  We present a novel characterization of complex networks, based on the
potential of an associated Schr\"odinger equation. The potential is designed so
that the energy spectrum of the Schr\"odinger equation coincides with the graph
spectrum of the normalized Laplacian. Crucial information is retained in the
reconstructed potential, which provides a compact representation of the
properties of the network structure. The median potential over several random
network realizations is fitted via a Landau-like function, and its length scale
is found to diverge as the critical connection probability is approached from
above. The ruggedness of the median potential profile is quantified using the
Higuchi fractal dimension, which displays a maximum at the critical connection
probability. This demonstrates that this technique can be successfully employed
in the study of random networks, as an alternative indicator of the percolation
phase transition. We apply the proposed approach to the investigation of
real-world networks describing infrastructures (US power grid). Curiously,
although no notion of phase transition can be given for such networks, the
fractality of the median potential displays signatures of criticality. We also
show that standard techniques (such as the scaling features of the largest
connected component) do not detect any signature or remnant of criticality.
