Large-scale Real-world Networks Research Articles

Modeling Multimodal Curbside Usage in Dynamic Networks

The proliferation of emerging mobility technology has led to a significant increase in demand for ride-hailing services, on-demand deliveries, and micromobility services, transforming curb spaces into valuable public infrastructure for which multimodal transportation competes. However, the increasing utilization of curbs by different traffic modes has substantial societal impacts, further altering travelers’ choices and polluting the urban environment. Integrating the spatiotemporal characteristics of various behaviors related to curb utilization into general dynamic networks and exploring mobility patterns with multisource data remain a challenge. To address this issue, this study proposes a comprehensive framework of modeling curbside usage by multimodal transportation in a general dynamic network. The framework encapsulates route choices, curb space competition, and interactive effects among different curb users, and it embeds the dynamics of curb usage into a mesoscopic dynamic network model. Furthermore, a curb-aware dynamic origin-destination demand estimation framework is proposed to reveal the network-level spatiotemporal mobility patterns associated with curb usage through a physics-informed data-driven approach. The framework integrates emerging real-world curb use data in conjunction with other mobility data represented on computational graphs, which can be solved efficiently using the forward-backward algorithm on large-scale networks. The framework is examined on a small network as well as a large-scale real-world network. The estimation results on both networks are satisfactory and compelling, demonstrating the capability of the framework to estimate the spatiotemporal curb usage by multimodal transportation. History: This paper has been accepted for the Transportation Science Special Issue on ISTTT25. Funding: This material is based upon work supported by the Office of Energy Efficiency and Renewable Energy, U.S. Department of Energy [Award DE-EE0009659]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2024.0522 .

A Simulation-Based Method for Optimizing Remote Park-and-Ride Schemes

Urbanization places greater demand on the link between downtown areas and suburbs, due to commuters’ long-distance and diverse trips. As an emerging form of park-and-ride (PNR) services, remote PNR (RPR) facilities have proved to be more economical and environmentally friendly, allowing travelers to park in a suburban area and travel to a rail station via bus. In this regard, a generalized simulation-based bilevel model for optimizing the locations and capacities of RPR facilities is developed in this article. A hybrid algorithm integrating Bayesian optimization, branch and bound, and trust region sequential quadratic programming is proposed to achieve an optimal solution. The proposed integrated method balances the desired efficiency and accuracy through the combination of machine learning-based technology and mathematical optimization methodology. The validity of the proposed model is tested on a large-scale real-world transportation network in Halle, Germany. Modeling and analyzing RPR schemes using the proposed framework may provide new insights into improving social welfare.

MPC-based framework incorporating pre-disaster and post-disaster actions and transportation network constraints for weather-resilient power distribution networks

Enhancing power system resilience to extreme weather requires an effective coordination of pre-disaster preventive response actions and post-disaster repair and restoration services. In this context, this article presents a two-stage framework to enhance resilience of power distribution networks to severe weather events. It considers operating and resource constraints in interdependent power and transportation networks based on official guidance from transport appraisal studies. It also incorporates sources of information available in a smart city context, such as energy measurements, traffic flows, and geographical information systems. In the first stage, a pre-disaster operational planning strategy is defined to prepare the grid for a high-risk outage scenario with the objective of minimizing the value of lost loads, formulated as a mixed-integer quadratic programming model. In the second stage, a post-disaster corrective strategy is implemented over a receding time horizon to compensate for deviations between the actions computed in the first stage and the actions required to minimize power outages, embedded into a model predictive control scheme. The effectiveness of the proposed framework is demonstrated on a real-world large-scale distribution network in the United Kingdom over a range of outage scenarios. Simulation results show that the proposed framework is capable of effectively minimizing first-stage operational costs and second-stage deviations between the projected and actual load demand supply. Numerical results obtained with the proposed framework indicate that the load energy unserved is at least twice smaller than with typical practices adopted by distribution network operators and computational times take two minutes or less. Therefore, it can be effectively used by distribution network operators to ensure an appropriate level of preparedness to power outages caused by extreme weather along with prompt restoration and repair services.

