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Evolutionary Events in a Mathematical Sciences Research Collaboration

Network

J.C. Brunsona,1, S. Fassinob,1,3, A. McInnesc,1,2,3, M. Narayand,1,3, B. Richardsonc,1,2,3, C. Francke,a,P. Ionf, R. Laubenbachera,∗

aVirginia Bioinformatics Institute, Washington St, MC 0477, Virginia Tech, Blacksburg, VirginiabDepartment of Mathematics, 227 Ayres Hall, University of Tennessee, Knoxville, TN 37996

cDepartment of Mathematics and Computer Science, Oakwood University, Cooper Complex Bld. B, 7000 Adventist Blvd,Huntsville, AL 35896

dLyman Briggs College, Michigan State University, 35 East Holmes Hall, East Lansing, MI 48825eLaboratory for Interdisciplinary Statistical Analysis, 212 Hutcheson Hall (0439), Blacksburg, VA 24061

fMathematical Reviews, P.O. Box 8604, Ann Arbor, MI 48107

Abstract

Collaboration is key to scientific research, and increasingly to mathematics. This paper contains alongitudinal investigation of mathematics collaboration and publishing using the proprietary databaseMathematical Reviews, maintained by the American Mathematical Society. The database contains pub-lications by several hundred thousand researchers over 25 years. Mathematical scientists became moreinterconnected, collaborative, and interdisciplinary over this interval, and twice the network experienceddramatic structural shifts. These events are examined and possible external factors are discussed. Smallersubject-specific subnetworks exhibit behavior that provides insight into the aggregate dynamics. Thedata are available upon request to the Executive Director of the AMS.

Keywords: mathematics research, collaboration networks, evolving networks2010 MSC: 91D30

1. Introduction

Collaboration networks have been studied ex-tensively in recent years, thanks to the availabil-ity of several excellent databases, e.g. [Gro02,New01b, BJN+02, FLC+04, AOL+07, TL07,

∗Corresponding author: Virginia Bioinformatics Insti-tute, Washington St, MC 0477, Blacksburg, VA 24060.Tel.: (540) 231-7506. Fax: (540) 231-2606.

Email addresses: jabrunso@vbi.vt.edu

(J.C. Brunson), sfassino@utk.edu (S. Fassino),antonio.mcinnes@oakwood.edu (A. McInnes),nadaraj2@msu.edu (M. Narayan), brya_08@yahoo.com(B. Richardson), chfranck@vbi.vt.edu (C. Franck),ion@ams.org (P. Ion), reinhard@vbi.vt.edu(R. Laubenbacher)

1These authors were partially funded by NSFAward:477855.

2These authors were partially funded byHHMI:52006309.

3These authors contributed equally to this project.1These authors were partially funded by NSF

Award:477855.2These authors were partially funded by

HHMI:52006309.3These authors contributed equally to this project.∗To whom correspondence should be addressed. Ad-

dress: Bioinformatics Facility, Washington Street, PostalCode 0477; phone: (540) 231-7506; fax: (540) 231-2606;email: reinhard@vbi.vt.edu.

Per10]. These studies have revealed a diversity oftopological structure, especially across disciplines,depicting typical ranges of basic graph-theoreticmetrics across real-world networks. While lon-gitudinal studies are increasingly common, theypredominantly take a cumulative approach; theyobserve network growth after a designated start-ing year, which may then be compared to evolvinggraph models [BJN+02]. However, evolving real-world networks can take decades to exhibit clearlong-term trends [RB10], and short-term changesin structure and behavior become obscured byaggregating information [TL07]. To strengthenmodels of scientific research collaboration, cumu-lative models must be supplemented by dynamicmodels that capture the effective relationshipsamong researchers [TL07, KW06]. Furthermore,while collaboration networks are often treated inthe larger context of complex networks, impor-tant differences exist between social networks andother real-world networks [NP03]. Most availablepublishing databases are too specialized (by disci-pline or region) to exhibit clear long-term trends.

