Real-Space Renormalization Group for Spectral Properties of Hierarchical Networks

We derive the determinant of the Laplacian for the Hanoi networks and use it to determine their number of spanning trees (or graph complexity) asymptotically. While spanning trees generally proliferate with increasing average degree, the results show that modifications within the basic patterns of design of these hierarchical networks can lead to significant variations in their complexity. To this end, we develop renormalization group methods to obtain recursion equations from which many spectral properties can be obtained. This provides the basis for future applications to explore the physics of several dynamic processes.