Impact in informetrics and beyond

TL;DR

This paper develops a mathematical framework for measuring impact in informetrics, analyzing various impact measures, bundles, and the Lorenz order to quantify influence of sources like authors or journals.

Contribution

It introduces a formal, multi-level mathematical approach to defining and comparing impact measures, including impact bundles and Lorenz order-based impact.

Findings

Impact measures based on rank-frequency functions are formalized.

Impact bundles provide a more comprehensive impact assessment.

Non-normalized Lorenz order offers a rigorous way to compare impact levels.

Abstract

The concept of impact is one of the most important concepts in informetrics. It is here studied mathematically. We first fix a topic for which we want to find influential objects such as authors or journals, and their production, such as publications generating citations. These objects are then said to have a certain degree of impact. We work on three levels. On the first level, we need a measure for these objects, represented by their rank-frequency function, describing the number of items per source (ranked in decreasing order of the number of items): an impact measure. These measures focus on the production of the most productive sources. The h-index is one example. In paper II we study a formal definition of impact measures based on these left-hand sides of the source-item rank-frequency functions representing these objects. The second level of impact investigation is using impact bundles (or sheaves) as in paper III. As an illustration, we mention that the h-index of a function Z is defined as x for which Z(x) = x, i.e., the abscissa of the intersection of the graph of Z with the line y = x. The h-bundle is defined in the same way but now the line y = x is replaced by an increasing line through the origin: y = $\theta$.x, $\theta$ > 0. So, we have a bundle of impact measures which is more powerful to measure the impact of an object Z. Impact bundles are characterized in paper III. A third level of impact investigations involves the non-normalized form of the Lorenz curve. In papers IV and V we study global impact measures as measures that respect the non-normalized Lorenz order between the rank-frequency functions representing objects Z. We say that object Z has more impact than object Y if Y is smaller than Z in the sense of the non-normalized Lorenz order. This is the highest level of impact treatment: a mathematical definition of the concept itself.