Fast Spread in Controlled Evolutionary Dynamics

We study the spread of a novel state in a network, in the presence of an exogenous control. The considered controlled evolutionary dynamics is a non-homogeneous Markov process that describes the evolution of the states of all nodes in the network. Through a rigorous analysis, we estimate the performance of the system by establishing upper and lower bounds on the expected time needed for the novel state to replace the original one. Such bounds are expressed in terms of the support and intensity of the control policy (specifically, the set of nodes that can be controlled and its energy) and of the network topology and establish fundamental limitations on the system’s performance. Leveraging these results, we are able to classify network structures depending on the possibility to control the system using simple open-loop control policies. Finally, we propose a feedback control policy that, using little knowledge of the network topology and of the system’s evolution at a macroscopic level, allows for a substantial speed up of the spreading process with respect to simple open-loop control policies. All these theoretical results are presented together with explanatory examples, for which Monte Carlo simulations corroborate our analytical findings.

💡 Research Summary

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The paper investigates the spread of a novel state (e.g., a new species, technology, or opinion) over a weighted undirected network when an external control can be applied to selected nodes. The underlying dynamics are modeled as a non‑homogeneous continuous‑time Markov jump process. Each edge ({i,j}) carries an independent Poisson clock of rate (W_{ij}); when the clock ticks and the incident nodes are in different states, a competition occurs. The novel state wins with probability (\beta) (assumed (>1/2) to give it an evolutionary advantage) and the original state wins with probability (1-\beta). In addition, a control vector (U(t)\ge 0) injects extra transition rate toward the novel state at the nodes where it is active.

The authors define two performance metrics: the spreading time (T=\inf{t\ge0: X(t)=\mathbf 1}) (time until all nodes are in the novel state) and the control cost (J=\int_0^{T}\mathbf 1^{\top}U(t),dt). Their goal is to bound the expectations (\mathbb{E}