A reduced model for complex network analysis of public transportation systems

Highlights

• Simple method to extract the skeleton of transportation-based networks.

• Close-form equations describing the influence of 2-degree nodes on networks.

• Similarities between different transportation modes.

• Their skeletons have hierarchical structure and small-world characteristics.

Abstract

Public transportation networks (PTNs) are represented as complex networks in order to analyze their robustness regarding node and link failures, to classify them into different theoretical network models, and to identify the characteristics of the underlying network. Usually, PTNs have a large amount of 1- and 2- degree nodes that blur the analysis and their characterization as complex networks. Subway and train-based transport networks present long single lines that connect central stations to far destinations differently from airport networks that usually have few large airports (hubs) connecting a significant number of small airports. By focusing on relevant network nodes and links and allowing comparisons between PTNs of different transportation modes, this paper proposes the Reduced Model as a simple method of network reduction that preserves the network skeleton (backbone structure) by properly removing 2-degree nodes of weighted and unweighted network representations. Different from other proposed methods, its simple formulation leads to mathematical expressions that show how the reduced model affects fundamental network metrics (degree, path length, and clustering coefficient distributions). The Reduced model is applied to four large real-world PTNs: (i) two Brazilian cities with bus-based transport; (ii) the Seoul metro network; (iii) a worldwide airport network. The results reveal a hub-based hierarchical structure when a large number of intermediary stops are present and small-world properties that emphasizes hub–hub connections after applying the Reduced model. Therefore, the reduced model emphasizes characteristics of the networks that could be difficult to identify without reduction.

Introduction

Complex networks are models used to represent physical, chemical, biological, and social systems whose elements can be represented by nodes (or vertices), and interactions between those elements can be represented by links (or edges). Efficient algorithms and statistical methods have been developed in order to extract hidden characteristics of large and very large networks [1], [2], sometimes relying on smaller versions of such networks [3].

Public Transportation Networks (PTNs) usually have low dimension in comparison to the aforementioned examples of complex networks, but they have been studied by the scientific community due to their inherit graph structure (stations as nodes and logical connections between the stations as links) and by their social impact [4], [5]. Several works have focused on statistically characterizing such networks and objectively comparing PTNs from different cities: classical complex network metrics like degree and path length distributions, clustering coefficient (or network transitivity), network betweenness, and network closeness, for example, have been used to topologically characterize the relationship between connected stations in cities from Poland [6], China [7], [8], [9], and Brazil [10].

PTNs can be modeled as complex networks by different graph representations like B, C, L, and P-Space representations [10], [11], [12]. They were conceived to answer particular questions about the PTNs they represent [6], [7], [9], [13], [14]. Particularly, the L-Space representation [15] of a transportation network is a graph where stations are nodes and a link between two given nodes is set if there are any number of routes connecting them. Different link weights (number of passengers [16], travel time [17], distance [18]) add physical meaning to the graph representation and lead to analysis of specific topics of PTNs [12], [15]. Alternatively, time series built from measured vehicles flows can be used to built realistic complex networks [19] whose metrics indicate the congestion state of routes (congested, free, or in a transition state).

Also, it is possible to build a multigraph L-Space representation where the number of links between two given nodes are equal to the number of bus routes connecting these nodes [6], [9], [13]. Perfect P-Space representation [20] (a type of multigraph) uses links with different weights, along with nodal weights, to better represent directional connectivity between stops in PTNs, particularly useful for bus-based ones. Such representation has Gaussian probability distribution functions describing the path lengths of weighted and unweighted PTN-derived networks.

One characteristic of both bus- and train-based networks is the presence of long routes of “2-degree” stations either branching out from stations generally located at city’s downtown area or connecting specific stations (like terminals). Such description is not usual in flight routes where air companies planned their routes in terms of hubs (or high-degree nodes). Additionally, circular patterns are observed in transport networks: a circular route is formed by two end stations ($a$ and $b$, for example) connected to each other by two different paths formed by sets of stations spatially distributed according to some policy where vehicles traverse from $a$ to $b$ in one path, and from $b$ to $a$ in the other one. In L-Space representations, these intermediate stations usually have degree equals to 2 because they only connect adjacent stations. Along with these circular routes [21], train-based PTNs (including subways) have shared stations for paths connecting $a$ and $b$ in both directions to increase connectivity and/or rationalize the use of resources. Circular routes are also observed in flight routes, generally connecting specific airports. By assuming the existence of a representative nucleus hidden inside the undirected network, the Core-Branch model [22] was proposed to obtain a simpler and more significant complex network for L-Space representation of train-based PTNs. This simpler network results from the removal of tree subgraphs that include all 1-degree nodes (stations) along with some 2- and 3-degree stations. However, it modifies the topological characteristics of the network, particularly for networks with large amount of tree subgraphs.

Thus, considering the presence of large number of 2-degree nodes in bus- and train-based PTNs, which directly affects complex network metrics (average network degree, average path length, and betweenness, for example), this work proposes a simple model (the Reduced model) that removes all these nodes from L-Space representations of PTNs (regardless the use of link weights) and keeps the network skeleton (or backbone structure) intact, which is more evident in hub-based PTNs. We also show that PTNs from different modes share a hidden topological structure, which is unveiled by the proposed model: the connectivity between significant stops has small-world characteristics and, particularly, if a reduced network has an hierarchical structure, the original network is a scale-free network or has significant characteristics of it. Finally, we show that iterative mathematical expressions can quantify the effects of the network reduction (or growing) over fundamental network metrics (i.e. degree, clustering coefficients, and path length).

The model was tested over bus-based PTNs from two major Brazilian cities (whose sizes are comparable to country networks [17]), Seoul metro, and the worldwide flight network. Theoretical complex networks were used to identify hidden and shareable topological structures in PTNs. Their weighted versions were built considering geographical distances traveled between stations through existing routes. The main contributions of this paper are: (i) a simplified complex network model (the Reduced model) of PTNs that preserves their topological characteristics for network analysis, and (ii) a mathematical approach to evaluate the influence of the network reduction on classical complex network metrics, allowing results comparison with other published works.

The paper is structured as follows: a brief review of complex network theory applied to model PTNs and complex network representation of PTNs with significant number of 1-degree and 2-degree nodes is presented. Then, the proposed model (the Reduced model) is described along with an analytical analysis of its influence over degree and clustering coefficient distributions, following a discussion of its influence on path length distribution and an analysis of the effects of the proposed model, and the Core-Branch model on theoretical networks. Finally, results and discussions about the topological effect of the Reduced model over real PTNs are presented.