A novel definition of volume dimension for a mass function based on a sigmoid asymptote is proposed; in particular, we extend the volume dimension of a mass function to define the volume dimensions for nodes and edges in complex networks. Furthermore, the relationship between the proposed volume dimension and the non-specificity term of the Deng entropy is shown, and the traditional volume dimension and volume dimension based on the node degree in complex networks are revisited. Our experiments show that in both real and synthetic complex networks, the volume dimension tends to follow a sigmoidal asymptote rather than the previously utilized power law asymptote.