In graph theory and network analysis, node degree is defined as a simple but powerful centrality to measure the local influence of node in a complex network. Preferential attachment based on node degree has been widely adopted for modeling network growth. However, many evidences exist which show deviation of real network growth from what a pure degree-based model suggests. It seems that node degree is not a reliable measure for predicting the preference of newcomers in attaching to the network, or at least, it does not tell the whole story. In this paper, we argue that there is another dimension to network growth, one that we call node "coreness". The new dimension gives insights on the global influence of nodes, in comparison to the local view the degree metric provides. We found that the probability of existing nodes attracting new nodes generally follows an exponential dependence on node coreness, while at the same time, follows a power-law dependence on node degree. That is to say, high-coreness nodes are more powerful than high-degree nodes in attracting newcomers. The new dimension further discloses some hidden phenomena which happen in the process of network growth. The power of node degree in attracting newcomers increases over time while the influence of coreness decreases, and finally, they reach a state of equilibrium in the growth. All these theories have been tested on real-world networks.