Demonstration-guided deep reinforcement learning for coordinated ramp metering and perimeter control in large scale networks

Effective traffic control methods have great potential in alleviating network congestion. Particularly, in an urban network consisting of heterogeneous roads (e.g., freeways and urban roads), how to integrate and coordinate control policies on different roads is a critical issue in large-scale networks. This study addresses this question from two aspects: modeling and control. From the modeling aspect, we formulate the hybrid traffic modeling in heterogeneous networks with the Asymmetric Cell Transmission Model (ACTM) for freeways and the generalized bathtub model for urban roads. For the control aspect, this study considers two representative control approaches: ramp metering for freeways and perimeter control for urban roads, and we aim to develop a deep reinforcement learning (DRL)-based coordinated control framework for large-scale networks. However, there are two significant challenges in the coordinated control in large-scale networks with DRL methods: non-stationary environment and large search space. To address both issues, we incorporate the demonstration to guide the DRL method for better convergence by introducing the concept of “teacher” and “student” models. The teacher models are traditional controllers that provide control demonstrations. For instance, ALINEA and Gating are two representative feedback controllers for ramp metering and perimeter control which can be “teacher” models. The student models are DRL methods, which learn from teachers and aim to surpass the teachers’ performance. Additionally, we develop a parallel training scheme to accelerate the proposed DRL method. To validate the proposed framework, we conduct two case studies in a small-scale network and a real-world large-scale traffic network in Hong Kong. Numerical results show that the proposed DRL method outperforms demonstrators as well as DRL methods, and the coordinated control is more effective than just controlling ramps or perimeters respectively. The research outcome reveals the great potential of combining traditional controllers with DRL for coordinated control in large-scale networks.

Leveraging Minimum Nodes for Optimum Key Player Identification in Complex Networks: A Deep Reinforcement Learning Strategy with Structured Reward Shaping

The problem of finding key players in a graph, also known as network dismantling, or network disintegration, aims to find an optimal removal sequence of nodes (edges, substructures) through a certain algorithm, ultimately causing functional indicators such as the largest connected component (GCC) or network pair connectivity in the graph to rapidly decline. As a typical NP-hard problem on graphs, recent methods based on reinforcement learning and graph representation learning have effectively solved such problems. However, existing reinforcement-learning-based key-player-identification algorithms often need to remove too many nodes in order to achieve the optimal effect when removing the remaining network until no connected edges remain. The use of a minimum number of nodes while maintaining or surpassing the performance of existing methods is a worthwhile research problem. To this end, a novel algorithm called MiniKey was proposed to tackle such challenges, which employs a specific deep Q-network architecture for reinforcement learning, a novel reward-shaping mechanism based on network functional indicators, and the graph-embedding technique GraphSage to transform network nodes into latent representations. Additionally, a technique dubbed ‘virtual node technology’ is integrated to grasp the overarching feature representation of the whole network. This innovative algorithm can be effectively trained on small-scale simulated graphs while also being scalable to large-scale real-world networks. Importantly, experiments from both six simulated datasets and six real-world datasets demonstrates that MiniKey can achieve optimal performance, striking a perfect balance between the effectiveness of key node identification and the minimization of the number of nodes that is utilized, which holds potential for real-world applications such as curbing misinformation spread in social networks, optimizing traffic in transportation systems, and identifying key targets in biological networks for targeted interventions.

Bifractality of fractal scale-free networks.

The presence of large-scale real-world networks with various architectures has motivated active research towards a unified understanding of diverse topologies of networks. Such studies have revealed that many networks with scale-free and fractal properties exhibit the structural multifractality, some of which are actually bifractal. Bifractality is a particular case of the multifractal property, where only two local fractal dimensions d_{f}^{min} and d_{f}^{max}(>d_{f}^{min}) suffice to explain the structural inhomogeneity of a network. In this work we investigate analytically and numerically the multifractal property of a wide range of fractal scale-free networks (FSFNs) including deterministic hierarchical, stochastic hierarchical, nonhierarchical, and real-world FSFNs. Then we demonstrate how commonly FSFNs exhibit the bifractal property. The results show that all these networks possess the bifractal nature. We conjecture from our findings that any FSFN is bifractal. Furthermore, we find that in the thermodynamic limit the lower local fractal dimension d_{f}^{min} describes substructures around infinitely high-degree hub nodes and finite-degree nodes at finite distances from these hub nodes, whereas d_{f}^{max} characterizes local fractality around finite-degree nodes infinitely far from the infinite-degree hub nodes. Since the bifractal nature of FSFNs may strongly influence time-dependent phenomena on FSFNs, our results will be useful for understanding dynamics such as information diffusion and synchronization on FSFNs from a unified perspective.