A specialized theory of evolving social networksis therefore required, and is underway. In thispaper we examine a large, longitudinal collabo-

Preprint submitted to Social Networks March 26, 2012

ration network. The American Mathematical So-ciety (AMS) maintains the proprietary databaseMathematical Reviews (MR), and we study thisdatabase across 1985–2009, during which nearly430,000 authors produced nearly 1.6 million pub-lications. MR aims to catalogue every mathemat-ical sciences publication each year, including bothprint and online journals, books, proceedings, andother publications [Jac97]. We therefore treat thedatabase as a census of the literature; however,we caution that the mathematics literature is it-self a fraction of the broader scientific literatureand highly entangled therewith.4 The network ismuch larger than most studied scientific collabo-ration networks, extends over a longer time, andis of consistently great size, which will allow us tocharacterize long-term trends and fluctuations.

2. Materials and Methods

Our data consist, for each publication, of en-coded author IDs, subject classifications from theAMS Mathematics Subject Classification Scheme[Soc11], and the year of publication. Whileauthors and publications, taken together, ex-hibit a bipartite structure, and bipartite mod-els that preserve this structure show promise[BMG03, GMY05, ZWL+08], the larger literatureand better-understood statistical toolkit on uni-partite models allows us to better contextualizeour network. We therefore adopt a unipartitemodel. In this model, nodes v1, . . . , vn correspondto authors and links vivj (m total) indicate coau-thorship. Each link vivj = vjvi receives a (collab-oration) weight wij given by the number of jointpublications by vi and vj [New04]. The graphevolves over time as authors begin and cease pub-lishing.

We investigated the evolving topology of thenetwork using several well-understood graph-theoretic metrics. To account for the publish-ing process underlying this structure while main-taining our unipartite perspective, we introducedpublication-sensitive analogs to the strictly graph-theoretic assortativity and clustering coefficients.These metrics reveal network properties not cap-tured by the originals and may warrant furtheruse.

4Our network not necessarily more bibliographicallycomplete or self-contained than previously studied collab-oration networks (such as the Los Alamos preprint archivein [New01b]); non-mathematician authors may appear onmathematics publications no less frequently than physi-cists who abstain from online databases collaborate withauthors who do not.

Table 1: The MR network over two intervals.MR network 1940–2000 1985–2009

years 61 25papers 1598 1599authors 337 429avg. authors/paper 1.45 1.75avg. papers/author 6.9 6.5collab. pairs 496 876avg. no. coauthors 2.9 4.1prop. in largest comp. .62 .75avg. separation 7.56 7.31global clustering coeff. .15 .14avg. clustering coeff. .34 .61assortativity .12 .069

Subject classifications within the MR databaseinclude two-digit prefixes from 01 to 97. We di-vided the literature coarsely into “pure” (03–58)and “applied” (60–95) subnetworks and for somespecific analyses into the similarly-sized subclas-sifications indicated in Fig. 2.5

To trace the effective structure of these net-works, we used, depending on the metric,nonoverlapping intervals of one year or of fiveyears or sliding windows of 5 years. The choiceof 5-year intervals offers meaningful comparisonsto [New01b]. In plots, we identify each windowby its last year; for instance, the year 1997 mayrefer to the interval 1993–7. Because the networkgrows most quickly from 1985 to 1989, and be-cause data is not complete in the most recentyears, we focused mainly on the period 1989–2007.The smaller subnetworks fluctuated widely, ob-scuring long-term trends, but their behavior illu-minates trends in the aggregate by distinguishingthe disciplines most reflective of, and plausiblyresponsible for, those trends.

3. Trends in Mathematical Publishing

We examined long-term trends exhibited by theMR network. We present the publishing data ina raw statistical analysis, emphasizing the rela-tionship of output rates to coauthorship and tomultidisciplinarity.Table 1 compares our network (all 25 years

taken together) with the MR network studied in[Gro02]. Several differences detectible in the ta-ble reflect long-term trends discussed below, in-cluding increased collaboration (rows 4, 6, 7) andgreater network connectivity (rows 8, 9, 11).

