Study on Base Station Topology in National Cellular Networks: Take Advantage of Alpha Shapes, Betti Numbers, and Euler Characteristics

Abstract

Faced with the ever-increasing trend of the cellular network scale, how to quantitatively evaluate the effectiveness of the large-scale deployment of base stations (BSs) has become a challenging topic. To this end, a deeper understanding of the cellular network topology is of fundamental significance to be achieved. In this paper, α-Shape, a powerful algebraic geometric tool, is integrated into the analysis of real BS location data for six Asian countries and six European countries, respectively. Firstly, the BS spatial deployments of both Asian and European countries express fractal features based on two different testifying metrics, namely the Betti numbers and the Hurst coefficients. Secondly, it is found out that the log-normal distribution presents the best match to the cellular network topology when the practical BS deployment is characterized by the Euler characteristics.

Fundamentals on Base Stations in Urban Cellular Networks: From the Perspective of Algebraic Topology

“…For the sake of verifying the fractal nature in the BS topology, the rescaled range analysis (R/S) method [17] is adopted to compute the Hurst coefficients of all the selected cities in this letter. Due to the space limitation, the details about the calculation of the Hurst coefficients will not be provided here, and all the relevant information can be found in [7]. As listed in Table III, the fractal nature is evidently affirmed because all the Hurst coefficients turn out to be very close to 1.…”

“…Moreover, the topological information can be extracted from the Betti numbers and the Euler characteristics as well, because the corresponding α-Shapes can be expressed in terms of these two features. Due to the space limitation, interesting readers could refer to [7] to find the details of these three topological notions and their relationships.…”