td>85</td><td>1.70</td><td>17</td><td>1.49</td></tr></tbody></table>

significantly to the TE for the days 2018-06-01 and 2018-06-02. As indicated in the table, the mode 0 predominates the TE on both days. Notably, the energy contribution from the first mode on 2018-06-01—the day exhibiting unexpected high TV—is lower than that on 2018-06-02. Apart from this, the distribution of contributions from the remaining modes appears consistent across both days.

Fig. 8 compares the coordinate distributions associated to the top four modes by TE contribution (the first one being excluded). The coordinates vary more for June 1 than for June 2, though the magnitudes follow similar patterns in both cases. High coordinate variation implies greater leg delay activation, resulting in high TV as expected from Eq. (9).

Let us now focus on the coordinates of second mode for each day, as this mode contributes most significantly to TV due to its nonzero associated eigenvalue. In Table III, we present the top five coordinates ranked by absolute value. The *Index* and *Coordinate* columns denote the position and value of each coordinate in the eigenvector, while the *Leg* column indicates the transportation leg associated with that index.

Both positive and negative coordinate values are observed in the table, which reflect distinct delay trends: legs with positive coordinates, such as ('LFPG', 'SBFZ'), ('LFPG', 'ZBAA'), and ('DTMB', 'LFPG'), exhibit similar delay behavior. These flights, respectively to Brazil, China, and from Tunisia, represent international connections. Conversely, legs with negative coordinates, such as the train leg ('CDG', 'Marne la Vallée') and the flight leg ('LFRB', 'LFPG'), show the opposite delay trend. Notably, ('LFRB', 'LFPG') is a French domestic flight, making it reasonable that this leg and the associated train leg exhibit comparable trends.

Examining the top five coordinates of the second mode on June 2, 2018, reveals an uneven partition between positive and negative values: only one value is positive, while the other four are negative. The absolute magnitude of the coordinate associated with ('UBBB', 'LFPG') stands out as at least twice as large as the others, which display relatively similar magnitudes.

Comparing the rankings of eigenvector coordinates for both days reveals distinct differences in their distributions: on June 1, the magnitudes are more uniformly distributed, and several coordinates contribute substantially to the TV. In contrast, on June 2, only the top coordinate (651st) makes a significant contribution. This difference partially explains the observed variation in TV between the two days.

This analysis highlights the power of spectral methods in

<figure><img src="image_15.png" /><figcaption>(a) Top eigenvectors for June 1, 2018.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(b) Top eigenvectors for June 2, 2018.</figcaption></figure>

Figure 8. Distribution of coordinates of the top four eigenvectors ranking highest (Table II) in TE contribution for June 1 and 2, 2018 (the first one being excluded): the ranking goes from the top (second) to the bottom (fifth)

<table><caption>TABLE III. TOP FIVE ABSOLUTE VALUE OF THE COORDINATES ASSOCIATED TO THE SECOND MODE (TABLE II) CONTRIBUTING THE MOST TE.</caption><thead><tr><th colspan="4">2018-06-01</th></tr><tr><th>Rank</th><th>Index</th><th>Coordinate</th><th>Leg</th></tr></thead><tbody><tr><td>1</td><td>505</td><td>0.40962</td><td>('LFPG', 'SBFZ')</td></tr><tr><td>2</td><td>686</td><td>-0.379122</td><td>('CDG', 'Marne la Vallée')</td></tr><tr><td>3</td><td>539</td><td>0.3693</td><td>('LFPG', 'ZBAA')</td></tr><tr><td>4</td><td>548</td><td>-0.254753</td><td>('LFRB', 'LFPG')</td></tr><tr><td>5</td><td>24</td><td>0.243857</td><td>('DTMB', 'LFPG')</td></tr></tbody></table>

<table><thead><tr><th colspan="4">2018-06-02</th></tr><tr><th>Rank</th><th>Index</th><th>Coordinate</th><th>Leg</th></tr></thead><tbody><tr><td>1</td><td>651</td><td>0.673802</td><td>('UBBB', 'LFPG')</td></tr><tr><td>2</td><td>333</td><td>-0.297954</td><td>('LFPG', 'HAAB')</td></tr><tr><td>3</td><td>311</td><td>-0.261981</td><td>('LFPG', 'FNLU')</td></tr><tr><td>4</td><td>609</td><td>-0.212666</td><td>('LYPG', 'LFPG')</td></tr><tr><td>5</td><td>356</td><td>-0.189828</td><td>('LFPG', 'KMEM')</td></tr></tbody></table>

<figure><img src="image_17.png" /></figure>
distinguishing between typical and exceptional delay patterns in complex multimodal networks. The unexpected high total variation observed on June 1, despite similar actual delay magnitudes to June 2, underscores the need to consider spatial distribution alongside aggregate metrics. Furthermore, the limited but notable correlations between flight and train legs suggest opportunities to enhance intermodal coordination and resilience.

