 Regular article
 Open Access
Complex delay dynamics on railway networks from universal laws to realistic modelling
 Bernardo Monechi^{1, 2}Email authorView ORCID ID profile,
 Pietro Gravino^{2, 3}View ORCID ID profile,
 Riccardo Di Clemente^{4, 5}View ORCID ID profile and
 Vito D. P. Servedio^{6}View ORCID ID profile
 Received: 22 June 2017
 Accepted: 30 August 2018
 Published: 17 September 2018
Abstract
Railways are a key infrastructure for any modern country. The reliability and resilience of this peculiar transportation system may be challenged by different shocks such as disruptions, strikes and adverse weather conditions. These events compromise the correct functioning of the system and trigger the spreading of delays into the railway network on a daily basis. Despite their importance, a general theoretical understanding of the underlying causes of these disruptions is still lacking. In this work, we analyse the Italian and German railway networks by leveraging on the train schedules and actual delay data retrieved during the year 2015. We use these data to infer simple statistical laws ruling the emergence of localized delays in different areas of the network and we model the spreading of these delays throughout the network by exploiting a framework inspired by epidemic spreading models. Our model offers a fast and easy tool for the preliminary assessment of the effectiveness of traffic handling policies, and of the railway network criticalities.
Keywords
 Complex systems
 Networks
 Delay dynamics
 Modelling
 Spreading
1 Introduction
Transportation networks are a critical infrastructure with an enormous impact on local, national, and international economies [1, 2]. At the microlevel, the commuting time impacts the economy and the shape of cities [3], directly influencing our life style and choices [4]. At the macrolevel the travel time regulates trade and stimulates economic activities [5]. Railroads, when correctly developed, foster interregional trades within the country, and increase income levels within the city boundaries [6]. Whether we are traveling in crowded coaches of the commuters rail during rush hours or in the longmedium high speed train trips, the reliability of the transportation system is a key factor in determining travel behavior [7]. The travel time unreliability in rail service can have substantial consequences for its users [8] and the growth of cities [9]. Delays are one of the causes that corrupt the travel time reliability of the transportation systems. To address delay propagation, traditional approaches apply either stochastic models [10] or use propagation algorithms on a timed event graph representation of a scheduled railway system [11]. Complex equations describing delay were obtained and solved by means of iterative refinement algorithms to predict positive delays in urban trains [12], while traffic control models were proposed to manage the safety of timetables after perturbations occur [13].
The development of models addressing the dynamics of trains over railway networks are usually focused either on delimited aspects, e.g., single lane dynamics [14] or the issuing of schedule [15, 16], or are comprehensively including a wide variety of different microscopic ingredients that can be hardly validated by real data [17, 18]. In the last years a new perspective bloomed to provide a different understanding of delay dynamics through the lens of Complex Systems. Railway networks have largely been studied and exhibit smallworld scaling properties [19], their topology have geospatial restriction [20] and structural redundancy [21]. On the other end, a railway delay dynamics framework within a complex systems approach is still lacking. Our aim is to develop a model capable of reproducing the actual delay structure and the emergence of congested areas, by exploiting some essential ingredients without relying on the detailed knowledge of the microscopic mechanisms underlying delay generation. A similar approach has already been successfully applied in the context of the Air Transport System of USA and Europe [22, 23], where interaction models were used to characterize and forecast the spreading of delays among flights. These modelling schemes are driven by the necessity of simple and novel frameworks to study transportation systems, allowing fast scenario simulations for schedule testing, which could be easily applied to resilience studies that are nowadays performed independently of the dynamics taking place over the network [21].
