Figure 1.
Circle representation of 32 metro networks in the world (using NodeXL [52]).
Figure 2.
Schematic graph of Lyon metro system and its adjacency matrix.
The left side is the sketch of the system where the shapes of the lines are kept even though the graph is isomorphic. Termini are illustrated by black circles and transfers stations by white circles. The right side of the graph shows the adjacency matrix (i.e. ‘1′ when a connection exist and ‘0′ otherwise). Note that link EK does not exist in real life, hence the greyed node E and the dotted line in the matrix.
Table 1.
Results for 28 metro networks.
Figure 3.
Evolution of average betweenness centrality CB with network size.
The regression fits a second degree polynomial and the statistical significance is surprisingly high; only Chicago does not fit the regression as well (perhaps due to its five-lined directed elevated section in the so called “loop” area).
Figure 4.
Cumulative distributions (CD) of normalized betweenness centrality for 28 metros.
Although it can be difficult to pinpoint one specific system, the main message here is that betweenness consistently becomes more distributed with network growth. The absence of a “winner takes all” paradigm is surprising considering it is often the case with other complex network properties (e.g., in scale-free networks).
Figure 5.
Evolution of quadratic coefficients |an| of normalized cumulative distributions with network size.
Figure 6.
Evolution of quadratic coefficients |ao| of cumulative distributions with network size.
Two clear and distinct regimes can be observed here. The high coefficient of Chicago and Stockholm, whilst being comparatively small, suggests the dominance of a radial feature. On the other hand, the lower coefficient of Paris, considering its size, suggests a dominant grid pattern. New York and London can be seen as hybrids, having fairly high coefficients whilst being large, which is quite intuitive (grid pattern in the center, joined with a radial pattern in the peripheries).
Figure 7.
Evolution of highest CBhi and lowest non-zero CBlo betweenness centralities with size.
While both centralities fit power law functions, the exponent of highest betweenness is much lower than the lowest betweenness, suggesting that the loss of share in betweenness from most central nodes does not decay as fast as for least central nodes.
Table 2.
Node betweenness and power law exponents.
Table 3.
Five most central stations and their betweenness centralities for metros with N ≥20.