5This scheme is imperfect. For instance, much of 60(Probability Theory and Stochastic Processes) might beclassified as pure mathematics, but this would split 60 from62 (Statistics).

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Figure 1: Across nonoverlapping 5-year intervals: (a)Numbers of authors with 1, 2, 3, and 4 publications. (b)Numbers of publications by 1, 2, 3, and 4 authors. (c)Numbers of authors with 0, 1, 2, and 3 coauthors. (d)Average number of publications by authors of a paper, asa function of the number of authors on the paper.

3.1. Publishing rates

We measured publishing rates individually andcollaboratively. Mathematics researchers havegrown more numerous and collaborative at accel-erating rates, though without becoming steadilymore prolific (Fig. 1 (a–c)). In fact, in recentyears highly collaborative projects have involvedauthors less prolific within mathematics, and av-erage prolificity has declined (Fig. 1 (d) and 3 (a–c)). While the number of more prolific authorshas accelerated, it has been outpaced by the num-ber of authors of only one publication, as we dis-cuss in the supplementary text. These trendswere starker in the applied network, which houseda greater proportion of less prolific authors, re-versed its trend from more to less prolific yearsearlier than the pure, and a greater surge in one-time authors (Fig. 3 (a–f)). Credit for declin-ing average publishing rates therefore rests largelywith such authors.This surge in less prolific authors reflects a ma-

jor event around 2001 that we will describe fur-ther. A closer look reveals another event yearsearlier: a surge in collaborative publishing after1995. From the interval 1989–95 to the interval1995–2009, rates of 2- to 6-author publicationsrose and rates of 7- and more-author publica-tions reversed from decline to rise (Fig. 2 (g)).Fluctuations in subject classification assignmentsand in graph-theoretic structure illuminated theseevents, as we discuss in the next section.

3.2. Multidisciplinarity

The literature grew steadily more multidisci-plinary, except for a brief period of specialization

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Figure 2: Across 1-year intervals, 1985–2007: (a) Numberof publications. (b) Number of authors. (c) Number ofpublications across subnetworks. (d) Number of authorsacross subnetworks. (e) Average number of authors perpublication. (f) Average number of subject classificationsper publication. (g) For each fixed number of authors(1–8), a histogram over years of publications attributedto that number of authors. Each histogram is scaled bythe total number of publications by that number of au-thors. We include four subdisciplines: algebra (◦; 08–22),differential equations (�; 34–35), computer science and in-formation (⋄; 68, 94), and classical physics (△; 70–86).

near 1995, as measured by the average number ofsecondary subject classifications 〈s〉 = 〈si〉i (as iranges across publications). Meanwhile the aver-age number of secondary authors per publication〈a〉 = 〈ai〉i (authors beyond the requisite one) in-creased monotonically (Fig. 2 (e,f)). While theapplied network exhibited larger 〈a〉 but smaller〈s〉, a regression model reveals a positive relation-ship between si and ai that is stronger in the moremultidisciplinary pure network (Fig. 3 (g–i)). Wefit to the combined pure and applied literature thelinear model

si = α0 + α1ai + α2ui + α3aiui + ǫi, (1)

where the indicator ui takes the value 0 if thepublication is classified as pure and 1 otherwise.The parameter α1 is then the effect of ai in thepure network, α1 + α3 that of ai in the ap-

3

plied, and α3 the interaction effect of ai andui. This coauthorship–multidisciplinarity rela-tionship weakened over time, but the subnetworksgrew variably similar and dissimilar over differ-ent intervals. Shifts in α3 coincided with the twoevents: the pure and applied networks grew simi-lar after the earlier event but dissimilar after thesecond.

