Real-World Networks are Low-Dimensional: Theoretical and Practical Assessment

TL;DR

This paper demonstrates that real-world networks are low-dimensional by analyzing GIRGs, providing a theoretical explanation for their properties and introducing an efficient algorithm to determine their dimensionality with high accuracy.

Contribution

The paper offers a theoretical analysis linking clustering coefficient decay to low dimensionality and presents a linear-time algorithm for accurately estimating the dimension of GIRGs.

Findings

Clustering coefficient scales inverse exponentially with dimensions in GIRGs.

The proposed algorithm accurately estimates the number of dimensions in GIRGs.

Experimental results on real-world networks validate the method's effectiveness.

Abstract

Detecting the dimensionality of graphs is a central topic in machine learning. While the problem has been tackled empirically as well as theoretically, existing methods have several drawbacks. On the one hand, empirical tools are computationally heavy and lack theoretical foundation. On the other hand, theoretical approaches do not apply to graphs with heterogeneous degree distributions, which is often the case for complex real-world networks. To address these drawbacks, we consider geometric inhomogeneous random graphs (GIRGs) as a random graph model, which captures a variety of properties observed in practice. Our first result shows that the clustering coefficient of GIRGs scales inverse exponentially with respect to the number of dimensions, when the latter is at most logarithmic in $n$. This gives a first theoretical explanation for the low dimensionality of real-world networks as observed by Almagro et al. in 2022. We further use these insights to derive a linear-time algorithm for determining the dimensionality of a given GIRG and prove that our algorithm returns the correct number of dimensions with high probability GIRG. Our algorithm bridges the gap between theory and practice, as it not only comes with a rigorous proof of correctness but also yields results comparable to that of prior empirical approaches, as indicated by our experiments on real-world instances.