Metro/Subway Comparison

Want to know how the subway network of your area compares to others?

This is a study comparing several subway networks. We have based the calculations on centrality and other measures from the field of networks, as well as some measurements defined in the study published in the journal Physica A entitled The complexity and robustness of metro networks (Sybil Derrible, Christopher Kennedy, The complexity and robustness of metro networks, Physica A: Statistical Mechanics and its Applications, Volume 389, Issue 17, 1 September 2010, Pages 3678-3691, ISSN 0378-4371).

The aim is both to include subway systems from Brazilian cities (not present in the original study), and to extend that study by including centrality measures in order to find out which are the top 5 stations in terms of betweenness (number of shortest paths from all vertices to all others that pass through that node).

Table 1 below shows, for some subway networks, how they perform in terms of these measures. It is divided in two tables: Table 1.a shows some characteristics of each network (number of edges, number of vertices, diameter, etc.), while Table 1.b shows, for each network, which are the top five vertices in terms of betweenness. This means that these five stations are those that carry the highest number of shortest paths between each arbitrary pair of origin and destination. Not surprisingly, they are important stations in each respective network.

Table 1 includes the subway systems of three Brazilian cities. Regarding São Paulo it includes lines 1 to 4 only, since line 5 (lilás) is not yet connected to the others, and the other lines are actually rail lines that were recently integrated to the subway system but are not really part of this system. It also includes the network of Rio de Janeiro. The subway network for Porto Alegre is hypothetical. It does not exist as such! The corresponding network topology can be found here.

The explanation for each column of Table 1 follows Table 1b.

After, we reproduce Table 2 from the above mentioned work by Derrible and Kennedy. Note some differences that appear are due to: i) we have not reproduced the measures that deal with the small-world and scale-free parts; ii) we have used a slightly different methodology to count vertices (basically we have always considered all vertices instead of diatonic ones). More on the methodology below.

Table 1.a

Table 1.b

Explanation of columns appearing in Table 1:

The following columns are straightforward: network name; number of lines that were considered in the study, number of stations (vertices), non-transfer stations, transfer stations, total number of links, ratio transfer stations to the total number of stations

other columns:

• • diatonic stations (5th column): sum of transfers and terminals

  • non-transfer terminals (6th column): number of terminals that are not transfers (these appear in the 7th column)

  • diatonic edges (8th): connect diatonic stations

  • multiple edges (9th): parallel links between two adjacent vertices; this measure differs from the one in Table 2, which considers only multiple edges between diatonic vertices

  • degree of connectivity (gamma): a measure of clustering

  • cyclomatic number (mu): calculates the number of cycles in a graph

  • robustness : main meesure proposed by Derrible and Kennedy, computes the net number of cycles, taking into account the size of the network

  • diameter: length of the longest geodesic (shortest path)

Methodology to find the top 5 stations in terms of betweenness (Table 1b):

• 1. a graph was generated for each subway network, in the so-called ncol format, which was then read by the igraph library/package in the R software

  2. igraph's betweeness was used to compute node/vertex/station betweenness; we have then picked the top 5 (station plus its betweenness value)

  3. igraph was also used for the total number of vertices, total number of edges/links, and to calculate the number of multiple edges

Disclaimer: the graphs were manually generated, i.e., they were basically created from visual inspection of official maps of each subway/metro network. This is an error-prone task thus graphs may (in fact they probably do) contain errors. However the fact that the top 5 stations in terms of betweenness are plausible in most cases is a good indication that errors were minor.

If you do find non-plausible or incorrect data, please report to us.

The table below is reproduced from Sybil Derrible, Christopher Kennedy, The complexity and robustness of metro networks, Physica A: Statistical Mechanics and its Applications, Volume 389, Issue 17, 1 September 2010, Pages 3678-3691, ISSN 0378-4371.