Figure 1

Density of eigenvalues for the graph with $\alpha =1/2, M/N=18, L/N=50$, and size $N=30000$. The straight line is the power $\propto z^{-3}$. In the inset, the cumulative degree distribution in the A set, for the same graph. The straight line is the power $\propto k^{-2}$.

Figure 2

Detail of the density of eigenvalues for the graph with $\alpha =1/2, M/N=18, L/N=10$.

Figure 3

Density of eigenvalues for the graph with $\alpha =1/2, N=30000, M/N=1, L/N=50$. The straight lines are the powers $\propto z^{-1/2}$ (dashed) and $\propto z^{-3}$ (solid). In the inset, the averaged logarithm of the inverse participation ratio for the graph with the same values of $\alpha , M/N$ and $L/N$ and sizes $N=30000$ ($\circ$), $N=10000$ ($\square$), $N=3000$ ($▵$), and $N=1000$ ($\diamond$).

Figure 4

Averaged logarithm of the inverse participation ratio for the graph with $\alpha =1/2, M/N=18$ and $L/N=50$ and sizes $N=30000$ ($\circ$), $N=10000$ ($\square$), $N=3000$ ($▵$), and $N=1000$ ($\diamond$).

Figure 5

Detail of the averaged logarithm of the inverse participation ratio, versus the rescaled eigenvalue $z/N^{\nu }$, with $\nu =0.3$. The graph parameters are $\alpha =1/2, M/N=18$ and $L/N=50$ and sizes $N=30000$ ($\circ$), $N=10000$ ($\square$), $N=3000$ ($▵$), and $N=1000$ ($\diamond$). The straight line is the dependence $3{log}_{10}(z/N^{\nu })+\mathrm{const}.$

Figure 6

Detail of the averaged logarithm of the inverse participation ratio for the graph with $\alpha =1/2, M/N=18$, and $L/N=50$ and sizes $N=30000$ ($\circ$), $N=10000$ ($\square$), $N=3000$ ($▵$), and $N=1000$ ($\diamond$).

Figure 7

Data collapse of the averaged logarithm of the inverse participation ratio using the formulas (6) and (7). The data used are identical to those in Fig. 6. The meaning of the symbols is also the same. In the inset, extrapolation of the size-dependent estimates $z_c\left(N\right)$ to infinite size. In this way the position of the mobility edge $z_{\mathrm{mob}}$ is established.

Figure 8

Detail of the averaged logarithm of the inverse participation ratio for the graph with $\alpha =1/2, M/N=18$, and $L/N=10$ and sizes $N=30000$ ($\circ$), $N=10000$ ($\square$), $N=3000$ ($▵$), and $N=1000$ ($\diamond$).

Figure 9

Dependence of the estimated position of the mobility edge on the parameter $\alpha$, for graphs with parameters $M/N=18$ and $L/N=50$.

Figure 10

Dependence of the estimated position of the mobility edge on the parameter $L/N$ for $\alpha =1/2$ and $M/N=18$ ($\circ$), $\alpha =1/2$ and $M/N=24$ ($\diamond$), $\alpha =1/2$, and $M/N=9$ ($▵$), $\alpha =3/4$ and $M/N=18$ ($\square$). In the inset, the dependence on $M/N$ for $\alpha =1/2$ and $L/N=50$.

Figure 11

This graph shows correlation between average degree $\langle d\rangle$ and inverse participation ratio. Each point corresponds to a single eigenvector in the lower part of the spectrum, $z<30$. The parameters of the graph are $\alpha =0.5, N=30000, M/N=18, L/N=50$.

Figure 12

Scaling exponents for the eigenvector moments ${\langle I_q\rangle }_z$. The averages are made over intervals of width 1 with midpoints at $z=8.5$ ($\circ$), $z=11.5$ ($\square$), $z=12.5$ ($▵$), $z=13.5$ ($\diamond$), and $z=15.5$ ($▿$). In the inset, detail of the same data. The term $q/2$ was subtracted just for better visibility of the nonlinear dependence on $q$.

Figure 13

Multifractal spectrum obtained numerically from the data shown in Fig. 12, with the same meaning of the symbols.

Figure 14

Density of eigenvalues obtained in the analysis of the empirical Amazon network. The size of the subset is $N=30000$ (solid line), $N=16000$ (dashed line), and $N=10000$ (dotted line). The straight line is the power $\propto z^{-2.2}$. In the inset, detail of the same data.

Figure 15

Averaged logarithm of the inverse participation

ratio, for the empirical Amazon network. The size of the subset is $N=30000$ ($▵$), $N=16000$ ($\square$), and $N=10000$ ($\circ$).

Figure 16

Detail of the data shown in Fig. 15. The size of the subset is $N=30000$ (solid line), $N=16000$ (dashed line), and $N=10000$ (dotted line).

Figure 17

Distribution of eigenvalue spacings, for eigenvalues within the interval $(20,22)$ ($\circ$, solid line in inset), $(101,111)$ ($+$, dashed line in inset), and $(200,210)$ ($\times$, dotted line in inset). In the inset, the same data are plotted in rescaled form, to show the behavior in the tail. The parameters of the graph are $\alpha =1/2, M/N=18, L/N=50$, and size $N=1000$. The full line in the main plot and the straight line in the inset are the Wigner surmise $P_{\mathrm{\Delta }}\left(\mathrm{\Delta }{z}_{\mathrm{norm}}\right)=(\pi \mathrm{\Delta }{z}_{\mathrm{norm}}/2)exp(-\pi {\left(\mathrm{\Delta }{z}_{\mathrm{norm}}\right)}^2/4)$.

Figure 18

Distribution of eigenvalue spacings, for eigenvalues within the interval $(8,9)$. The parameters of the graph are $\alpha =0.5, M/N=18, L/N=50$ and size $N=1000$ ($+$, solid line in the inset), $N=3000$ ($\times$, dashed line in the inset). In the inset, the same data are plotted in logarithmic scale, to show the behavior in the tail. The full line in the main plot and the straight line in the inset is the dependence $exp(-\mathrm{\Delta }{z}_{\mathrm{norm}})$.

Figure 19

Spacing variation parameter, averaged over eigenvalues within the interval of width 1 centered at $z$. The parameters of the graph are $\alpha =1/2, M/N=18, L/N=50$ and sizes $N=30000$ ($\circ$), $N=10000$ ($\square$), $N=3000$ ($▵$), and $N=1000$ ($\diamond$). The horizontal arrows indicate the values which correspond to Gaussian orthogonal ensemble and to the Poisson placement of eigenvalues.

Figure 20

Relative variance of the eigenvalue spacing distribution within the interval of width 1 centered at $z$. The parameters of the graph are $\alpha =1/2, M/N=18, L/N=50$ and sizes $N=30000$ ($\circ$), $N=10000$ ($\square$), $N=3000$ ($▵$), and $N=1000$ ($\diamond$). The horizontal arrows indicate the values which correspond to Gaussian orthogonal ensemble and to the Poisson placement of eigenvalues.