Measuring importance in complex networks

A variety of centrality measures can be defined on a network to determine the global `importance' of a node $i$. However, the inhomogeneity of complex networks implies that not all nodes $j$ will consider $i$ equally important. In this talk, we use a linearized form of the Generalized Erdos numbers [Morrison and Mahadevan EPL 93 40002 (2011)] to define a pairwise measure of the importance of a node $i$ from the perspective of node $j$ which incorporates the global network topology. This localized importance can be used to define a global measure of centrality that is consistent with other well-known centrality measures. We illustrate the use of the localized importance in both artificial and real-world networks with a complex global topology.