A segmented fractal model associated with the spatial distribution characteristics of urban rail transit network

In view of the limitations of fractal models with single fractal dimension in representing the spatial non-uniform distribution of urban rail transit network, a segmented fractal model is established by summarizing the variation patterns of inflection points in network length curves and the definition of binary classification in spatial distribution analysis. This model adheres to the countable additivity definition of the measurement for disjoint fractal sets and retains the inherent measurement relationship in fractal theory. By examining the results of spatial distribution analysis derived from typical urban rail transit network data, the correlation between model parameters and spatial distribution characteristics is investigated. This process validates the effectiveness of the model while also exploring the physical meaning of its parameters. The results show that the segmented point in this model divides the network into two domains. The fractal dimensions corresponding to the first and second domains are relatively independent and can be utilized to characterize the spatial heterogeneous growth rate of the network. Segmented point in this model is identified as the main parameter that exhibits significant positive correlations with the spatial distribution characteristics, including the standard deviation distance, semi-major axis and semi-minor axis. The correlation coefficients are 0.89, 0.90, and 0.75, respectively. These results indicate that the network located within the first domain demonstrates an aggregation distribution characteristic, whereas that within the second domain exhibits a dispersion distribution characteristic. Besides, the parameters in the model have been found to inadequately reflect the intensity of directional distribution within the network. However, the segmented point within these model parameters can serve as an indicator for the coverage range of a directionally distributed network along both its semi-major and semi-minor axes.