Following a similar approach, we develop a new data driven model of delay propagation among trains diffusing over a railway network. We start by inferring universal laws governing the emergence of delays over the railway network as a consequence of the occurrence of adverse conditions that are not depending on the (possible) interactions between trains. Hence, inspired by models for epidemic spreading [24], we introduce the Delay Propagation Model as a novel framework to asses and test how such emerging delays spawn and spread over the network. We name the first kind of delays as “exogenous delays”, while the second ones stemming from the interaction between trains as “endogenous delays”. While the occurrence of exogenous delays can be detected and analysed from our data, the endogenous delays, on the contrary, since the (supposed) direct interaction between trains cannot be inferred from the dataset we collected, will be modeled with a one parameter interaction. The effects and consistency of such endogenous delay modelling scheme will be checked a posteriori by means of numerical simulations. Our model is close to an SIS model of epidemic spreading [24], since trains can either get infected by delay, recover from it and then get infected again. We show that this model is capable of reproducing the empirical distribution of delays measured in the data as well as the emergence of large congested areas. Moreover, we show that the removal of the delay propagation mechanism prevents the modeled system from generating large disruptions, hence strongly suggesting the existence of this kind of interaction in the real system. Finally, we propose an application of our model in studying a scenario where the propagation of a localized delay leads to the emergence of a vast nonfunctioning area in the Italian Railway Network.
The paper is structured as follows: in the first section we describe the dataset and some empirical findings that inspired the assumptions used in the development of the model; in the second section we discuss the model in detail and will show its capability of reproducing empirical findings such as the distribution of delays and the size of congested areas; we will also demonstrate the usefulness of the model as a scenario simulation tool; in the last section we will summarize our findings and discuss possible future developments.
2 Methods
2.1 Data
The clustering and the betweenness outline the typical small word topology of transportation networks [19]. German stations show a distribution between 10 and 100 trains per day, narrower that the train distribution of the Italian stations, which instead display a broader distribution, suggesting a more heterogeneous handling of train traffic load (Fig. 1F). Finally, Fig. 1G shows the histogram of the average delays of trains aggregated by stations for the two nations. Both distributions show a peak and heavy tail, whereas compared to the German distribution, the Italian distribution is clearly shifted towards higher delays. The topological similarities between the two networks suggest that the differences in delays and traffic load might be the result of differences in the trains dynamics.
2.2 Train interaction on railways networks
2.3 Defining possible interactions
 (i)
links entering A, i.e. trains moving towards the last station crossed by i;
 (ii)
links entering B, i.e. trains travelling towards the same station i is currently travelling to;
 (iii)
links exiting from A, i.e. trains departed from the last station crossed by i;
 (iv)
links exiting from B, i.e. trains leaving the station i is currently travelling to.
2.4 Exogenous generation of delay
2.5 Generation of delay at departure
Departure delays, i.e. the delay a train acquires right before leaving the first station on its route, cannot be considered in principle completely exogenous. In other words, due to the fact that different trains in our datasets can actually be the same physical train (e.g., the same convoy travelling back and forth along the same path on the railway network – this is denoted as “rotation” –), the delay at departure might suffer from the influence of the traffic. However, railway administrators envisage suitable time buffers at the endpoints of the paths of each train so that it is reasonable to assume, at least as crude approximation, that departure delay is exogenous in character. It has already been shown that this kind of delay can be described by a qexponential distribution in [32]. However, the dependence on the parameters of the obtained distributions with respect to the topological properties of the network has not been investigated yet. Following the same spirit of the previous paragraph for the exogenous delay on links, we divide the nodes in the network (the train stations), with respect to their outdegree. The outdegree \(k_{\mathrm{out}}\) represents roughly the number of different railway lines starting from a certain stations and hence can be considered as a proxy for the complexity of the station itself. Once the nodes of the networks have been divided according to \(k_{\mathrm{out}}\), we fitted these distributions using a qexponential following the procedure defined in [32] (see Fig. S12, Fig. S13 and Fig. S14 of Additional file 1).
In Germany the departure delay can be considered constant and independent on the size of the station. In Italy the \(P(\delta t>0  k_{\mathrm{out}})\) grows linearly with \(k_{\mathrm{out}}\) meaning that stations with high degree are generating larger disruptions in the network.
3 Modelling a realistic rail transport system
The analyses proposed so far suggest the existence of two sources of delays: exogenous delays spontaneously occurring due to external adverse conditions at departure and during travel, depending on the topological properties of the Railway Network; endogenous delays resulting from the interaction between trains. While in the first case we were able to characterize the statistical laws governing the emergence of delays, we cannot directly investigate the mechanisms of interaction between trains. Following the ideas exploited to model real world epidemics [34, 35] which have already been proven effective on air traffic delay modeling [22], we will define a propagation process for delays i.e. we will suppose that trains can spread their delays from one another according to some fixed probability and in certain conditions. Such probability will be derived by comparing the results from the simulations with the model and the empirical data.
