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

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.