On Tuesday this week, Pete posted an interesting comment on my posting on Collaboration in Mathematics. His thesis is that mathematicians are to an increasing extent organising themselves into research groups or teams. Each team includes a range of expertise and publishes many papers. Please read it if you have not already done so.

While I contend that examples like Andrew Wiles and Grigori Perelman demonstrate (in their work on Fermat’s Last Theorem and the Poincaré conjecture respectively) that the old paradigm of solitary genius writing world-changing paper still exists, there is no doubt that he is right.

By some kind of synchronicity, that afternoon there was a talk by Ralph Kenna in the Dynamical Systems and Statistical Physics seminar. The talk was entitled Phase transitions in the growth of groups. I was attracted by the title because “growth of groups” is a very important area in group theory, where there are many different kinds of growth that can be considered: the number of elements expressible as a word of length *n* in a given generating set; the number of subgroups of some specific type of index *n*; and (in my own work) the number of orbits on *n*-sets or *n*-tuples of a permutation group. Some very striking phenomena occur in these situations, concerning gaps in the possible growth rates, and I was interested to get any new insight about this.

But this was not what was meant. The abstract of the talk makes clear that the word “groups” is here used in the non-mathematical sense of which “groups of researchers” is an example. The abstract reads:

Groups of interacting nodes (such as research groups of interacting scientists) are considered as many-body, complex systems and their cooperative behaviour is analysed from a statistical-physics, mean-field viewpoint. Contrary to the naive expectation that group success is an accumulation of the strengths of its nodes, it is demonstrated that intra-group interactions play a dominant role. These drive the growth of groups and give rise to phenomena akin to phase transitions, where the relationship between group quality and size reduces. The hitherto intutitive notion of critical mass is quantified and measured for research groups in academia.

So, having a small gap in my timetable, and having just been thinking about groups of researchers as a result of Pete’s comment, I decided to go along and hear what he had to say.

And very entertaining it was too, if perhaps not entirely credible…

He had an extremely simple model of what determines the strength of a research group. He presented it in three stages. *N* is the size of the group, and *s* is the “strength” per head, where strength is measured by something like RAE income. Since detailed information on the 2008 RAE results is available, this provided an ideal test bed for theories.

- Assume first that the total strength of a group is just the sum of the strengths of its individuals. Then
*s*will be constant as*N*increases. - Next, assume that there are two contributions, one from individuals and one from pairs (a collaboration effect). Then
*s*should increase linearly:*s*=*aN*+*b*. - Finally, assume that there is a largest group size for optimal collaboration, say
*N*. Then_{c}*s*is given by a broken line:*s*=*a*_{1}*N*+*b*_{1}for*N*<*N*, and_{c}*s*=*a*_{2}*N*+*b*_{2}for*N*>*N*, where the slope decreases at the break:_{c}*a*_{2}<*a*_{1}.

He stopped here, invoking Occam’s razor against making the model more complex, largely because of what happened next. First, one more bit of theory: the “critical mass” of a group turns out to be *N _{c}*/2 in this model. Call a group “large”, “medium” or “small” according as its size is greater than

*N*, between

_{c}*N*/2 and

_{c}*N*, or smaller than

_{c}*N*/2. Then the optimal way to distribute a small additional amount of resource turns out to be to give it first to the medium-size groups, next to the large groups, and last to the small groups.

_{c}The amazing thing, which he showed us in a vast range of pictures, is that this model fits the RAE data astonishingly well. Moreover, it makes predictions for the optimal research group size which are mostly well in line with one’s expectations. A couple of interesting outliers were the exceptions that prove the rule:

- In computer science, the fit was not very good, and several competing pictures with quite different group size all gave about the same goodness of fit. This could be explained if computer science were not really one subject but several; this is probably true.
- In pure mathematics, the optimal group size was too small to be measured by the model. The smallest group submitted to the RAE had four people, and was already a “large” group in his classification. So even one or two mathematicians form a “large” group, which suggests that most pure mathematics is done by individuals or collaborating pairs.

His final conclusion was: Collaboration is crucial. Things like our common room are of very great importance!

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