Controllability of Fractional-Order Directed Complex Networks, with Self Loop and Double Edge Structure

For that the conclusion of maximum matching is an important basic theory for controllability of complex networks, we first study the validity of maximum matching for fractional-order directed complex networks. We also develop a new analytical tool to study the controllability of an arbitrary fractional-order directed complex directed network with self loop by identifying the set of driver nodes with time-dependent control that can guide the system's entire dynamics. Through analyzing a mass of typical examples, we propose a new theory named "variant maximum matching" which is superior to the old one. Finally, we present some typical examples to prove the correctness of our conclusions.