  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.
