  The investigations of financial markets from a complex network perspective
have unveiled many phenomenological properties, in which the majority of these
studies map the financial markets into one complex network. In this work, we
investigate 30 world stock market indices through their visibility graphs by
adopting the visibility algorithm to convert each single stock index into one
visibility graph. A universal allometric scaling law is uncovered in the
minimal spanning trees, whose scaling exponent is independent of the stock
market and the length of the stock index. In contrast, the maximal spanning
trees and the random spanning trees do not exhibit universal allometric scaling
behaviors. There are marked discrepancies in the allometric scaling behaviors
between the stock indices and the Brownian motions. Using surrogate time
series, we find that these discrepancies are caused by the fat-tailedness of
the return distribution, the nonlinear long-term correlation, and a coupling
effect between these two influence factors.
