Observational Equivalence in System Estimation: Contractions in Complex Networks
For researchers designing or analyzing complex networks, this work provides insights into how network topology influences observability and sensor placement, though the results are incremental as they extend known contraction concepts to specific network models.
This paper establishes a link between observability in complex networks and the concept of contractions, showing that for observable network tracking, at least one node per contraction must be measured. Analyzing contraction distributions in power-law and small-world networks, they find that contraction size correlates with clustering coefficient and degree heterogeneity, affecting the number of measurements needed and observability recovery under sensor failure.
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.