  Revealing the structural features of a complex system from the observed
collective dynamics is a fundamental problem in network science. In order to
compute the various topological descriptors commonly used to characterize the
structure of a complex system (e.g. the degree, the clustering coefficient), it
is usually necessary to completely reconstruct the network of relations between
the subsystems. Several methods are available to detect the existence of
interactions between the nodes of a network. By observing some physical
quantities through time, the structural relationships are inferred using
various discriminating statistics (e.g. correlations, mutual information,
etc.). In this setting, the uncertainty about the existence of the edges is
reflected in the uncertainty about the topological descriptors. In this study,
we propose a novel methodological framework to evaluate this uncertainty,
replacing the topological descriptors, even at the level of a single node, with
appropriate probability distributions, eluding the reconstruction phase. Our
theoretical framework agrees with the numerical experiments performed on a
large set of synthetic and real-world networks. Our results provide a grounded
framework for the analysis and the interpretation of widely used topological
descriptors, such as degree centrality, clustering and clusters, in scenarios
where the existence of network connectivity is statistically inferred or when
the probabilities of existence $\pi_{ij}$ of the edges are known. To this
purpose we also provide a simple and mathematically grounded process to
transform the discriminating statistics into the probabilities $\pi_{ij}$ .
