Controllability robustness of Henneberg-growth complex networks

It is important for a networked-system to have both good controllability and robustness, where the former measures the ability that the networked system can be properly steered via control to any target state form any initial state within a finite time duration, while the later reflects how well this system can regain its controllability after being destructively attacked. Empirical observations suggest that multi-loop structures are beneficial for enhancing the controllability robustness. The Henneberg-growth mechanism in social networks offers a natural manner to generate multi-clique structures that can be developed to multi-loops by assigning proper edge directions. In this paper, a series of random polygon networks are generated, forming random triangle, rectangle, pentagon and hexagon networks. Then, their controllability robustness is investigated. A realistic measure is designed for characterizing their controllability robustness, which can be used to filter out trivial performances after the network is severely destructed, so that the computation and analysis become much more efficient. Extensive simulation results suggest that, for random polygon networks, 1) non-loop polygon structures are inferior to the loop polygons, confirming that the multi-loop structures are indeed beneficial for controllability robustness; 2) polygons with more sides possess better controllability robustness; and 3) the correlation between controllability robustness and connectivity robustness is weak, implying that the two objectives cannot be enhanced in the same way.