3.1 The delay propagation model
 Departure delay :

This delay is assigned at the beginning of the path (originating station) of each train and is considered exogenous and unrelated with the current traffic conditions at the departing stations or in the nearby links. We assign to a train either a positive or negative delay according to the empirically found law of the corresponding probabilities \(p_{\mathrm{dep}}^{+}=P(\delta t_{\mathrm{dep}} > 0\,  k_{\mathrm{out}})\) and \(p_{\mathrm{dep}}^{}=P(\delta t_{\mathrm{dep}} < 0\,  k_{\mathrm{out}})\) respectively (and no delay with complementary probability \(1  p_{\mathrm{dep}}^{+}p_{\mathrm{dep}}^{}\)) (see Fig. S15 of Additional file 1). Once the sign of the delay has been decided, we assign a positive or negative delay value so that \(\delta t_{\mathrm{dep}} \sim e_{q(k_{\mathrm{out}}), b(k_{\mathrm{out}})}(\delta t)\), i.e. distributed according to a qexponential distribution with the parameters q and b depending on \(k_{\mathrm{out}}\) and on the sign of the delay itself as in Fig. 6.
 Exogenous Link Delay :

This delay is assigned whenever a train starts travelling on a link for the first time (Fig. 7 reports an exemplification of the model). Considering a train i passing from the link AB to the link BC, we adopt the same modelling scheme as in the departure delay case assigning to the train a positive delay with probability \(p_{\mathrm{exo}}^{+}=P(\delta t_{\mathrm{exo}} > 0\,  d(BC))\), a negative one with probability \(p_{\mathrm{exo}}^{}=P(\delta t_{\mathrm{exo}} < 0\,  d(BC))\) and no delay with \(1  p_{\mathrm{exo}}^{+}p_{\mathrm{exo}}^{}\), where \(d(BC)\) is the geodesic length of the links BC (see Fig. S10 of Additional file 1 for the corresponding law of probability). Having decided the sign of the delay, we fix the parameter of a qexponential distribution q and b according to \(d(BC)\) (Fig. 5) and we extract the magnitude of the delay so that \(\delta t_{\mathrm{exo}} \sim e_{q(d(BC)), b(d(BC))}(\delta t)\).
 Delay Propagation :

We model the interaction between trains with a mechanism of propagation of delay from other trains to train i. The investigations performed on the datasets suggest that such interaction can occur only when nearby trains are in the Backward propagation configuration of Fig. 3.
Following the picture of the model in Fig. 7, when the train i starts travelling on the link BC leading to the C station is susceptible of propagation from the train j, which is travelling from C towards D. Inspired by the SIS models of epidemic spreading [24], in which agents can get infected and recover from the illness with a fixed probability, we introduce the delay propagation parameter \(\beta \in [0,1]\) describing the probability that the delay of the train j propagates to i. We assume that the propagation occurs at the time i starts travelling onto BC. When train i travels on BC from time \(t_{1}\) to time \(t_{2}\), we check whether delayed trains are travelling on links in the Backward propagation configuration with respect to BC in \([t_{1},t_{2}]\). Thus, we randomly pick one of those trains, say j, and with probability β its delay \(\delta t_{j}\) is added to the delay of i.
4 Results
The model is based on the exogenous sources of delay that represent the spontaneous emergence of disruptions due to the aggregation of a finite number of external causes (trains malfunctions, accidents, bad weather, etc.). These sources are modelled according to a universal probabilistic law, whose parameters are inferred from the data at our disposal. The propagation parameter β modulates the mutual interaction between trains. For simplicity, we chose this parameter to be uniform all over the network, which is a strong assumption since the propagation might depend on the considered part of the railway system.
4.1 Reproducing the emergence of delays and congestion
We identify, at each time step, the “clusters of congested stations” as the connected components of the railway network left once we removed all the functioning stations from the network. (i.e. the stations whose congestion might be correlated with one another because of the propagation of delay).
