Today I’ve been preparing tomorrow’s particle physics lecture on the Cabibbo mechanism for quark mixing, which inspired me to go back to Paul Crowther’s guest post of a couple of days ago to present the data in a slightly different way.

The centrepiece of Paul’s post was the following graph which shows the distribution of two different bibliometric measures for the UK astronomical community. There is the h-index (which is the number h such that the author has h papers cited at least h times) and a normalised version of h in which each paper’s citations are divided by the number of authors of that paper before the index is formed; I call this index h_{norm}. The results are shown below:

Generally speaking the two indices track each other fairly well, but there are clearly some individuals for whom they diverge. These correspond to researchers whose main mode of productivity is through large consortia and for whom h is correspondingly much larger than h_{norm}.

The “outliers” are more easily identified by forming the ratio

which is plotted in the graph below kindly provided by Paul Crowther.

Notice that the “lurker index” is constructed to normalise out any general trend with h and the data do seem consistent with a constant mean across the ranked list. There is, however, a huge spread even among the top performers.

If this were particle physics rather than astronomy the results wouldn’t be presented in terms of a ratio like but as a *mixing angle *like the Weinberg angle or the Cabibbo angle. In this scheme we envisage each researcher’s output publication list as involving a mixture of “solo” and “collaborator” basis states, i.e.

|output>=cos(θ) |solo>+sin(θ) |collaborator>

The angle θ gives a quantitative indication of an author’s inclination to lurk in other people’s publication lists. If θ=0 then the individual’s papers are going to be all single-author affairs with no question marks over attribution of impact. If θ=90° then the individual does primarily collaborative research – perhaps he/she is a good mixer? Most researchers lie somewhere between these two extremes.

I therefore suggest that we should measure bibliometric productivity and impact not just through one “amplitude”, say h, but by the addition of a mixing angle, i.e. the whole output should be summarised as (h,θ). One could estimate the relevant angle fairly straightforwardly as

,

but alternative definitions are possible and a more complete understanding of the underlying process is needed to make this more rigorous.

Stephen Hawking has a particularly small mixing angle (~5.7°); many members of the astronomical Premiership have much larger values of this parameter. The value of θ corresponding to the average value of is about 23.5° and my own angle is about 8.6°.

And here, courtesy of the ever-reliable Paul Crowther, is a graph of mixing angle versus raw h-index for the whole crowd shown in the above diagram.

P.S. If you thinking this application of mixing angle is daft, then you should read this post.