A Spatiotemporal Traffic Speed Prediction Method Combining Attention and Multivariate Graph Convolution Fusion

Spatiotemporal prediction is widely used in the fields of neuroscience, climate, and transportation, and traffic speed prediction is one of the typical research areas. Since traffic networks are irregular grid structures with complex nonlinear spatiotemporal dependencies between nodes, traditional single-feature prediction methods are difficult to adapt to complex road conditions. In order to improve the prediction accuracy of traffic speed, this paper proposes a spatiotemporal traffic speed prediction method combining attention and multivariate graph convolution fusion (CAMGCF). It constructs a spatiotemporal model, specifically, the network uses multivariate graph convolution to fuse external factor features and traffic speed features and uses an attention mechanism to adaptively fuse feature information from different time series to achieve stability in long-term prediction. Finally, the framework is evaluated on three real-world large-scale road network traffic datasets, and the experimental results show that the prediction accuracy of this paper has more accurate results compared with the benchmark method.

Structured prediction of sparse dependent variables for traffic state estimation in large-scale networks

Currently, one of the biggest challenges in modern traffic engineering is related to traffic state estimation (TSE). Although many machine learning and domain models can be used for TSE, they do not consider the sparsity and spatial dependence of traffic state variables. In this paper, we propose a hybrid soft computing model of two Gaussian conditional random field (GCRF) models for the inference of traffic speed, which is a relevant variable for TSE and travel information systems. The proposed model can infer the traffic state variables in large-scale networks whose nodes are geographically dispersed. Moreover, by combining a Gaussian conditional random field binary classification model (GCRFBC), which classifies traffic regimes as free-flow or potentially congested, and a regression GCRF model for the prediction of traffic speed in potentially congested traffic regimes, the model addresses two specifics of the problem: sparsity in traffic data, and the fact that observations are not independent. The proposed model was tested on two large-scale real-world networks in Serbia, namely an arterial E70-E75 335 km long highway stretch and the major ski resort Kopaonik with 55 km of ski slopes. In addition, the proposed model showed better prediction performance than several other unstructured and structured models.

Dynamic pedestrian traffic assignment with link transmission model for bidirectional sidewalk networks

Planning assessment of urban walking infrastructure requires appropriate modeling methodologies that capture the time-dependent and unique microscopic characteristics of bidirectional pedestrian streams. In this paper, we develop a simulation-based dynamic pedestrian traffic assignment (DPTA) model specifically formulated for walking networks (e.g. sidewalks) with bidirectional links. The model consists of a dynamic user equilibrium (DUE) based walking route choice and a link transmission model (LTM) for network loading. The formulated DUE adopts a pedestrian volume delay function (pVDF) taking into account the properties of bidirectional pedestrian streams such as self-organization. The adopted LTM uses a three-dimensional triangular bidirectional fundamental diagram as well as a generalized first-order node model. The applicability and validity of the model is demonstrated in hypothetical small networks as well as a real-world large-scale network of sidewalks in Sydney. The model successfully replicates formation and propagation of shockwaves in walking corridors and networks due to bidirectional effects.

Large-Scale Maintenance and Rehabilitation Optimization for Multi-Lane Highway Asphalt Pavement: A Reinforcement Learning Approach

Pavement maintenance and rehabilitation (M&R) optimization is of great importance to the sustainable development of roadway infrastructure. Various models have been developed for supporting M&R decision-making. However, there is still a lack of research that can provide lane-specific M&R strategies for large-scale pavement networks which consider uncertainty in optimization to achieve management flexibility. This study proposes an innovative M&R optimization approach for multi-lane highway pavement based on a reinforcement learning (RL) method. Life cycle assessment (LCA) and life cycle cost analysis (LCCA) were integrated to assess the environmental and economic impact of M&R decisions, respectively. The uncertainty of pavement deterioration was considered by constructing an RL simulation environment that contains several probabilistic pavement performance models. The proposed method was applied to a large-scale real-world highway network as demonstration, and compared with the state-of-the-practice hierarchical threshold-based approach (HT). The results show that the RL model saved about 26.59% of the cost in comparison to the HT approach, which was equal to 18147.27 million CNY. It could keep the long-term pavement performance within an acceptable range in a cost-effective manner. The RL model tends to select less rehabilitations and more preventive maintenance than the HT model. It was also found that incorporating uncertainty into optimization allows the model to balance the expected return and the negative (risk) and positive (opportunity) uncertainty of the solution. The outcomes of this study are expected to improve the current pavement management practice and demonstrate the potential of RL in pavement M&R optimization.