In order to check whether our model is able to reproduce the emergence of these areas, we focused on two measures: (i) the size of the clusters in terms of number of stations and (ii) the relation between the size and the diameter of the clusters. This latter measure is capable of giving insights about the topology of the congested clusters. Bottom panels of Fig. 8A show real and simulated cumulative distributions of cluster sizes for both countries. We find a strong accordance between model and reality when β is set to the optimal value, pointing out how the model is also able to reproduce this aspect of delays dynamics. Similarly to the arrival delay distribution, here β affects the tail of the distribution so that \(\beta=0\) (i.e. no propagation of endogenous delays) implies the absence of large congested areas. We study the diameter of the cluster defined as the distance between the farthest couple of nodes in the cluster. We compute the diameter both in the case where just the topological distance between nodes is considered and the case where each link is weighted with the geodesic distance between the stations it connects. To panels of Fig. 8B shows the dependence of the cluster diameter computed with both distances on the cluster size. In the cases where the diameter is computed by using the geodesic distance, the dotted line represents the diameter of a cluster of size n, assuming that all the links of the clusters are as long as the average link length of the network. Such pattern corresponds to the case where all the clusters are randomly sampled from the whole network without any constraint. From the lower panels of Fig. 8B we can see that while clusters do have a pathlike structure, they cannot be considered randomly distributed over the networks. Instead, they seem to be deployed in areas where the links are larger than the average links length of the corresponding network, resulting in the same dependence of the dotted lines but shifted upwards. The fact that the geographical diameter of large clusters grows up to hundreds of kilometers indicates that disturbances can propagate between far away parts of the network.
The topological measure in the top panels of Fig. 8B exhibits strong deviations from the pathlike behaviour especially when large clusters are involved. The mismatch between the two ways of characterizing clusters could be due to the fact that the geographical distance may hide, by stretching them, non pathlike structures that are actually present in the delay propagation patterns. The appearance of these clusters, and their features, are typical outcomes of a nontrivial complex dynamics arising from train interactions. For this reason it is a remarkable result that, as can be neatly be seen in Fig. 8, the model proposed, despite its simplicity, captures this behaviour correctly. The relations between cluster size and cluster diameter is show very good agreements with the empirical measures for both nations and, perhaps more importantly, for both clusters diameter definition: the topological and the spatial one. This clear adherence with reality of the results from a model with only one parameter trained on a so small training set (only one week) is a strong proof of the fact that the model hypothesis are very likely to be correct. Our model is capable of reproducing some global patterns of the delay dynamics in railway systems. However, some discrepancies at a more microscopic level can be observed. In particular our model fails to correctly describe the behaviour of stations with low traffic. For these stations, the hypothesis of a constant in time and homogeneous coupling parameter β may not be well justified. We refer to Sect. 5 of Additional file 1 for a thorough discussion of this point.
4.2 Scenario simulation: prediction of the effects of a strong localized disturbance
In order to check this hypothesis, we ran a simulation in which a large delay of 100 minutes is assigned with probability 1 to each of the trains crossing the “NAPOLI CENTRALE–AVERSA” link in that period of time. Due to the nondeterministic nature of the model, we performed the simulation of 200 different realizations in order to have a set of scenarios to be compared with the real data. This comparison is shown in Fig. 9B. We can see that with this simple modification and not considering other possible correlations between the occurrences of external adverse conditions in nearby links, we are able to reproduce a pattern which is qualitatively similar to the one observed in the data, clearly pointing out that the “NAPOLI CENTRALE–AVERSA” link played a major role in the congestion of the network. Moreover, our scenarios indicates the probability of having a minor congestion in the line also in the early morning, probably due to usual minor disruption occurring on other links. The proposed scenario simulation could be easily extended to less localized adverse occurrences, which might comprehend spatially correlated disruptions due to natural events or strikes. Moreover, the same approach could be used to study more dramatic effect of node, link removal or the positive effects due to the introduction of more resilient and optimized schedules. In this cases, it is sufficient to use the same structure of the model and modify the input schedule and/or the Railway Network itself.
