Multifractality in random networks with power-law decaying bond strengths

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 A_{mn}∝|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 I_{2} as well as the scaling of the probability distribution functions P(lnI_{2})] the existence of the critical value μ_{c}≡μ_{c}(α) for α<1. Moreover, we characterize 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}〉.