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 hnorm. 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 hnorm.

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

$l= \frac{h-h_{\rm norm}}{h+h_{norm}}$

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

Notice that the “lurker index” $l$ 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 $l$ 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

$\sin\theta = l= \frac{h-h_{\rm norm}}{h+h_{norm}}$,

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 $l$ 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.

### 8 Responses to “What’s your mixing angle?”

1. Anton Garrett Says:

The point of measuring someone’s quality cum quantity of output is to enable ordering, which means coming up with a single (real!) number. So although I follow the arguments for differing metrics with interest, I don’t think much of (h,θ) as the tag for each person.

2. as peter is trying to measure two things (productivity and impact) – i think he is allowed two numbers… but it would be nice to see what the distribution of (h,θ) looks like (coded on h-rank or age or subject).

• Ian

i’ve produced a plot of (raw) h-index versus mixing angle for all 509 individuals;

Paul (39, 20.2)

3. telescoper Says:

There is another way of incorporating two numbers in the description, but it would probably be too complex

…however I should say I didn’t intend this as a particularly serious suggestion, it just seemed moderately amusing after I’d spent the afternoon working out example cross-sections using the Cabibbo mixing formalism. I also managed to prove something that I’d vaguely wondered about before – i.e. why you only need one real parameter (the angle) for two generations, while for three you need a complex 3-by-3 matrix. I would include the proof here, but it’s too boring to fit into this comments box.

Another thing I should say is that by no means am I trying to say that people who are good at collaborating should be thought of as “inferior”. It’s just a different type of approach to that of the soloist. The point is that it’s difficult to compare people from these different categories using a single number. Or two, or three, or …

• Anton Garrett Says:

On the contrary, it’s very easy to compare people using a single number (which is not the case with more DOF). It’s just difficult to *interpret* the comparison.
Yours pedantically etc

4. Of course, in writing papers as well, interference needs to be accounted for.

5. Woken Postdoc Says:

The scatter plot narrows towards the right (greater raw-h). Is the taper consistent with mere randomness in the central limit theorem, or is there an underlying signal that says something about success, groupthink and conformity?

• Woken Postdoc Says:

Oops, I see now. The plot of “lurker index” vs rank looks like there’s no trend. But isn’t it surprising that there is no clear link between rank and lurker index? Wouldn’t we intuitively expect “star” astronomers to have a higher “l”? Big Players tend to attract more camp-followers, and would therefore be expected to become more “collaboratively” oriented.