4. Evolution of the Coauthorship Graph

We adopt a graph-theoretic approach to studyconnectivity, correlations, and clustering in termsof coauthorship. We made use of several graph-theoretic metrics. We performed calculations onlargest connected components unless otherwisenoted, for two reasons: (1) The same fluctuationsare visible in time series for entire (disconnected)graphs, though often subdued. (2) The steadilyshrinking proportion of nodes outside largest com-ponents affects statistics sensitive to the presenceof isolated authors and to highly connected, in-dependent teams (two common forms that smallconnected components take).

4.1. Individual and network connectivity

We measured connectivity three principal ways.The number ki of coauthors of an author vi is thatauthor’s degree, a measure of individual connec-tivity. With an increase in average degree comesan increase in graph density D = m/ 1

2n(n − 1),

the proportion of possible node–node links thatare realized. We may also measure global connec-tivity by the proportion of nodes subsumed bythe largest connected component itself. Finally,we gain insight into the efficiency of this connec-tivity from the mean node–node separation withinthis component. We adopted the harmonic meanseparation 〈ℓ〉 between pairs of authors defined by

〈ℓ〉−1 =∑

i,j

ℓij−1/ 1

2n(n− 1),

rather than the arithmetic mean, to place empha-sis on local connections [LM01]. (The metrics arenonetheless highly correlated. Taking their resid-uals from linear fits each year as ordered pairsproduces r = .995, though the arithmetic meanvaries more about its fit.)While the average degree 〈k〉 = 2m/n in-

creased, the rate of increase over 1994–2009 wasalmost double that over 1989–1994, predomi-nantly due to applied publications (Fig. 1 (c)and 3 (d,e)). The change in pace of average de-gree after 1994, especially in the applied network,is consistent with the surge in collaboration ob-served above. Meanwhile, the largest component

of the aggregate network absorbed greater propor-tions of authors, from 37% (1989) to 65% (2009),though this trend decelerated. These proportionsspan the typical range for collaboration networks[Gro02, New01b, BJN+02, TL07, Per10], suggest-ing that the proportional rise will continue todecelerate as the networks approach a practicalupper limit on collaboration network cohesion.The pure and applied subnetworks conglomeratedsimilarly, though they exhibited different meanseparation, with the applied network consistentlymore dispersed (Fig. 3 (f)). Generally, 〈ℓ〉 de-creases as D increases, and while residuals fromlinear fits of these metrics exhibited some corre-lation (r = −.63), the pure network was consis-tently tighter despite the greater density of theapplied. This implies that the structure of col-laboration varies in important ways, among disci-plines and over time, and we studied this structurethrough coauthor correlations and clustering.

4.2. Correlations among collaborators

A network is assortative, or exhibits assortativemixing, when similar pairs of nodes are preferen-tially linked, disassortative when linking is prefer-entially dissimilar, and nonassortative otherwise[New03]. The normalized degree correlation coef-ficient rcol measures assortative mixing by num-ber of collaborators. We supplemented rcol witha measure rpub of assortative mixing by numberof publications. (See the supporting informationfor a formal definition.)

Collaboration networks are known to be assor-tative by collaborators (.1 < rcol < .4) but pre-vious studies indicate that mathematics networksare less so [TL07, New03]. We also found rcol tobe positive but low in the aggregate, pure, and ap-plied networks, though some subdisciplines werelargely nonassortative (Fig. 4 (a,d)). Mathemat-ics researchers were more strongly correlated bypublishing rate (.3 < rpub < .6). The applied net-work and subnetworks exhibited stronger correla-tions by both metrics, signifying more hierarchicalorganization.

The events come into sharper focus throughthese correlation coefficients. Around 1995 thenetwork shifted from progressively disassorta-tive mixing to progressively assortative, mostlywith respect to collaborators and predominantlyamong applied researchers. After 2001 this trendreversed again as coauthors became less corre-lated with respect both to collaborators and topublications, and in the latter case earlier in thepure network.

