Papers › Multifractality in random networks with power-law decaying bond strengths
Multifractality in random networks with power-law decaying bond strengths
Didier A. Vega-Oliveros, J. A. Méndez-Bermúdez, Francisco A. Rodrigues
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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=μ_c=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(lnI₂)] the existence of the critical value μ_c≡μ_c(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 D_q, that we compute from the finite network-size scaling of the typical eigenfunction participation numbers exp⟨lnI_q ⟩.
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