  In this paper, we propose to study on sufficient control of complex networks
which is to control a sufficiently large portion of the network, where only the
quantity of controllable nodes matters. To the best of our knowledge, this is
the first time that such a problem is investigated. We prove that the
sufficient controllability problem can be converted into a minimum cost flow
problem, for which an algorithm can be easily devised with polynomial
complexity. Further, we study the problem of minimum-cost sufficient control,
which is to drive a sufficiently large subset of the network nodes to any
predefined state with the minimum cost using a given number of controllers. It
is proved that the problem is NP-hard. We propose an ``extended
$L_{\mathrm{0}}$-norm-constraint-based Projected Gradient Method" (eLPGM)
algorithm which may achieve suboptimal solutions for the problems at small or
medium sizes. To tackle the large-scale problems, we propose to convert the
control problem into a graph algorithm problem, and devise an efficient
low-complexity ``Evenly Divided Control Paths" (EDCP) algorithm to tackle the
graph problem. Simulation results on both synthetic and real-life networks are
provided, demonstrating the satisfactory performance of the proposed methods.
