The Visibility Graphs of Correlated Time Series Violate the Barthelemy's Conjecture for Degree and Betweenness Centralities

TL;DR

This paper investigates the relationship between degree and betweenness centralities in visibility graphs derived from correlated time series, revealing violations of a longstanding conjecture in certain models due to anomalous scaling behaviors.

Contribution

It demonstrates that Barthelemy's conjecture does not hold for scale-free visibility graphs of specific correlated time series models, highlighting new anomalous scaling phenomena.

Findings

Violations of Barthelemy's conjecture in SF visibility graphs for BTW and FBM models.

Large fluctuations in the b-k relation lead to hyperscaling relation violations.

Discovery of super-universal behavior in degree distribution functions.

Abstract

The problem of betweenness centrality remains a fundamental unsolved problem in complex networks. After a pioneering work by Barthelemy, it has been well-accepted that the maximal betweenness-degree ($b$-$k$) exponent for scale-free (SF) networks is $\eta_{\text{max}}=2$, belonging to scale-free trees (SFTs), based on which one concludes $\delta\ge\frac{\gamma+1}{2}$, where $\gamma$ and $\delta$ are the scaling exponents of the distribution functions of the degree and betweenness centrality, respectively. Here we present evidence for violation of this conjecture for SF visibility graphs (VGs). To this end, we consider the VG of three models: two-dimensional (2D) Bak-Tang-Weisenfeld (BTW) sandpile model, 1D fractional Brownian motion (FBM) and, 1D Levy walks, the two later cases are controlled by the Hurst exponent $H$ and step-index $\alpha$, respectively. Specifically, for the BTW model and FBM with $H\lesssim 0.5$, $\eta$ is greater than $2$, and also $\delta<\frac{\gamma+1}{2}$ for the BTW model, while Barthelemy's conjecture remains valid for the Levy process. We argue that this failure of Barthelemy's conjecture is due to large fluctuations in the scaling $b$-$k$ relation resulting in the violation of hyperscaling relation $\eta=\frac{\gamma-1}{\delta-1}$ and emergent anomalous behaviors for the BTW model and FBM. A super-universal behavior is found for the distribution function for a generalized degree function identical to the Barabasi-Albert network model.