\section*{Introduction}
Measuring the research performance of scholars plays a critical role
in the allocation of scholarly resources at all levels
``quantitative'' means of measurement has long been through the use of
citations \cite{Garfield55,desollaprice}.  Citations are routinely
used to evaluate the research productivity of individuals
citations to measure research performance involves several confounding
factors which tend to become
more important as the degree of aggregation decreases.  For the
evaluation of individuals, important challenges are:
\begin{description}
\item[Discipline]  Citation practices vary widely among various fields.
Citation rates can vary between disciplines by an order of
magnitude \cite{leydes}; among sub-disciplines in the same discipline they can
vary by a factor of two.

\item[Co-Authorship] A paper can have an arbitrary number of authors, from
one to several thousand.  Should an author of a single authored paper
receive the same credit for a citation as someone who has co-authors?

\item[Age] The number of citations accrued by an individual scales
with the square of his/her career length \cite{hindex,kurtz05b};
thus, a person with a career length of 10 years will have half the
citations of an equal person with a career length of 14.14 years. This
age effect problem is exacerbated by the fact that the two
aforementioned challenges are time dependent. For example, in the
field of astrophysics, both the mean number of references and the mean
number of authors have approximately doubled in the last 20 years.
\cite{Henneken2011,Schulman1997}
\end{description}

Some of the lesser challenges associated with using citations to measure research productivity of individuals are:
\begin{description}
\item[Self-Citation] If an author cites papers by him/herself should they
count as much as citations from papers by others?

\item[Curation] In addition to having a database of articles and
citations, one must clean and curate its data. For example, 
an analysis of an individual's productivity requires that
one be able to exactly identify the articles written by that
individual. Name changes (e.g., due to marriage) and homonyms (name clashes, where different people have the same name) can make this a serious
problem.

\item[Shot Noise] Sometimes an individual can, almost entirely by
chance, become an author of one or more very highly cited papers,
perhaps as a student.  The citation distribution is a Zipf like power
law, whereby some articles are cited thousands of times more than the
median; clearly, there can be circumstances where a direct count of
citations is not a fair representation of impact.
\end{description}

In a highly influential paper, Hirsch \cite{hindex} proposed a pair of
citation-based measures (\textit{h, m}) which: solve the shot-noise
problem, substantially improve the age problem, and help with the
curation difficulty, discussed above. The Hirsch index, \textit{h}, is
the position in a citation ranked list where the rank equals the
number of citations; absent shot noise {\it h} is obviously proportional
to the square root of the total number of citations, which grows
linearly with career length \cite{hindex,kurtz05b}.  The \textit{m}
quotient is \textit{h} divided by career length, and is a constant
throughout the career of an individual with constant productivity in a
constant environment.

The \textit{h}-index is by far the most widely used
indicator of personal scientific productivity. As such, it has been greatly
reviewed and criticized in specialized literature and innumerable alternatives have been
proposed \cite[for a review]{egghes}. Some notable substitutes of the h-index include: 
the mean number of citations per paper \cite{Lehmann:2006fk}, the \textit{e}-index which complements the h-index for excess citations \cite{eindex}, the \textit{g}-index, similar to {\it h}, but differs for it accounts for the averaged citation count an author has accrued \cite{gindex}, and the highly cited publications indicator \cite{inconsistency}.
Two normalizations of the {\it h}-index which have been proposed in the literature with promising results are by the number of article co-authors \cite{batista}, and by the average number of citations per article per
discipline \cite{radicchi}. The measures proposed in this article use both of these normalizations, combined.

While the \textit{h}-index is a valuable, simple, and effective
indicator of scholarly performance, we find that it is inadequate for
cross-disciplinary and historical comparisons of
individuals. Comparing two scholars from different disciplines or from
different time periods, or with differing co-authorship practices,
based on their \textit{h}-index would very likely yield erroneous
results, simply because citation and authorship practices have changed
(and constantly change) across disciplines and through time.