5 Conclusions
Railways and the railway transport systems have been historically of utmost importance for the development of modern countries. Nowadays their importance is on the rise again due to their relevance in the reduction of CO_{2} emissions and their competitiveness in short and middle range movements, so that huge investment have been made in the European Union to improve their efficiency. However, the emergence of large disruptions and the intrinsic inefficiencies seem to be endemic in this kind of transportation. The understanding of the universal features governing the dynamics of the system from a theoretical point of view could give an important contribution to the solution of such problems. The development of a universal model explaining the emergence of delays in Railway Network would allow for a quantification of the causes behind the occurrence of large disruptions and for a more direct link between the effects that localized interventions might have on the global system. In this work, we developed a novel model of delay emergence and propagation between trains moving on Railway Networks, partly based on empirical laws inferred from the data governing the accidental emergence of delays from the influence of adverse occurrences. Remarkably, the statistical models used for the description of this “exogenous delays” showed how these adverse occurrences are the result of the same finite number of causes (e.g., bad weather conditions and malfunctions) independently from the topology, as opposed to the scale of these disruptions, which are largely influenced by the part of the network the trains are travelling on. We find, in fact, that both the complexity of a station as measured by the number of different routes originating from it and the length of a route, are connected to the magnitude of the microscopic delays composing the overall delay by simple statistical laws with two parameters, whose value is the same all over the network and can be inferred and fixed by data. This kind of universality is somehow surprising and opens the possibility of easily simulating the behaviour of any railway system once the complete schedules and the network structure are known. These emergent delays can then spread from one train to another by means of a delay propagation probability β, which is in fact the only free parameter of our model. Despite the high level of abstraction used in our approach, our model is capable of reproducing the empirical patterns found in data when the delayspreading probability β is fine tuned to an optimal value. The model reproduces correctly the final delay distribution and the distribution of the sizes of congested areas. Moreover, the model grasps the topological properties of such congested areas, which appears to be spreading in some cases for many kilometres through the network. According to our model, the emergence of large disruptions is the result of the interplay between the occurrence of localized exogenous delays and the propagation of such delays between trains. In other words, turning off the delay propagation mechanism prevents the system from generating extremely large delays and congested areas, pointing out that interactions are the driving force behind the emergence of major spontaneous adverse occurrences. The modelling scheme seems quite promising so far, but it still suffers from some major assumptions that might limit its predictive power. In fact, the interactions between trains could occur in longer ranges and not just between neighboring links and the propagation probability could not be uniform all over the Railway Network but instead depending on specific operational conditions. Moreover, the modelization of exogenous delays could be refined by taking into account more static topological feature of the Railway Networks or by introducing dynamical variations due to different traffic conditions. Another interesting possibility would be to increase the number of Railway Systems under study (e.g., the French SNFC sharing similar characteristics with respect to the Italian and German systems whose data is publicly available [36]), in order to understand the reasons behind the quantitative differences observed, e.g. larger delays in the Italian case. In the final part of the paper, we show how the model can be applied to study the capability of functioning of the system after a localized large disruption occurring in a single node of the network. This approach can be easily extended to more complex case studies of distributed disruptions, the occurrence of strikes, the removal of trains or parts of the network, also including the possibility to introduce delay management strategies to increase the resilience of the system. The model is also useful in order to have a fast test of changes in the overall schedule of trains so to have a more precise assessment of their global effects. Finally, despite its simplicity the model is open an increase of complexity that might lead to a better adherence to the empirical findings such as longrange interactions between trains and a more detailed model of exogenous delays including correlations between nearby link and the dependence on traffic conditions.
Declarations
Acknowledgements
The authors acknowledge Fabio Lamanna for the initial discussion about the datasets to be used for the work.
Availability of data and materials
The datasets supporting the conclusions of this article are included within the article (and its additional files).
Funding
This work has been supported by the KREYON Project, funded by the John Templeton Foundation under contract n. 51663; VDPS acknowledges financial support from the Austrian Research Promotion Agency FFG under grant #857136. RDC as Newton International Fellow of the Royal Society acknowledges support from The Royal Society, The British Academy and the Academy of Medical Sciences (Newton International Fellowship, NF170505).
Authors’ contributions
BM and RDC conceived and designed the analysis; BM analised the datasets; BM and VDPS designed the model; BM performed the simulations; BM, PG, RDC and VDPS wrote the manuscript. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare no competing interests.
Consent for publication
Not applicable.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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