Multifractality in random networks with power-law decaying bond strengths

Abstract

In this paper we demonstrate numerically that random networks whose adjacency matrices $\mathbf{A}$ 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 $\mu$ as $A_{mn}\propto |m-n{|}^{-\mu }$; while the sparsity is driven by the average network connectivity $\alpha$: for $\alpha =0$ the vertices in the network are isolated and for $\alpha =1$ 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 $\mu ={\mu }_c=1$, we clearly show [from the scaling of the relative fluctuation of the participation number $I_2$ as well as the scaling of the probability distribution functions $P(\ln I_2)$] the existence of the critical value ${\mu }_c\equiv {\mu }_c(\alpha )$ for $\alpha <1$. Moreover, we characterise the multifractality of the eigenfunctions of our random network model by the use of the corresponding multifractal dimensions $D_q$, that we compute from the finite network-size scaling of the typical eigenfunction participation numbers $\exp ⟨\ln I_q⟩$.

arXiv comment7 pages, including 5 figures. Submitted for publication

References in corpus (51)

• Characterization of complex networks: A survey of measurements
• Anderson Transitions
• Self-similarity of complex networks
• Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications
• Statistics of energy levels and eigenfunctions in disordered systems
• Origins of fractality in the growth of complex networks
• Spectra of "Real-World" Graphs: Beyond the Semi-Circle Law
• Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices
• How to calculate the fractal dimension of a complex network: the box covering algorithm
• Continuous-Time Quantum Walks: Models for Coherent Transport on Complex Networks
• Localization and centrality in networks
• Fluctuations of the inverse participation ratio at the Anderson transition
• Multifractality and critical fluctuations at the Anderson transition
• Skeleton and fractal scaling in complex networks
• Scaling theory of transport in complex networks
• New Class of Random Matrix Ensembles with Multifractal Eigenvectors
• Fractality in complex networks: critical and supercritical skeletons
• Scaling theory of the Anderson transition in random graphs: ergodicity and universality
• Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary
• Statistics of eigenstates near the localization transition on random regular graphs
• Quantum transport on small-world networks: A continuous-time quantum walk approach
• Quantum Google in a Complex Network
• Multifractality of wave functions on a Cayley tree: From root to leaves
• Determination of multifractal dimensions of complex networks by means of the sandbox algorithm
• Fluctuations of the correlation dimension at metal-insulator transitions
• Weighted Fractal Networks
• Critical statistics in a power-law random banded matrix ensemble
• Multifractality of complex networks
• Fluctuation of the Correlation Dimension and the Inverse Participation Number at the Anderson Transition
• Fractal and multifractal properties of a family of fractal networks
• Energy level dynamics in systems with weakly multifractal eigenstates: equivalence to 1D correlated fermions
• Anderson localization on the Cayley tree : multifractal statistics of the transmission at criticality and off criticality
• Universality in the spectral and eigenfunction properties of random networks
• Spectral properties of the Google matrix of the World Wide Web and other directed networks
• Monitoring the localization-delocalization transition within a 1D model with non-random long-range interaction
• Probing the eigenfunction fractality with a stop watch
• Trapped-ion quantum simulation of excitation transport: disordered, noisy, and long-range connected quantum networks
• Fractal scale-free networks resistant to disease spread
• Delocalization transition for the Google matrix
• Localization of eigenvectors in random graphs
• Localization properties of a tight-binding electronic model on the Apollonian network
• Role of fractal dimension in random walks on scale-free networks
• Spectral Properties of Directed Random Networks with Modular Structure
• Universality at Breakdown of Quantum Transport on Complex Networks
• Scattering at the Anderson transition: Power--law banded random matrix model
• Multifractal dimensions for critical random matrix ensembles
• Genuine localisation transition in a long-range hopping model
• Scattering and transport properties of tight-binding random networks
• Parametric invariant Random Matrix Model and the emergence of multifractality
• Quantum percolation in power-law diluted chains
• Dynamics of swollen fractal networks

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