  The Griffiths phase has been proposed to induce a stretched critical regime
that facilitates self-organizing of brain networks for optimal function. This
phase stems from the intrinsic structural heterogeneity of brain networks, such
as the hierarchical modular structure. In this work, we extend this concept to
modified hierarchical networks with small-world connections based on Hanoi
networks. Through extensive simulations, we identify the role of an exponential
distribution of the inter-moduli connectivity probability across hierarchies
determining the emergence of the Griffiths phase. Numerical results and the
complementary spectral analysis on the relevant networks can be helpful for a
deeper understanding of the essential structural characteristics of finite
dimensional networks to support the Griffiths phase.
