We consider the Dyson hierarchical graph G, that is a weighted fully-connected graph, where the pattern of weights is ruled by the parameter σ ∈ (1/2, 1]. Exploiting the deterministic recursivity through which G 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 continuous-time quantum walk embedded in G, the relaxation times of a polymer whose structure is described by G, and the community structure of  in terms of modularity measures.

The exact Laplacian spectrum for the Dyson hierarchical network / Agliari, Elena; Tavani, Flavia. - In: SCIENTIFIC REPORTS. - ISSN 2045-2322. - ELETTRONICO. - 7:(2017), p. 39962. [10.1038/srep39962]

The exact Laplacian spectrum for the Dyson hierarchical network

AGLIARI, ELENA;TAVANI, FLAVIA
2017

Abstract

We consider the Dyson hierarchical graph G, that is a weighted fully-connected graph, where the pattern of weights is ruled by the parameter σ ∈ (1/2, 1]. Exploiting the deterministic recursivity through which G 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 continuous-time quantum walk embedded in G, the relaxation times of a polymer whose structure is described by G, and the community structure of  in terms of modularity measures.
Multidisciplinary
01 Pubblicazione su rivista::01a Articolo in rivista
The exact Laplacian spectrum for the Dyson hierarchical network / Agliari, Elena; Tavani, Flavia. - In: SCIENTIFIC REPORTS. - ISSN 2045-2322. - ELETTRONICO. - 7:(2017), p. 39962. [10.1038/srep39962]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/944374
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