Collaborative scalar-gain estimators for potentially unstable social dynamics with limited communication

In this paper, we study the estimation of potentially unstable social dynamics—e.g., social and political movements, environmental and health hazards, and global brands; when they are observed by a geographically distributed set of agents. We are interested in scenarios when the information exchange among the agents is limited. This paper considers a generalization of distributed estimation to vector (non-scalar) and dynamic (non-static) cases. As we will show, when the state-vector evolves over time, the information flow over the communication network may not be fast enough to track this evolution. In this context, the key questions we address are: (i) can a distributed estimator with limited communication track an unstable system? and; (ii) what is the cutoff point beyond which the given observations and the agent topology may not result into a bounded estimation error? To address these questions, we present a scalar-gain estimator and characterize the relation between the system instability and communication/observation infrastructure. We derive and analyze the aforementioned cutoff point as the Scalar Tracking Capacity, and further show that unstable vector systems can be distributedly estimated with bounded error.