  Graph signal processing (GSP) deals with the representation, analysis, and
processing of structured data, i.e. graph signals that are defined on the
vertex set of a generic graph. A crucial prerequisite for applying various GSP
and graph neural network (GNN) approaches is that the examined signals are
smooth graph signals with respect to the underlying graph, or, equivalently,
have low graph total variation (TV). In this paper, we develop GSP-based
approaches to verify the validity of the smoothness assumption of given signals
(data) and an associated graph. The proposed approaches are based on the
representation of network data as the output of a graph filter with a given
graph topology. In particular, we develop two smoothness detectors for the
graph-filter-output model: 1) the likelihood ratio test (LRT) for known model
parameters; and 2) a semi-parametric detector that estimates the graph filter
and then validates its smoothness. The properties of the proposed GSP-based
detectors are investigated, and some special cases are discussed. The
performance of the GSP-based detectors is evaluated using synthetic data, data
from the IEEE 14-bus power system, and measurements from a network of light
intensity sensors, under different setups. The results demonstrate the
effectiveness of the proposed approach and its robustness to different
generating models, noise levels, and number of samples.
