The analysis of psychological networks in previous research has been limited to the inspection of centrality measures and the quantification of specific global network features. The main idea of this paper is that a psychological network entails more potentially useful and interesting information that can be reaped by other methods widely used in network science. Specifically, we suggest methods that provide clearer picture about hierarchical arrangement of nodes in the network, address heterogeneity of nodes in the network, and look more closely at network's local structure. We explore the potential value of minimum spanning trees, participation coefficients, and motif analyses and demonstrate the relevant analyses using a network of 26 psychological attributes. Using these techniques, we investigate how the network of different psychological concepts is organized, which attribute is most central, and what the role of intelligence in the network is relative to other psychological variables. Applying the three methods, we arrive at several tentative conclusions. Trait Empathy is the most "central" attribute in the network. Intelligence, although peripheral, is weakly but equally related to different kinds of attributes present in the network. Analysis of triadic configurations additionally shows that the network is characterized by relatively strong open triads and an unusually frequent occurrence of negative triangles. We discuss these and other findings in the light of possible theoretical explanations, methodological limitations, and future research.