## V. CONCLUSION

This paper presents an application of Graph Signal Processing techniques to analyze delay patterns within a Hub-and-Spoke air-rail intermodal network. To characterize expected cascading delays, we introduce a graph delay network where edges represent Pearson correlations of delays between transportation legs. This approach effectively identifies clusters of legs whose delays influence each other and serves as a valuable indicator of anticipated delay propagation in disruption scenarios. To detect unexpected delays, we employ two spectral measures: total variation and total energy. Signals exhibiting low TV are considered smooth and represent typical delay patterns, whereas signals with high TV reveal abnormal, non-smooth distributions indicative of atypical disruptions. Distinguishing between expected and unexpected delay patterns is crucial for enhancing operational efficiency, reliability, and system resilience. Understanding these patterns empowers transport operators to proactively manage disruptions through targeted interventions, optimized scheduling, and resource allocation, ultimately improving passenger travel reliability while reducing operational costs.

Although our methodology was validated using real flight data combined with simulated rail delay data, it primarily demonstrates the potential of the approach to study air-rail network dynamics. The insights derived should be interpreted with caution, as meaningful conclusions about the real network require extensive analysis of fully observed, real-world data. Furthermore, analyzing the eigenvector coordinates enables the identification of legs exhibiting unexpected delay variability, highlighting the need for further investigation to uncover the underlying causes of such anomalies. Finally, future work could explore the sensitivity of our spectral analysis results to alternative delay distribution models, such as Weibull or lognormal, which can better capture heavy tails or multimodal characteristics observed in some datasets. This would help validate the robustness of the methodology and provide a deeper understanding of how different delay dynamics influence the spectral metrics used for identifying irregular delay patterns.

## VI. ACKNOWLEDGMENT

This research was co-funded by SESAR Joint Undertaking and European Commission, within the project "Implemented Synergies, data sharing contracts and goals between transport modes and air transportation" (SIGN-AIR) under grant number 101114845.

## REFERENCES

[1] E. Commission, D.-G. for Research, Innovation, D.-G. for Mobility, and Transport, *Flightpath 2050 – Europe’s vision for aviation – Maintaining global leadership and serving society’s needs*. Publications Office, 2011.

[2] S. J. Undertaking and U. P. de Catalunya, “Sign-air: Implemented synergies, data sharing contracts and goals between transport modes and air transportation,” https://www.sign-air.eu/, 2024, project funded by the European Union under grant agreement No. 101114845 (HORIZON-SESAR-2022-DES-IR-01).

[3] M. Hrušovský, E. Demir, W. Jammernegg, and T. van Woensel, “Real-time disruption management approach for intermodal freight transportation,” *Journal of Cleaner Production*, 2021.

[4] Z. He, K. Navneet, W. V. Dam, and P. Mieghem, “Robustness assessment of multimodal freight transport networks,” *Reliab. Eng. Syst. Saf.*, vol. 207, p. 107315, 2021.

[5] N. Ippolito and O. Cats, “Multi-modal and multi-layer robustness analysis of the european rail and air networks,” *Scientific Reports*, vol. 14, 2024.

[6] M. S. van der Tuin and A. Pel, “The disruption transport model: computing user delays resulting from infrastructure failures for multimodal passenger & freight traffic,” *European Transport Research Review*, vol. 12, 2020.

[7] M. R. Siegrist, B. Büchel, and F. Corman, “Evaluating the effects of disruptions on the behavior of travelers in a multimodal network utilizing agent-based simulation,” *Transportation Research Record*, vol. 2676, pp. 436–445, 2022.

[8] Y. Xu, S. Wandelt, and X. Sun, “Immuner: Integrated multimodal mobility under network disruptions,” *IEEE Transactions on Intelligent Transportation Systems*, vol. 24, pp. 1480–1494, 2023.

[9] F. Sun, S. Wang, Z. Hu, Y. Zhang, and L. Zhong, “Air-rail intermodal collaborative decision making for aircraft recovery from airport station disruption,” *Research in Transportation Business & Management*, 2024.

[10] A. Marzuoli, E. Boidot, P. Colomar, M. Guerpillon, E. Feron, A. Bayen, and M. Hansen, “Improving disruption management with multimodal collaborative decision-making: A case study of the asiana crash and lessons learned,” *IEEE Transactions on Intelligent Transportation Systems*, vol. 17, pp. 2699–2717, 2016.

[11] S. Freitas, D. Yang, S. Kumar, H. Tong, and D. H. Chau, “Graph vulnerability and robustness: A survey,” *IEEE Transactions on Knowledge and Data Engineering*, vol. 35, no. 6, pp. 5915–5934, 2023.

