Multifractality in random networks with power-law decaying bond strengths

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 ${\mu }_c\equiv {\mu }_c\left(\alpha \right)$ for $\alpha <1$. 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 $exp〈ln{I}_q〉$.

Figure 1

Relative fluctuation of the participation number, $\eta$, as a function of $\mu$ for the dPBRM model for selected values of the sparsity: (a) $\alpha =0.06$, (b) $\alpha =0.08$, (c) $\alpha =0.1$, (d) $\alpha =0.4$, (e) $\alpha =0.6$, and (f) $\alpha =0.8$. In each panel we show curves for different network sizes (arrows indicate increasing $N)$.

Figure 2

(a) The critical value ${\mu }_c$ as a function of the sparsity $\alpha$ for the dPBRM model. Here, we define error bars as the width of the $\mu$ region where the curves of Fig. 1 cross. (b) Relative fluctuation of the participation number, $\eta$, at $\alpha \left({\mu }_c\right)$.

Figure 3

Probability distribution functions of the participation number $P(ln{I}_2)$ for the dPBRM model with sparsity $\alpha =0.3$ (upper row), $\alpha =0.6$ (middle row), and $\alpha =0.9$ (lower row). We used $\mu =0.9<{\mu }_c$ (left column), $\mu ={\mu }_c$ (middle column), and $\mu =1.2>{\mu }_c$ (right column). Each histogram was constructed from $2^{15}$ data values. Arrows indicate increasing $N$.

Figure 4

Logarithm of the typical participation numbers $〈ln{I}_2〉$ as a function of the logarithm of $N$ for the dPBRM model with sparsity $\alpha$. Dashed lines are linear fittings to the data used to extract the following correlation dimensions: $D_2(\alpha =0.3)=0.6084\pm 0.0179,D_2(\alpha =0.6)=0.7010\pm 0.0037$, and $D_2(\alpha =0.9)=0.7475\pm 0.0073$.

Figure 5

(a) Multifractal dimensions $D_q$ as a function of $q$ for the dPBRM model with sparsity $\alpha =0.3$, 0.5, 0.7, and 0.9 (from bottom to top). Red symbols correspond to the multifractal dimensions of the PBRM model, i.e., $\alpha =1$. (b) $D_q$ as a function of $\alpha$ for selected values of $q$.