An important prerequisite for the study of network controllability is to prove that the system is controllable. The controllability of the network means that the entire network is controlled by applying proper external inputs or adjusting inputs to achieve the desired state. The traditional way to calculate the control input nodes of directed network is to solve the maximum matching of the bipartite graphs corresponding to the network. However

since this method does not impose restrictions on the matching manner of the network node

the node control chain is too long and the information transfer of the control input is delayed

which affects the whole controllable performance of the network. The Kalman criterion and the PBH criterion can prove whether the system is controllable within a certain period of time. However

the relationship between nodes in the network becomes more complicated as the size of the network increases

so the simple utilization of such methods increases the complexity of the operation. Therefore

the lower bound algorithm for the controllability index K (KMLA) was proposed by combining the Kalman rank criterion. Control node cluster for control input can be quickly determined by determining the lower bound of the network’s controllability index. Then based on in-degree of networks

the minimum upper bound algorithm for the controllability index K (KMUA) was proposed. It was found that the KMUA algorithm proposed in this paper could make the upper bound of the K value closer to the K value when the network reached the K-step controllable. The upper and lower bounds of the controllability index K were verified by the specific network model and the real network. Results show that the algorithm can optimize the control chain length of the driven node by combining the upper and lower bounds of the controllability index.