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Figure 3: Across 5-year sliding windows, 1985–9 to 2005–9: Average number of (a) publications per author, (b) collaborativepublications per author, and (c) publications per pair (zeros omitted from each average). (d) Average degree 〈k〉. (e)Residuals of 〈k〉 from best linear fits. (f) Harmonic average separation. (g–i) Estimates of α1, α1 + α3, and α3 in theregression model (1).

4.3. Scale-freenessRecently the graph-theoretic statistic s(g) =

∑

kikj , a sum taken over edges of graph g,has been used to quantify “scale-freeness” amonggraphs with a common (scaling) degree sequence[LADW05]. This s-metric is greatest where high-degree nodes are linked preferentially, producinga highly connected “hub-like” core. The metric

S(g) =s(g)− smin

smax − smin

(refined in [Li07]) normalizes s over the range of s-values across graphs of the same degree sequenceas g, and therefore has range [0, 1]. S may alsobe interpreted as a similar normalization of rcolacross this collection of graphs.The S-metric is best understood across graphs

with a power law degree sequence, and whilethe degree sequences of collaboration networksare not well-modeled by power laws [LADW05,New01c], power-law approximations are popular[Gro02, New01b] and helpful in distinguishing col-laboration networks from other categories of net-works [ASBS00]. Recent studies apply S to sev-eral model networks [LADW05, THL06, THLH07,BC08] but applications to social networks are lim-ited [Hsi09]. We observed .48 < S < .58. Thetime series for S reveals that fluctuations in rcolmay be interpreted in the context of gradually di-minishing scale-freeness (Fig 4 (c)).

4.4. ClusteringWhereas 〈k〉, rcol, rpub, and S measure individ-

ual and pairwise structure, clustering measures

structure concerning triples. Among triples of au-thors a, b, and c where a and b collaborated and aand c collaborated, the (global) clustering coeffi-cient C expresses the proportion for whom b and calso collaborated. Disassortative graphs permit areduced number of possible triangles among nodesof different degrees, and thus admit a smallerrange of values for C. This may be globally ac-counted for using relative probabilities [New01a],which we consider in the supplementary text, butwe principally adopted a clustering coefficient Cdesigned to correct for this locally [SV05]. Theaggregate network showed low clustering, .22 <C < .27, compared to other coauthorship graphs[New01c, BJN+02, TL07, Per10]. The correctiondoubled the range to .46 < C < .53, as observedin other networks [SV05].

We also introduced an exclusive clustering co-efficient: Among triples of authors where a and bcollaborated without c, a and c collaborated with-out b, and both b and c published at least twice,C× is the proportion for whom b and c collabo-rated without a. C× detects changes in coauthor-ship that cannot be explained by team collabo-ration, and distinct pairwise publications suggeststronger, transitive relationships than single com-mon publications.6

Locally, the clustering coefficient ci of an au-thor vi is the proportion of the ki(ki − 1)/2 pairs

6While clustering coefficients have been introducedfor bipartite author–publication graphs [RA04, ZWL+08],they do not address this issue directly.

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Figure 4: Across 5-year sliding windows, 1985–9 to 2005–9: (a) Assortative mixing by number of coauthors, rcol. (b)Assortative mixing by number of publications, rpub. (c) The S-metric. (d) rcol across subdisciplines. (e) rpub across

subdisciplines. (f) S across subdisciplines. (g) Global clustering C corrected for degree assortativity: (h) Global clusteringC× based on pairwise exclusive publications. (i) Average local clustering corrected for degree assortativity across authorsof fixed degree 4, 7, and 10.

of their collaborators who have themselves col-laborated. Again we adopted the correction cifrom [SV05]. We used the network-wide aver-age 〈c〉 in our change point analysis (Table 2),and we stratified authors by degree in Fig. 4 (i)to compare clustering across differently-connectedresearchers.

