Information about the relative importance of nodes and edges in a graph can be obtained through centrality measures, widely used in disciplines like Sociology. For example, eigenvector centrality uses the eigenvectors of the Adjacency Matrix corresponding to a network, to determine nodes that tend to be frequently visited. Formally established measures of centrality are degree centrality, closeness centrality, betweenness centrality, eigenvector centrality, subgraph centrality and Katz centrality. The purpose or objective of analysis generally determines the type of centrality measure to be used. For example, if one is interested in dynamics on networks or the robustness of a network to node/link removal, often the dynamical importance Characterizing the Dynamical Importance of Network Nodes and Links 
of a node is the most relevant centrality measure. For a centrality measure based on k-core analysis see ref.[ A model of Internet topology using k-shell decomposition]