## Abstract

This study examines long-term trends and shifting behavior in the collaboration network of mathematics literature, using a subset of data from *Mathematical Reviews* spanning 1985–2009. Rather than modeling the network cumulatively, this study traces the evolution of the “here and now” using fixed-duration sliding windows. The analysis uses a suite of common network diagnostics, including the distributions of degrees, distances, and clustering, to track network structure. Several random models that call these diagnostics as parameters help tease them apart as factors from the values of others. Some behaviors are consistent over the entire interval, but most diagnostics indicate that the network’s structural evolution is dominated by occasional dramatic shifts in otherwise steady trends. These behaviors are not distributed evenly across the network; stark differences in evolution can be observed between two major subnetworks, loosely thought of as “pure” and “applied”, which approximately partition the aggregate. The paper characterizes two major events along the mathematics network trajectory and discusses possible explanatory factors.

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## Notes

- 1.
These classifications are increasingly often suggested by authors and reviewers but are ultimately decided upon by the editors.

- 2.
See the MSC itself at http://www.ams.org/mathscinet/msc/msc2010.html for finer detail.

- 3.
Our data from 2009 is incomplete and so is omitted from the 1-year plots. We include it in 5-year plots and analyses with the expectation that the impact of the missing data on the 5-year calculations will be slight.

- 4.
The numbers are accelerating more rapidly than an exponential growth model can account for, given that the model assumes that \(\lim_{t\to-\infty}x=0\).

- 5.
Because authors, unlike publications, recur over time, comparisons like these become problematic between intervals of different duration.

- 6.
While always near zero, whether it is positive or negative depends on window size.

- 7.
We may interpret the second expression (1) as the expected productivity of a researcher chosen (uniformly) at random from those attributed by a randomly-chosen publication having given cooperativity

*a*, but*not*as the expected productivity of a researcher chosen at random from the collection of researchers who have been attributed by*some*publication of cooperativity*a*. - 8.
The first plot may be contrasted with Fig. 2 of (Glänzel 2002), which depicts a decline in productivity associated with especially high cooperativity in the mathematics literature (obtained through the SCI), in contrast to the two other scientific literatures in the same study.

- 9.
We compute the ratio, rather than the difference or another single-value comparison, to better account for the changing size and density of the network. Optimally, one would compute a test statistic like the

*Z*-score (e.g. Maslov and Sneppen 2002), but this correctly requires first generating and then running the same (expensive) statistics on a collection of random graphs. - 10.
We construct \(G_1^{\prime}\) from the subset of the literature having primary MSC ranging from 03 to 94. The analysis of this unipartite projection of

*G*_{2}onto*P*rather than*N*was another open question from (Grossman 2002). - 11.
We investigated the relationship of multidisciplinarity to cooperativity across publications, analogously to (1) though taken over publications rather than attributions, but found no substantive relationship.

- 12.
This factor derives from Newman et al. (2001) as \(1-\sum_k\frac{n_k}{n}u^k\), where

*u*is the solution to the equation 2*mu*= ∑_{ k }*n*_{ k }*ku*^{k-1}(recall that*m*is the number of edges). - 13.
Whiskers are omitted. When bound to the median by some small multiple of the interquartile range, the diameter in each case reduced the meaning of the whisker to precisely this bound; while whiskers allowed to extend to 1 and to the diameter in each interval crowd out the boxes for vertical space in the plot.

- 14.
Comparison to actual NSW models indicates that this is not an artifact of increased total size.

- 15.
The bipartite model predicts different connectivity distributions than what we observe in

*G*_{1}, so degree-dependent comparisons to this model would require somewhat deeper discussion. - 16.
It is possible for measured clustering to be lower than that predicted by the NSW model, as in Newman et al. (2001) (company directors), should very little clustering be due to distinct pairwise collaborations and many highly cooperative publications share a common pool of authors, which publications would in the model be attributed to distinct teams of researchers.

- 17.
Our code in R uses the nls function to locate maximum-likelihood estimators, i.e. those that minimize \(SSE=\sum_t{\epsilon_t}^2\).

- 18.
The threshold for cooperativity is chosen to be the values for which publication counts

*decreased*until the mid-90s. The other two thresholds are chosen so that the proportion of researchers removed to obtain the second and third alternatives most closely resembles the proportion of publications removed to obtain the first. - 19.
While we do not include it here, degree–degree correlations, measured as assortativity (Newman 2003), varies similarly.

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## Acknowledgements

The authors are grateful to the American Mathematical Society for providing access to the *MR* database and for agreeing to make the data publicly available (by request to the Executive Director). The authors thank Sastry Pantula and Philippe Tondeur for helpful information, and Sid Redner, Betsy Williams, and participants of the Summer 2010 REU in Modeling and Simulation in Systems Biology for helpful conversations and support. J. C. Brunson, S. Fassino, A. McInnes, M. Narayan, and B. Richardson were partially funded by NSF Award:477855. A. McInnes and B. Richardson were partially funded by HHMI:52006309. S. Fassino, A. McInnes, M. Narayan, and B. Richardson contributed equally.

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Brunson, J.C., Fassino, S., McInnes, A. *et al.* Evolutionary events in a mathematical sciences research collaboration network.
*Scientometrics* **99, **973–998 (2014). https://doi.org/10.1007/s11192-013-1209-z

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### Keywords

- Mathematics research
- Collaboration networks
- Evolving networks

### Mathematics Subject Classification

- 91D30
- 05C82