Eco-Friendly Disaster Evacuation Framework for 6G Connected and Autonomous Vehicular Networks

Crowd and evacuation management have been active areas of research and study over recent years. Connected Autonomous Vehicles (CAVs) could provide safe and vital life-saving reinforcement when evacuating populations from a large-scale disaster site. However, navigation in disaster-affected environments is vulnerable to the presence of unpredictable moving obstacles. Existing solutions perform optimal route determination while considering the presence of a single rescue vehicle. To optimize path planning and enhance energy efficiency in the presence of a fleet of rescue vehicles and Unmanned Aerial Vehicles (UAVs), multiple sources and target destinations, we propose a Multi-Agent Deep Reinforcement Dijkstra (MADRD) algorithm in a sixth generation (6G) assisted environment to achieve cooperative navigation in the Route Planning Framework. The MADRD algorithm allows vehicles to learn from the experiences and behavior of other CAVs to obtain an optimal path from the source to the affected target site. Route selection and throttle parameters are optimized to achieve efficient fuel consumption, reduced travel time and lower traffic congestion. We simulate the Route Planning Framework on a large-scale real-world road network. The experimental results indicate that MADRD achieves 40% less fuel consumption, 45% higher road throughput and 34% less evacuation time than its state-of-the-art counterparts.

Simulation-based optimization of large-scale dedicated bus lanes allocation: Using efficient machine learning models as surrogates

Dedicated Bus Lanes (DBLs) have been implemented in many cities to boost buses’ reliability and to alleviate traffic congestions. However, how to correctly allocate DBLs across a large-scale real-world network is challenging. Simulation-Based Optimization (SBO) methods were utilized in this work to resolve this optimal allocation problem. Traditional discrete SBO methods are intractable when handling high-dimensional, costly, simulation-based Transportation Network Design Problems (TNDPs) using a limited computational budget. Thus, several efficient Machine Learning (ML)-based surrogate models and a Jackknife uncertainty estimator were introduced to existing SBO framework in this work. A number of comparative experiments between proposed methods and frequently-used Gaussian Process (GP)-SBO methods were conducted. A mesoscopic simulation and Dynamic Traffic Assignment (DTA) tool was adopted to evaluate the network performance.The results of numerical studies show that the optimization efficiency of proposed method is significantly higher than that of commonly used GP-based method when dealing with high dimensional problems. A real-world DBLs allocation case study in Guiyang, Guizhou Province, China again proves that efficient ML-based SBO method is capable to take much less CPU runtime to obtain a better solution than traditional method. The optimal DBLs allocation scheme found by one of the efficient ML-based methods raises the network performance by 5.05 %. A total of 1,376 h travel time is saved, and the average travel time per traveler drops by 0.75 min. In conclusion, efficient ML-based SBO method proposed in this study is more promising to handle large-scale, discrete, costly simulation-based DBLs allocation problems within a limited computational budget than common GP-based SBO methods.

COSINE: Compressive Network Embedding on Large-Scale Information Networks

There is recently a surge in approaches that learn low-dimensional embeddings of nodes in networks. However, for large-scale real-world networks, it’s inefficient for existing approaches to store amounts of parameters in memory and update them edge by edge. With the knowledge that nodes having similar neighborhoods will be close to each other in the embedding space, we propose COSINE (COmpresSIve Network Embedding) algorithm, which reduces the memory footprint and accelerates the training process by parameter sharing among similar nodes. COSINE applies graph partitioning algorithms to networks and builds parameter sharing dependency of nodes based on the results of partitioning. In this way, COSINE injects prior knowledge about high-order structural information into models, which makes network embedding more efficient and effective. COSINE can be applied to any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">embedding lookup</i> method and learn high-quality embeddings with limited memory and less training time. We conduct experiments on multi-label classification and link prediction, where baselines and our model have the same memory usage. Experimental results show that COSINE improves baselines by up to 23 percent on classification and 25 percent on link prediction. Moreover, the training time of all representation learning methods using COSINE decreases by 30 to 70 percent.

