Relation between the degree and betweenness centrality distribution in complex networks

The centrality measures, like betweenness b and degree k in complex networks remain fundamental quantities helping to classify them. It is realized from Barthelemy's paper [Eur. Phys. J. B 38, 163 (2004)10.1140/epjb/e2004-00111-4] that the maximal b-k exponent for the scale-free (SF) networks is η_{max}=2, belonging to SF trees, based on which one concludes δ≥γ+1/2, where γ and δ are the scaling exponents for the distribution functions of the degree and the betweenness centralities, respectively. This conjecture was violated for some special models and systems. Here we present a systematic study on this problem for visibility graphs of correlated time series, and show evidence that this conjecture fails in some correlation strengths. We consider the visibility graph of three models: two-dimensional Bak-Tang-Weisenfeld (BTW) sandpile model, one-dimensional (1D) fractional Brownian motion (FBM), and 1D Levy walks, the two latter cases are controlled by the Hurst exponent H and the step index α, respectively. In particular, for the BTW model and FBM with H≲0.5, η is greater than 2, and also δ<γ+1/2 for the BTW model, while the Barthelemy's conjecture remains valid for the Levy process. We assert that the failure of the Barthelemy's conjecture is due to large fluctuations in the scaling b-k relation resulting in the violation of hyperscaling relation η=γ-1/δ-1 and emergent anomalous behavior for the BTW model and FBM. Universal distribution function of generalized degree is found for these models which have the same scaling behavior as the Barabasi-Albert network.