The exact Laplacian spectrum for the Dyson hierarchical network
Abstract
We consider the Dyson hierarchical graph , that is a weighted fullyconnected graph, where the pattern of weights is ruled by the parameter σ ∈ (1/2, 1]. Exploiting the deterministic recursivity through which is built, we are able to derive explicitly the whole set of the eigenvalues and the eigenvectors for its Laplacian matrix. Given that the Laplacian operator is intrinsically implied in the analysis of dynamic processes (e.g., random walks) occurring on the graph, as well as in the investigation of the dynamical properties of connected structures themselves (e.g., vibrational structures and relaxation modes), this result allows addressing analytically a large class of problems. In particular, as examples of applications, we study the random walk and the continuoustime quantum walk embedded in , the relaxation times of a polymer whose structure is described by , and the community structure of in terms of modularity measures.
 Publication:

Scientific Reports
 Pub Date:
 January 2017
 DOI:
 10.1038/srep39962
 arXiv:
 arXiv:1604.05864
 Bibcode:
 2017NatSR...739962A
 Keywords:

 Physics  Data Analysis;
 Statistics and Probability
 EPrint:
 A slightly revised version has been published in Scientific Reports (2017)