  Observability of complex systems/networks is the focus of this paper, which
is shown to be closely related to the concept of contraction. Indeed, for
observable network tracking it is necessary/sufficient to have one node in each
contraction measured. Therefore, nodes in a contraction are equivalent to
recover for loss of observability, implying that contraction size is a key
factor for observability recovery. Here, using a polynomial order contraction
detection algorithm, we analyze the distribution of contractions, studying its
relation with key network properties. Our results show that contraction size is
related to network clustering coefficient and degree heterogeneity.
Particularly, in networks with power-law degree distribution, if the clustering
coefficient is high there are less contractions with smaller size on average.
The implication is that estimation/tracking of such systems requires less
number of measurements, while their observational recovery is more restrictive
in case of sensor failure. Further, in Small-World networks higher degree
heterogeneity implies that there are more contractions with smaller size on
average. Therefore, the estimation of representing system requires more
measurements, and also the recovery of measurement failure is more limited.
These results imply that one can tune the properties of synthetic networks to
alleviate their estimation/observability recovery.
