Functional Connectivity Networks for Transportation Delay Analysis: from Theory to Software

TL;DR

The paper tackles the propagation of delays in transportation networks by adopting functional networks that treat airports or stations as nodes and propagate connections as edges inferred from time-series delay data. It formalizes a five-step framework (data preparation, detrending, connectivity analysis, network reconstruction, and network analysis) and introduces delaynet, a modular Python package that unifies these steps, supports synthetic data generation, and provides a suite of macroscale and centrality metrics. Key contributions include a detailed discussion of the methodological choices and pitfalls, an implementation that enables reproducible end-to-end analyses, and a case study on Swiss rail delays demonstrating dense, directed propagation patterns and corridor-like community structure. The practical impact lies in offering researchers a robust, extensible toolkit for analyzing delay propagation across transportation modes, improving reliability, comparability, and validation of functional-network-based insights.

Abstract

Within the endeavour of modelling and understanding the propagation of delays in transportation networks, an approach that has attracted increasing interest in the last decade is the creation of functional network representations. These graphs map elements of interest (e.g. airports or stations) as nodes, and derive pairwise propagation patterns from their dynamics through correlation and causality tests. In spite of multiple notable results, this approach still lacks a coherent framework, with decisions related to many fundamental steps being left to the judgement of the researcher. We here provide an introduction to the theory behind functional networks for transportation systems, detailing the main steps and the associated pitfalls. We further introduce a Python package, delaynet, designed to support the researcher in the reconstruction and analysis of such networks. We finally present an analysis of the propagation of delays in the Swiss train system; and discuss future research steps.

Figures (11)

• Figure 1: Overview of the delay analysis framework implemented in delaynet. The framework transforms raw delay data through five sequential stages: (1) data preparation and preprocessing; (2) detrending methods, to remove systematic patterns and seasonality that could mask genuine propagation signals; (3) connectivity analysis, quantifying statistical relationships between delay time series; (4) network reconstruction, which applies thresholding strategies to build functional networks from pairwise connectivity values; and (5) network analysis, to extract structural insights about delay propagation patterns, critical nodes, and system vulnerabilities.
• Figure 2: Determination of optimal lag using $p$-values in a Granger Causality analysis between two synthetic time series. The vertical red dashed line indicates the minimum $p$-value obtained. See main text for details.
• Figure 3: Comparison of different connectivity measures as a function of the time series length, on synthetic data generated using the delayed causal networks (see Sec. \ref{['sec:data_preparation']}). See the legend for a list of measures, with the corresponding estimator within parentheses. Results correspond to $200$ randomly generated time series.
• Figure 4: Comparison of different connectivity measures as a function of the time series length, on real data for the Atlanta (ATL) and Denver (DEN) airports. See the legend for a list of measures, with the corresponding estimator within parentheses. Results correspond to $500$ randomly extracted time series.
• Figure 5: Propagation networks corresponding to the top-$15$ US airports, using data for year 2019. Left, centre and right networks have been reconstructed using respectively Granger Causality, COP, and Transfer Entropy (with a box kernel). The top pictures depict the network, with node sizes proportional to the out-degree, and the colour of link indicating the optimal lag (from 1 hour, light shade, to 6 hours, dark shade). The bottom graphs report the corresponding adjacency matrices; black squares indicate the presence of a causal connection between the corresponding airports, white squares the absence thereof.