Energy cost study for controlling complex social networks with conformity behavior

Target control of complex networks: How to save control energy

Controlling complex networks has received much attention in the past two decades. In order to control complex networks in practice, recent progress is mainly focused on the control energy required to drive the associated system from an initial state to any final state within finite time. However, one of the major challenges when controlling complex networks is that the amount of control energy is usually prohibitively expensive. Previous explorations on reducing the control energy often rely on adding more driver nodes to be controlled directly by external control inputs, or reducing the number of target nodes required to be controlled. Here we show that the required control energy can be reduced exponentially by appropriately setting the initial states of uncontrollable nodes for achieving the target control of complex networks. We further present the energy-optimal initial states and theoretically prove their existence for any structure of network. Moreover, we demonstrate that the control energy could be saved by reducing the distance between the energy-optimal states set and the initial states of uncontrollable nodes. Finally, we propose a strategy to determine the optimal time to inject the control inputs, which may reduce the control energy exponentially. Our conclusions are all verified numerically, and shed light on saving control energy in practical control.

How zealots affect the energy cost for controlling complex social networks

The controllability of complex networks may be applicable for understanding how to control a complex social network, where members share their opinions and influence one another. Previous works in this area have focused on controllability, energy cost, or optimization under the assumption that all nodes are compliant, passing on information neutrally without any preferences. However, the assumption on nodal neutrality should be reassessed, given that in networked social systems, some people may hold fast to their personal beliefs. By introducing some stubborn agents, or zealots, who hold steadfast to their beliefs and seek to influence others, the control energy is computed and compared against those without zealots. It was found that the presence of zealots alters the energy cost at a quadratic rate with respect to their own fixed beliefs. However, whether or not the zealots' presence increases or decreases the energy cost is affected by the interplay between different parameters such as the zealots' beliefs, number of drivers, final control time regimes, network effects, network dynamics, and number and configurations of neutral nodes influenced by the zealots. For example, when a network dynamics is linear but does not have conformity behavior, it could be possible for a contrarian zealot to assist in reducing control energy. With conformity behavior, a contrarian zealot always negatively affects network control by increasing energy cost. The results of this paper suggest caution when modeling real networked social systems with the controllability of networked linear dynamics since the system dynamical behavior is sensitive to parameter change.