  Recent works have established a novel viewpoint that treats the eigenvalue
spectra of disordered quantum systems as time-series, and corresponding
algorithms such as singular-value-decomposition has proven its advantage in
studying subtle physical quantities like Thouless energy and non-ergodic
extended regime. On the other hand, algorithms from complex networks have long
been known as a powerful tool to study highly nonlinear time-series. In this
work, we combine these two ideas together. Using the particular algorithm
called visibility graph (VG) that transforms the eigenvalue spectra of a random
spin system into complex networks, it's shown the degree distribution of the
resulting network is capable of signaturing the eigenvalue evolution during the
thermal to many-body localization transition, and the networks in the thermal
phase have a small-world structure. We further show these results are robust
even when the eigenvalues are incomplete with missing levels, which reveals the
advantage of the VG algorithm.
