  The multifractal detrended fluctuation analysis of time series is able to
reveal the presence of long-range correlations and, at the same time, to
characterize the self-similarity of the series. The rich information derivable
from the characteristic exponents and the multifractal spectrum can be further
analyzed to discover important insights about the underlying dynamical process.
In this paper, we employ multifractal analysis techniques in the study of
protein contact networks. To this end, initially a network is mapped to three
different time series, each of which is generated by a stationary unbiased
random walk. To capture the peculiarities of the networks at different levels,
we accordingly consider three observables at each vertex: the degree, the
clustering coefficient, and the closeness centrality. To compare the results
with suitable references, we consider also instances of three well-known
network models and two typical time series with pure monofractal and
multifractal properties. The first result of notable interest is that time
series associated to proteins contact networks exhibit long-range correlations
(strong persistence), which are consistent with signals in-between the typical
monofractal and multifractal behavior. Successively, a suitable embedding of
the multifractal spectra allows to focus on ensemble properties, which in turn
gives us the possibility to make further observations regarding the considered
networks. In particular, we highlight the different role that small and large
fluctuations of the considered observables play in the characterization of the
network topology.
