AUTHOR=Boettcher Stefan 

TITLE=Renormalization Group for Critical Phenomena in Complex Networks

JOURNAL=Frontiers in Physiology

VOLUME=volume 2 - 2011

YEAR=2011

URL=https://www.frontiersin.org/journals/physiology/articles/10.3389/fphys.2011.00102

DOI=10.3389/fphys.2011.00102

ISSN=1664-042X

ABSTRACT=
We discuss the behavior of  statistical models on a novel class
of complex ``Hanoi'' networks. Such modeling is often the cornerstone
for the understanding of many dynamical processes in complex
networks. Hanoi networks are special because they integrate
small-world hierarchies common to many social and economical
structures with the inevitable geometry of the real work these
structures exist in. In addition, their design allows \emph{exact}
results to be obtained with the venerable renormalization group
(RG). Our treatment will provide a detailed, pedagogical introduction
to RG. In particular, we will study the Ising model with RG, for which
the fixed-points are determined and the RG-flow is analyzed.  We show
that the small-world bonds result in non-universal behavior. It is
shown that a diversity of different behaviors can be observed with
seemingly small changes in the structure of hierarchical networks
generally, and we provide a general theory to describe our findings.