Multifractality in random networks with power-law decaying bond strengths

Abstract

In this paper we demonstrate numerically that random networks whose adjacency matrices A are represented by a diluted version of the Power-Law Banded Random Matrix (PBRM) model have multifractal eigenfunctions. The PBRM model describes one-dimensional samples with random long-range bonds. The bond strengths of the model, which decay as a power-law, are tuned by the parameter µ as Amn ∝ |m − n| −µ ; while the sparsity is driven by the average network connectivity α: for α = 0 the vertices in the network are isolated and for α = 1 the network is fully connected and the PBRM model is recovered. Though it is known that the PBRM model has multifractal eigenfunctions at the critical value µ = µc = 1, we clearly show [from the scaling of the relative fluctuation of the participation number I2 as well as the scaling of the probability distribution functions P (ln I2)] the existence of the critical value µc ≡ µc(α) for α < 1. Moreover, we characterise the multifractality of the eigenfunctions of our random network model by the use of the corresponding multifractal dimensions Dq, that we compute from the finite network-size scaling of the typical eigenfunction participation numbers exp ln Iq .

Geometrical and spectral study of β -skeleton graphs

“…It is important to stress that the localized phase corresponds to mostly isolated vertices while the delocalized phase identifies mostly complete graphs, see e.g. [11,19,25]. We characterize the delocalization-to-localization transition by means of topological and spectral properties; we use the standard average degree as topological measure and, within a random matrix theory approach, the nearestneighbor energy-level spacing distribution and the entropic eigenvector localization length as spectral measures.…”

“…See for example Refs. [23][24][25] where multifractality of eigenvectors has been reported in random graph models. Moreover the independence of both ln I 2 and ln I 2 on N for β = 2 in the circle-based proximity rule confirms that the corresponding eigenvectors are in the localized regime; that is, D 2 ≈ 0.…”

Topological versus spectral properties of random geometric graphs

“…It is fair to mention that both IPR and S, which quantify the extension of eigenvectors in a given basis, have been widely used to study the localization characteristics of the eigenvectors of random graphs and network models. Among the vast amount of studies available in the literature, as examples of recent studies were the IPR and S were applied on graphs studies, we can mention that: (i) the IPR facilitated the introduction of the concept of layer localization in multilayer random networks, this new concept was shown to have relevant implications in the dynamics of desease contagion in multiplex systems [45,46]; also (ii) the IPR allowed to demonstrate that the eigenvectors of random networks with power-law decaying bond strengths are multifractal objects [47]; while (iii) S was used to define universal parameters able to scale the eigenvector properties of multiplex and multilayer networks [24] and bipartite graphs [25]. In contrast, r has been scarcely used in graph studies; for a recent exception see Ref.…”

Localization of light in three dimensions: A mobility edge in the imaginary axis in non-Hermitian Hamiltonians

“…Moreover, the distribution P (P R q /P R typ q ), where P R typ q = exp ln(P R q ) , is invariant at criticality in the large system size limit. This implies that the variance of the distribution P (ln(P R q )) is independent of the size at criticality [52][53][54][55][56][57][58][59], allowing for a precise identification of the critical point.…”