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

Ying Chen, Rongpeng Li+2 more|arXiv (Cornell University)|Aug 16, 2018

Advanced MIMO Systems Optimization36 references3 citations

TL;DR

This paper applies algebraic topology tools—α-shapes, Betti numbers, and Euler characteristics—to real base station (BS) location data from 12 countries, revealing fractal topological structures in cellular networks and demonstrating that Euler characteristics follow a log-normal distribution. The findings provide a robust topological framework for optimizing large-scale BS deployments.

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.

Motivation & Objective

• To address the challenge of quantitatively evaluating large-scale base station (BS) deployments in cellular networks.
• To uncover intrinsic topological features of real-world BS spatial configurations beyond traditional density-based models.
• To investigate whether topological invariants such as Betti numbers and Euler characteristics can reveal universal patterns in cellular network topology.
• To determine the best-fitting statistical distribution for Euler characteristics derived from real BS deployments.

Proposed method

• Employed α-shapes—a computational topology tool—to model the geometric and topological structure of real BS location point sets.
• Calculated Betti numbers (β₀ and β₁) from α-shapes to quantify topological features such as connected components and loops.
• Used the Euler-Poincaré formula (χ = β₀ − β₁) to compute Euler characteristics for each α-shape configuration.
• Applied rescaled range (R/S) analysis to estimate Hurst coefficients and test for fractal behavior in BS spatial distributions.
• Fitted the empirical probability density function (PDF) of Euler characteristics to candidate heavy-tailed distributions, including log-normal, Lévy, and Weibull.
• Evaluated goodness-of-fit using root mean square error (RMSE), with the lowest RMSE indicating the best match.

Experimental results

Research questions

• RQ1 Do the spatial deployments of base stations in cellular networks exhibit fractal characteristics as measured by Betti numbers and Hurst coefficients?
• RQ2 What is the statistical distribution that best fits the Euler characteristics of real-world cellular network topologies?
• RQ3 Are there universal topological patterns in BS deployments across geographically and culturally diverse countries?
• RQ4 Can topological invariants such as Betti numbers and Euler characteristics serve as reliable metrics for optimizing base station deployment strategies?

Key findings

• The Betti number curves for both Asian and European countries exhibit ripple-like patterns, indicating complex topological structures with multiple levels of connectivity and loop formation.
• All Hurst coefficients computed across 12 countries are very close to 1 (ranging from 0.94 to 0.99), confirming strong long-range dependence and fractal behavior in BS spatial configurations.
• The log-normal distribution provides the best fit to the empirical PDF of Euler characteristics, with RMSE values consistently one order of magnitude lower than other candidates like Lévy and Weibull distributions.
• For every country studied—both in Asia and Europe—the log-normal distribution yields the smallest RMSE in fitting the Euler characteristic PDF, indicating a universal statistical pattern.
• Despite geographical, cultural, and historical differences, the Euler characteristics of BS deployments across countries conform to the same statistical distribution, suggesting a universal topological law.
• The combination of α-shapes, Betti numbers, and Euler characteristics enables a robust, data-driven characterization of cellular network topology beyond conventional spatial density models.