Real-Space Renormalization Group for Spectral Properties of Hierarchical Networks

Abstract

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.

14 pages, RevTex, 2 pdf figures. Numerous corrections and improvements. For related work, see http://www.physics.emory.edu/faculty/boettcher/

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• Hierarchical structure and the prediction of missing links in networks
• Spatial search by quantum walk
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• The Griffiths Phase on Hierarchical Modular Networks with Small-world Edges
• Evaluation of Spectral Zeta-Functions with the Renormalization Group
• Quantum Ultra-Walks: Walks on a Line with Hierarchical Spatial Heterogeneity