MFF-Net: A multi-feature fusion network for community detection in complex network

Community detection is a crucial research orientation in complex network owing to its practical applications. Recently, the convolutional neural network (CNN) based edge classification community detection approach achieved impressive performance as a representative graph embedding method. However, existing methods rely heavily on the manually defined relationship between nodes and local feature representations of the edges, leading to the potential limitation of feature representation in complex real-world networks. To tackle this issue, in this paper, we propose a novel multi-feature fusion network for community detection, namely MFF-Net, which can embed edges of the network into comprehensive feature representations according to the intrinsic property of the network. Specifically, instead of utilizing local features of the edges only, we first propose to simultaneously consider local and non-local relationships of the edges through the local neighbors of the nodes and random node sequences sampled from a customized random walk. Then, we propose to convert local and non-local feature representations of the edges into grayscale images for edge classification via introducing a quantitative relationship between nodes. More importantly, those obtained feature representations are fused with an implicit manner in a latent space, which can leverage more comprehensive relationships for complex real-world networks. Extensive experiments on the computer-generated networks, small-scale and large-scale real-world networks, show that the proposed MFF-Net can achieve better performance compared with existing methods, in terms of quantitative and visualized results.

Efficient Computation of Betweenness Centrality by Graph Decompositions and Their Applications to Real-World Networks

Betweenness centrality is one of the most significant and commonly used centralities, where centrality is a notion of measuring the importance of nodes in networks. In 2001, Brandes proposed an algorithm for computing betweenness centrality efficiently, and it can compute those values for all nodes in O(nm) time for unweighted networks, where n and m denote the number of nodes and links in networks, respectively. However, even Brandes' algorithm is not fast enough for recent large-scale real-world networks, and therefore, much faster algorithms are expected. The objective of this research is to theoretically improve the efficiency of Brandes' algorithm by introducing graph decompositions, and to verify the practical effectiveness of our approaches by implementing them as computer programs and by applying them to various kinds of real-world networks. A series of computational experiments shows that our proposed algorithms run several times faster than the original Brandes' algorithm, which are guaranteed by theoretical analyses.

Optimization of hybrid cable networks with dampers and cross-ties for vibration control via multi-objective genetic algorithm

For long-span cable-stayed bridges, interconnecting adjacent cables using cross-ties and connecting each cable to bridge deck become a promising solution for cable vibration mitigation, leading to hybrid cable networks. However, optimizing the system parameters and configuration for best vibration control effects is still challenging. Therefore, this study proposes an optimization method based on three important ingredients, i.e., a general and efficient numerical model of cable networks, a reasonable definition of the optimization objectives, and a multi-objective optimization method. The cable network is modeled using the component mode synthesis method with modal expansion and quasi-static correction, the optimization objectives consider both multi-mode damping effects and stiffness enhancement, and the multi-objective genetic algorithm is then employed. The proposed method is first verified by comparing with analytical methods when applying to relatively simple cable–damper systems. Subsequently, a five-cable network is studied using the method for optimal design of cross-tie and damper parameters. Furthermore, the method is applied to optimize cross-ties and dampers for the eight longest cables in the Sutong Bridge, considering different numbers, locations and anchoring conditions of the cross-ties. The results have demonstrated the effectiveness of the proposed method in dealing with real-world large-scale cable networks. It is shown that the optimization can ensure that vibrations in local modes of the cable network are dominated by cable segments with a damper installed in the span and hence are sufficiently damped. Besides, damping ratios of hybrid cable networks are shown to be sensitive to the cross-tie location and parameters while they are relatively insensitive to the damper coefficients.

Finite-size scaling of geometric renormalization flows in complex networks.

