Fractal power law and polymer-like behavior for the metro growth in megacities

Abstract

The paper analyzes the correlation between the population of megacities and the size of the metro in these megacities. The correlation was found only for a sampling of large cities with an area of more than 1000 km² with a number of metro stations of more than 41. For the first time, it was shown that for such a sampling of the largest megacities, consisting of 56 cities, there is a correlation between the number of metro stations St and the population of the city P of the form of the power law $St\approx {(P/P_{\ast })}^{{\alpha }_m}$, where $P_{\ast }$ is the average number of people served by one station, ${\alpha }_m\approx 4/3$ is the exponent. It is shown that all cities in the sampling can be divided into 4 groups. Each group has approximately the same average number of people served by one station. It varies from $P_{\ast }\approx 120$ to $P_{\ast }\approx 415$ thousand people per station. The common features of cities included in different groups are discussed. It is assumed that the discovered correlation is of a fractal nature. It is shown that the fractal dimension of the external perimeter for a two-dimensional percolation cluster has a the same value. A qualitative model is proposed that can explain such fractal behavior for metro networks in megacities. It is shown that the discovered exponent ${\alpha }_m$ in the correlation is close in value to the fundamental percolation constant $C_f$ (${\alpha }_m\approx C_f\approx 1,327\approx 4/3$), which characterizes the most general topological properties of fractals, primarily such as connectivity close to critical point (Milovanov, 1997). A possible relation between the structure of large metro networks and this percolation constant is discussed. An analogy is shown between large metro networks and transport routes formed by ants in large anthills, as well as with branched polymer molecules.