Controlling for size in centrality scores

All measures of centrality in graphs seem to be correlated with degree, the sheer number of connections of a position. There are occasions in which one wants a measure that is not necessarily related to degree but whose relationship to degree is an empirical finding. Existing corrections, which force a lack of correlation, or which have no statistical justification, are inadequate for this purpose. Based on an algorithm developed by Snijders (1991) [Snijders, T.A.B., 1991. Enumeration and simulation methods for 0-1 matrices with given marginals. Psychometrika 56, 397-417.], for generating random graphs with fixed marginals, we suggest a measure of centrality that is logically but not necessarily empirically independent of degree. We examine the measure using data from Davis (1941) [Davis, A., Gardner, B., Gardner, M.R., 1941. Deep South. Univ. of Chicago Press, Chicago.] and Oliver (1993) [Oliver, A., 1993. New Biotechnology Firms: A Multilevel Analysis of Interorganizational Relations in an Emerging Industry. PhD dissertation, Univ. of California, Los Angeles.].