[12] G. Currie and C. Muir, “Understanding passenger perceptions and behaviors during unplanned rail disruptions,” *Transportation Research Procedia*, vol. 25, pp. 4392–4402, 2017, world Conference on Transport Research - WCTR 2016 Shanghai. 10-15 July 2016. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S2352146517306294

[13] M. Kramarz, E. Przybylska, and M. Wolny, “Reliability of the intermodal transport network under disrupted conditions in the rail freight transport,” *Research in Transportation Business & Management*, vol. 44, p. 100686, 2022, challenges and solutions for current freight transport and logistics. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S2210539521000699

[14] A. Marzuoli, E. Boidot, P. Colomar, M. Guerpillon, E. Feron, A. Bayen, and M. Hansen, “Improving disruption management with multimodal collaborative decision-making: A case study of the asiana crash and lessons learned,” *IEEE Transactions on Intelligent Transportation Systems*, vol. 17, no. 10, pp. 2699–2717, 2016.

[15] Y. Lei and H. Mu, “Analysis and optimization of a stochastic petri net for air-rail intermodal transportation,” *PLOS ONE*, vol. 19, 2024.

[16] M. Lu, E. J. Perez, K. Mason, and M. Z. Li, “A hierarchical spatial and temporal optimisation of the air-high speed rail intermodal network,” *Journal of Transport Geography*, 2025.

[17] W. Yiqiao, L. Qiaoyi, X. Cao, X. Zhou, V. Latora, L. Tong, and W. Du, “Travel time analysis in the chinese coupled aviation and high-speed rail network,” *Chaos Solitons & Fractals*, vol. 139, p. 109973, 2020.

[18] D. Chen, D. Song, and Z. Yang, “Optimization of an interregional flight network based on the multimodality of flight and high-speed rail: A case study in china,” *SSRN Electronic Journal*, 2022.

[19] L. Zhu, J. Sun, Z. He, and Q. Shao, “Research on irregular flight recovery based on air-rail spatiotemporal network model,” *IEEE Access*, vol. 12, pp. 147350–147363, 2024.
[20] Z. Chen, Y. Wang, and L. Zhou, "Predicting weather-induced delays of high-speed rail and aviation in china," *Transport Policy*, vol. 101, pp. 1–13, 2021.

[21] J. Pineda-Jaramillo and F. Viti, "Identifying the rail operating features associated to intermodal freight rail operation delays," *Transportation Research Part C: Emerging Technologies*, 2023.

[22] W. Burgholzer, G. Bauer, M. Posset, and W. Jammernegg, "Analysing the impact of disruptions in intermodal transport networks: A micro simulation-based model," *Decis. Support Syst.*, vol. 54, pp. 1580–1586, 2013.

[23] R. Ishfaq and C. Sox, "Design of intermodal logistics networks with hub delays," *Eur. J. Oper. Res.*, vol. 220, pp. 629–641, 2012.

[24] A. Ziliaskopoulos and W. Wardell, "An intermodal optimum path algorithm for multimodal networks with dynamic arc travel times and switching delays," *Eur. J. Oper. Res.*, vol. 125, pp. 486–502, 2000.

[25] J. A. Deri and J. M. F. Moura, "Spectral projector-based graph fourier transforms," *IEEE Journal of Selected Topics in Signal Processing*, vol. 11, pp. 785–795, 2017.

[26] R. Shafipour, A. Khodabakhsh, G. Mateos, and E. Nikolova, "A directed graph fourier transform with spread frequency components," *IEEE Transactions on Signal Processing*, vol. 67, pp. 946–960, 2018.

[27] H. H. Behjat, C. Westin, R. Ossenkoppele, and D. Ville, "Signal-adapted decomposition of graph signals," 2025.

[28] A. Raza, M. M. Munir, and M. Hussain, "Optimizing network insights: Ai-driven approaches to circulant graph based on laplacian spectra," *Physica Scripta*, vol. 99, 2024.

[29] S. Zhou and Y. Zhang, "A novel and simple spectral method for nonlocal pdes with the fractional laplacian," *ArXiv*, vol. abs/2311.07814, 2023.

[30] M. Z. Li, K. Gopalakrishnan, H. Balakrishnan, and K. Pantoja, "A spectral approach towards analyzing air traffic network disruptions," 2019.

[31] Y. Yang, P. Huang, Q. Peng, J. Li, and C. Wen, "Statistical delay distribution analysis on high-speed railway trains," *Journal of Modern Transportation*, pp. 1–10, 2019.

[32] B. Monechi, P. Gravino, R. Di Clemente, and V. D. Servedio, "Complex delay dynamics on railway networks from universal laws to realistic modelling," *EPJ Data Science*, vol. 7, no. 1, p. 35, 2018.

[33] S. Mobilités, "Sncf mobilités: Rapport financier 2018," Groupe SNCF, Rapport financier, 2018, consulté en septembre 2025. [Online]. Available: https://www.groupe-sncf.com/medias-publics/2024-03/sncf_mobilites_rapport_financier_2018.pdf

<figure><img src="image_18.png" /></figure>