  Network analysis has emerged as a key technique in communication studies,
economics, geography, history and sociology, among others. A fundamental issue
is how to identify key nodes, for which purpose a number of centrality measures
have been developed. This paper proposes a new parametric family of centrality
measures called generalized degree. It is based on the idea that a relationship
to a more interconnected node contributes to centrality in a greater extent
than a connection to a less central one. Generalized degree improves on degree
by redistributing its sum over the network with the consideration of the global
structure. Application of the measure is supported by a set of basic
properties. A sufficient condition is given for generalized degree to be rank
monotonic, excluding counter-intuitive changes in the centrality ranking after
certain modifications of the network. The measure has a graph interpretation
and can be calculated iteratively. Generalized degree is recommended to apply
besides degree since it preserves most favourable attributes of degree, but
better reflects the role of the nodes in the network and has an increased
ability to distinguish among their importance.
