Entropy Analysis of a Railway Network’s Complexity

Abstract

Abstract:Railway networks are among the many physical systems that reveal a fractal structure. This paper studies the Portuguese railway system, and analyzes how it evolved over time, namely what concerns the structure of its different levels, and its distribution over the territory. Different mathematical tools are adopted, such as fractal dimension, entropy and state space portrait. The results are consistent with the historical evolution of the network.

Complexity Analysis of Global Temperature Time Series

“…A fractal is a geometrical object that displays identical patterns at different scales. The is a measure of how much the fractal fills the space as it is magnified from larger to smaller scales [ 54 , 55 , 56 ]. The box-counting method is often adopted for estimating the , due to its straightforward numerical implementation.…”

“…The box-counting method is often adopted for estimating the , due to its straightforward numerical implementation. For a fractal represented by a binary image, with non-zero pixels belonging to the fractal, and zero pixels forming the image background, the box counting algorithm is as follows [ 56 ]: Pad the image with background pixels so that its dimensions are at a power of 2; Cover the fractal object with a grid of squares with size (in the first iteration, there is just one square of equal size to the size of the image); Count the number of boxes (i.e., squares), , needed to cover the object; If , then make and repeat step 2. Estimate the as the slope of the log-log plot, versus , calculated by means of the least squares method.…”

Complex network model for railway timetable stability optimisation

“…Feng et al presented a multi‐layer model to analyse traffic flow patterns in subway networks and generate a timetable for the Beijing Subway System [26]. Valério et al perform the entropy analysis work on the railway network complexity [27]. Shao and Li designed a complex network model to analyse railway accidents based on the maximal information coefficient [28].…”

Extracting the Relationship and Evolutionary Rule Connecting Residents’ Travel Demand and Traffic Supply Using Multisource Data

“…Benguigui [ 12 ] first identified the fractal features of an urban transport network and carried out case studies on the London Underground, Paris Metro, Moscow Metro, and Rhine Subway (Germany). Batty [ 13 , 14 ] linked transport analysis to urban geography on the basis of fractal dimensions, after which fractal approaches have been widely applied to different parts of cities including the city center [ 15 ], the whole scope of a city [ 16 ], and conurbations [ 11 ]. Different modes of transport have also been analyzed using fractal methods including traditional surface transport [ 15 ], subway [ 16 ], and the whole public transport network.…”

“…Batty [ 13 , 14 ] linked transport analysis to urban geography on the basis of fractal dimensions, after which fractal approaches have been widely applied to different parts of cities including the city center [ 15 ], the whole scope of a city [ 16 ], and conurbations [ 11 ]. Different modes of transport have also been analyzed using fractal methods including traditional surface transport [ 15 ], subway [ 16 ], and the whole public transport network. Several urban issues are discussed in the research in relation to transport networks such as the complexity and space filling capacity of urban transport networks [ 17 ], city morphology [ 18 ], and the consistency of transportation and the built-up environment [ 19 ].…”