Fast Spread in Controlled Evolutionary Dynamics

We study a controlled evolutionary dynamics that models the spread of a novel state in a network where the exogenous control aims to quickly spread the novel state. We estimate the performance of the system by analytically establishing upper and lower bounds on the expected time needed for the novel state to replace the original one. Such bounds are expressed as functions of the control policy adopted and of the network structure, and establish fundamental limitations on the system's performance. Leveraging these results, we classify network structures depending on the possibility of achieving a fast spread of the novel state (i.e., complete replacement in a time growing logarithmically with the network size) using simple open-loop control policies. Finally, we propose a feedback control policy that using little knowledge of the network and of the system's evolution at a macroscopic level allows for a substantial speed up of the spreading process, guaranteeing fast spread on topologies where simple open-loop control policies are not sufficient. Examples and simulations corroborate our findings.