This paper introduces the concept of Lyapunov V-stability for complex dynamical networks. Under the new framework, each dynamical node is associated with a passivity degree, which indicates to what extent an effort is required for stabilizing the node. From this approach, the network stability problem is converted to measuring the negative definiteness of one simple matrix that characterizes the topology of the network. Pinning control is then suggested and investigated based on the new V-stability formulation. As an illustrative example, a network with different node dynamics and non-uniform coupling strengths is simulated to verify the analytic results. Moreover, a comparison study for three different kinds of networks is provided to further illustrate the novelty and efficacy of the proposed V-stability criterion and stabilization scheme. (C) 2007 Elsevier Ltd. All rights reserved.