Optimal Control of Temporal Networks With Variable Input and Node-Source Connection

Many networked systems built upon real-life physical or social interactions have time-varying connections among individual units, where the temporal changes in connectivity and/or interaction strength lead to complicated dynamics. The temporal network model was proposed in the form of controlled linear dynamical systems acting in an ordered sequence of time intervals. One of the core challenges in network science is the control of networks and the optimization of the control strategy. However, most canonical frameworks for solving optimal control problems were established for static networks featuring constant topology. New theories and techniques are yet to be developed for the temporal networks, with an important case being that the input and the source-node connection are both variables. In this work, by formulating a quadratic energy cost without solving the Riccati differential equation, we show that the control effort can be reduced substantially by improving either the system trajectories or the input matrices. The two approaches are further combined in a coordinate descent framework, integrating linearly constrained quadratic programming, and a projected gradient descent method. Taken together, the results underline the potential of temporal networks as energy-efficient control systems and present strategies to improve the control input. Moreover, the proposed algorithms can serve as a starting point for future engineering of real-world temporal networks.