Renormalization Group for Critical Phenomena in Complex Networks - Stefan eBoettcher (Frontiers in Physiology, 2011)
We discuss the behavior of statistical models on a novel classof complex ``Hanoi'' networks. Such modeling is often the cornerstonefor the understanding of many dynamical processes in complexnetworks. Hanoi networks are special because they integratesmall-world hierarchies common to many social and economicalstructures with the inevitable geometry of the real work thesestructures exist in. In addition, their design allows emphexactresults to be obtained with the venerable renormalization group(RG). Our treatment will provide a detailed, pedagogical introductionto RG. In particular, we will study the Ising model with RG, for whichthe fixed-points are determined and the RG-flow is analyzed. We showthat the small-world bonds result in non-universal behavior. It isshown that a diversity of different behaviors can be observed withseemingly small changes in the structure of hierarchical networksgenerally, and we provide a general theory to describe our findings.