Observational Equivalence in System Estimation: Contractions in Complex Networks

TL;DR

This paper investigates how contraction properties in complex networks influence their observability and measurement requirements, revealing relationships with network topology and implications for system estimation and recovery.

Contribution

It introduces a polynomial contraction detection algorithm and analyzes how contraction size relates to network properties like clustering and degree heterogeneity.

Findings

Contraction size correlates with clustering coefficient and degree heterogeneity.

Power-law networks with high clustering have fewer small contractions, easing measurement needs.

Small-World networks with high degree heterogeneity have more small contractions, increasing measurement and recovery challenges.

Abstract

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.