Some characteristics of complex networks need to be derived from global knowledge of the network topologies, which challenges the practice for studying many large-scale real-world networks. Recently, the geometric renormalization technique has provided a good approximation framework to significantly reduce the size and complexity of a network while retaining its "slow" degrees of freedom. However, due to the finite-size effect of real networks, excessive renormalization iterations will eventually cause these important "slow" degrees of freedom to be filtered out. In this paper, we systematically investigate the finite-size scaling of structural and dynamical observables in geometric renormalization flows of both synthetic and real evolutionary networks. Our results show that these observables can be well characterized by a certain scaling function. Specifically, we show that the critical exponent implied by the scaling function is independent of these observables but depends only on the structural properties of the network. To a certain extent, the results of this paper are of great significance for predicting the observable quantities of large-scale real systems and further suggest that the potential scale invariance of many real-world networks is often masked by finite-size effects.

Supervised temporal link prediction in large-scale real-world networks

Link prediction is a well-studied technique for inferring the missing edges between two nodes in some static representation of a network. In modern day social networks, the timestamps associated with each link can be used to predict future links between so-far unconnected nodes. In these so-called temporal networks, we speak of temporal link prediction. This paper presents a systematic investigation of supervised temporal link prediction on 26 temporal, structurally diverse, real-world networks ranging from thousands to a million nodes and links. We analyse the relation between global structural properties of each network and the obtained temporal link prediction performance, employing a set of well-established topological features commonly used in the link prediction literature. We report on four contributions. First, using temporal information, an improvement of prediction performance is observed. Second, our experiments show that degree disassortative networks perform better in temporal link prediction than assortative networks. Third, we present a new approach to investigate the distinction between networks modelling discrete events and networks modelling persistent relations. Unlike earlier work, our approach utilises information on all past events in a systematic way, resulting in substantially higher link prediction performance. Fourth, we report on the influence of the temporal activity of the node or the edge on the link prediction performance, and show that the performance differs depending on the considered network type. In the studied information networks, temporal information on the node appears most important. The findings in this paper demonstrate how link prediction can effectively be improved in temporal networks, explicitly taking into account the type of connectivity modelled by the temporal edge. More generally, the findings contribute to a better understanding of the mechanisms behind the evolution of networks.

Modified Lomax model: a heavy-tailed distribution for fitting large-scale real-world complex networks

Real-world networks are generally claimed to be scale-free, meaning that the degree distributions follow the classical power-law, at least asymptotically. Yet, closer observation shows that the classical power-law distribution is often inadequate to meet the data characteristics due to the existence of a clearly identifiable non-linearity in the entire degree distribution in the log-log scale. The present paper proposes a new variant of the popular heavy-tailed Lomax distribution which we named as the Modified Lomax (MLM) distribution that can efficiently capture the crucial aspect of heavy-tailed behavior of the entire degree distribution of real-world complex networks. The proposed MLM model, derived from a hierarchical family of Lomax distributions, can efficiently fit the entire degree distribution of real-world networks without removing lower degree nodes as opposed to the classical power-law based fitting. The MLM distribution belongs to the maximum domain of attraction of the Frechet distribution and is right tail equivalent to Pareto distribution. Various statistical properties including characteristics of the maximum likelihood estimates and asymptotic distributions have also been derived for the proposed MLM model. Finally, the effectiveness of the proposed MLM model is demonstrated through rigorous experiments over fifty real-world complex networks from diverse applied domains.

Reducing traffic congestion and increasing sustainability in special urban areas through one-way traffic reconfiguration

In the contemporary sustainable urban set up, one of the critical issues adversely affecting the quality of life in urban areas and inflicting immense costs on cities is traffic congestion. Traffic congestion is an outgrowth of increased traffic flow in certain locations of large cities. Recently, urban decision-makers and transportation planners resort to one-way traffic system as an effective traffic management strategy, which has a profound effect on reducing traffic congestion and improving traffic flow, leading to urban sustainability. In the present paper, the authors endeavored to develop a novel methodological framework based on optimization techniques in order to mitigate traffic congestion through one-way traffic network reconfiguration. To test the efficacy of the proposed methodological framework, two cases were analyzed and evaluated: the Sioux Falls transportation network as a medium-sized one and a real-world large-scale transportation network of the city of Isfahan in Iran. Through the first instance, it can be seen that the proposed method can effectively reduce the total travel time in the area of interest by approximately 9%. The numerical results for the transportation network of Isfahan, justified the practical value of the model and solution method through converting proper links to one-way. As the calculation results of two cases have demonstrated, the solution can effectively reduce the total travel time of travelers in a certain congested urban area.