Sharpest possible clustering bounds using robust random graph analysis

TL;DR

This paper develops tight bounds for clustering and correlation in complex networks using only basic degree distribution statistics, providing insights into the extremal properties of scale-free networks.

Contribution

It introduces a robust method to bound network measures based on mean, range, and dispersion of degrees, independent of full degree distribution fitting.

Findings

Power-law networks with diverging variance are extremal for correlation and clustering.

Tight bounds are derived for all networks sharing the same degree summary statistics.

Robust laws describe how correlation and clustering evolve with network size and degree.

Abstract

Complex network theory crucially depends on the assumptions made about the degree distribution, while fitting degree distributions to network data is challenging, in particular for scale-free networks with power-law degrees. We present a robust assessment of complex networks that does not depend on the entire degree distribution, but only on its mean, range and dispersion: summary statistics that are easy to obtain for most real-world networks. By solving several semi-infinite linear programs, we obtain tight (the sharpest possible) bounds for correlation and clustering measures, for all networks with degree distributions that share the same summary statistics. We identify various extremal random graphs that attain these tight bounds as the graphs with specific three-point degree distributions. We leverage the tight bounds to obtain robust laws that explain how degree-degree correlations and local clustering evolve as function of node degrees and network size. These robust laws indicate that power-law networks with diverging variance are among the most extreme networks in terms of correlation and clustering, building further theoretical foundation for widely reported scale-free network phenomena such as correlation and clustering decay.