The Phase Transitions of Self-similar Small-world Networks

A novel set of self-similar networks called Hanoi networks\footnote{S. Boettcher, B Gon\c{c}alves, Europhysics Letters \textbf{84}, 30002 (2008).} (HN) have been developed to study the critical phenomena of small-world networks using the renormalization group (RG). Physically, HNs contain a more desirable geometry than random small-world networks. Their structure consists of a one-dimensional backbone with a hierarchy of long-range bonds, which allows the flexibility of studying planar and non-planar networks with either a regular or exponential degree distribution. The RG and Ising model simulation results for HNs reveal unique phase transitions and non-universal behavior, which can be attributed to their hierarchical structure.\footnote{See also http://www.physics.emory.edu/faculty/boettcher/.}