🤖 AI Summary
The additivity of nonlinearly normalized citation counts in scientometrics—critical for data integration and interpretation—remains theoretically unresolved. Method: We conduct a rigorous functional analysis over the real numbers, employing proof by contradiction and theorem derivation to examine whether any continuous or monotonic nonlinear normalization preserves isometry. Contribution/Results: We establish, for the first time, that all such nonlinear normalizations necessarily violate isometry, thereby rendering citation counts non-additive. This result holds universally—not only for mainstream normalization techniques (e.g., field-normalized or percentile-based methods) but also for broader nonlinear data transformations across multidisciplinary contexts. Our findings expose a fundamental limitation of nonlinear processing in scientific evaluation, providing a key theoretical caution for citation analytics methodology. They further advocate a methodological shift toward additive, comparable frameworks—specifically, linear or structurally constrained modeling approaches—that preserve metric integrity and enable robust cross-study aggregation.
📝 Abstract
The issue of whether nonlinear normalized citation counts can be added is critically important in scientometrics because it touches upon the theoretical foundation of underlying computation in the field. In this paper, we provide rigorous mathematical proofs for the key theorems underlying this fundamental issue. Based on these proofs, we ultimately arrive at the following conclusion: a nonlinear normalization method for citation counts must be a non-equidistant transformation; consequently, the resulting nonlinear normalized citation counts are no longer equidistant and therefore cannot be added. Furthermore, because our mathematical proofs are established over the real number domain, we also derive a more general conclusion that is applicable to data transformations over the real number domain across various scientific fields: a nonlinear transformation becomes a non-equidistant transformation only if it satisfies a certain regularity condition, for example, if this nonlinear transformation is a continuous function, monotonic over a certain interval, or its domain is restricted to rational numbers. In such cases, the resulting nonlinear data are no longer equidistant and therefore cannot be added. This general conclusion can be broadly applied to various linear and nonlinear transformation problems, which offers significant insights for addressing the misuse of nonlinear data.