  In the present paper, we analyze the fractal structures in magnitude time
series for a set of unprecedented sample extracted from the National Earthquake
Information Center (NEIC) catalog corresponding to 12 Circum-Pacific subduction
zones from Chile to Kermadec. For this end, we used the classical Rescaled
Range ($R/S$) analysis for estimating the long-term persistence signature
derived from scaling parameter so-called Hurst exponent, $H$. As a result, we
measured the referred exponent and obtained all values of $H>0.5$, indicating
that a long-term memory effect exists. The main contribution of our paper, we
found a possible fractal relationship between $H$ and the $b_{s}(q)$-index
which emerges from nonextensive Gutenberg-Richter law as a function of the
asperity, i.e., we show that the values of $H$ can be associated with the
mechanism which controls the abundance of magnitude and, therefore, the level
of activity of earthquakes. Finally, we concluded that dynamics associated with
fragment-asperity interactions can be emphasized as a self-affine fractal
phenomenon.
