Continuous, weighted Lorenz theory and applications to the study of fractional relative impact factors

Abstract

This paper introduces weighted Lorenz curves of a continuous variable, extending the discrete theory as well as the non-weighted continuous model. Using publication scores (in function of time) as the weights and citation scores (in function of time) as the dependent variables, we can construct an “impact Lorenz curve” in which one can read the value of any fractional impact factor, i.e. an impact factor measured at the time that a certain fraction of the citations is obtained or measured at the time a certain fraction of the publications is obtained. General properties of such Lorenz curves are studied and special results are obtained in case the citation age curve and publication growth curve are exponential functions. If g is the growth rate and c is the aging rate we show that ln c ln 1 g determines the impact Lorenz curve and also we show that any two situations give rise to two non-intersecting (except in (0, 0) and (1, 1)) Lorenz curves. This means that, for two situations, if one fractional impact factor is larger than the other one, the same is true for all the other fractional impact factors. We show, by counterexample that this is not so for “classical” impact factors, where one goes back to fixed time periods. The paper also presents methods to determine the rates c and g from practical data and examples are given.