  A variety of metrics have been proposed to measure the relative importance of
nodes in a network. One of these, alpha-centrality [Bonacich, 2001], measures
the number of attenuated paths that exist between nodes. We introduce a
normalized version of this metric and use it to study network structure,
specifically, to rank nodes and find community structure of the network.
Specifically, we extend the modularity-maximization method [Newman and Girvan,
2004] for community detection to use this metric as the measure of node
connectivity. Normalized alpha-centrality is a powerful tool for network
analysis, since it contains a tunable parameter that sets the length scale of
interactions. By studying how rankings and discovered communities change when
this parameter is varied allows us to identify locally and globally important
nodes and structures. We apply the proposed method to several benchmark
networks and show that it leads to better insight into network structure than
alternative methods.