We found patterns of clustering to reaffirm thatthe two events were driven by larger teams of col-laborators. While C× ≤ C ≤ C by definition,in our network CX was comparatively tiny, with.0041 < C× < .0051 (Fig. 4 (g,h)). This in-dicates that highly collaborative projects droveoverall clustering behavior. Clustering increasedafter 1995, at both local and global scales andby both graph-theoretic and exclusive definitions.In particular, better-connected authors exhibitedgreater clustering earlier than less-connected au-thors. After 2001, however, graph-theoretic clus-tering surged while exclusive clustering plum-meted (Fig. 4 (g–i)). After 2001 the pureand applied networks grew increasingly dissimilar,with greater graph-theoretic clustering in the ap-plied but greater exclusive clustering in the pure.This suggests a connection to the more promi-nent disassortative mixing in the applied network,and indeed the propagation of highly collabora-tive projects by disassortative short-lived researchteams would explain both dissimilarities. Fur-thermore, C× mimicked collaboration weight 〈w〉

and rpub, suggesting that autonomous collabora-tions are better forged among similarly prolific au-thors.

5. Events and Change Point Models

While long-term trends varied widely, fluctu-ations in our metrics, as revealed by residu-als from linear fits, were often highly correlated(Fig. 5). Similar fluctuations suggest mathemati-cal or sociological dependencies among properties;we grouped together metrics with strongly corre-lated time series and identify these groups by sym-bol in Table 2 and Fig. 6 (c,d). We used a changepoint model7 to arrange these shifts chronologi-cally.Our change point model fits a continuous,

piecewise-linear curve with one corner to a setof ordered pairs (xi, yi), subject to error froma fixed distribution, analogously to a linear fit.The model takes the slopes, intercept, and changepoint to be unknown and the errors to come froma normal distribution with unknown variance:

yi = β0+β1xi+β2(xi−c)δxi>c+ǫi, ǫi ∼ N(0, σ)

7An recent bibliography of change point problems byKhodadadi and Asgharian [KA08] traces change pointmodels to a 1954 discussion by Page [Pag54] on piecewisecontinuous models.

6

pkanmCc

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Figure 5: Correlation matrix of time series of key statisticsacross 5-year sliding windows. Top/right to bottom/left:p, no. publications; k, 〈k〉; a, 〈a〉; n, n; m, m; C, C; c, 〈c〉;Ss, avg. no. subject classifications per author; s, 〈s〉; w,〈w〉; X, C×; rp, rpub; z, avg. no. publications per author;Z, avg. no. collab. publications per coauthor; K, 〈κ〉; S,S; rc, rcol; a3, α3.

The indicator δx>c takes the value 1 when x > cand 0 otherwise. The parameters β0, β1, β2, c en-code the two slopes β1 and β1+β2, the y-interceptβ0, and the change point c. Our code in R usesiterative methods to find estimators for the pa-rameters that minimize SSE =

∑

i ǫi2. We opti-

mized this model over intervals visually centeredabout the dramatic shift of each time series to ob-tain the dates in Table 2. We used intervals of 11years when possible, shortening to 10 years in caseour algorithm failed, for consistency with slidingwindows. We exhibit code and all change pointfits to time series in the supplementary materials.

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Figure 6: Across 5-year sliding windows, 1985–9 to 2005–9:(a) Residuals from best linear fit of the number of pub-lications, with change point fits about the mid-90s eventoverlaid. (b) Residuals from best linear fit of the num-ber of authors, with change point fit about the mid-2000sevent overlaid. (c–d) Differences in best-fit change pointsfrom the aggregate network to the few-author network atthe both events.

Table 2: Change points in time series around both events.

statistic (symbol groups as in Fig. 6) change

N avg. collab. publications/coauthor 1992.13• correlation by no. coauthors (rcol) 1993.13

� global SV-clustering coeff. (C) 1993.17• scale-freeness (S) 1993.49� avg. no. coauthors (〈k〉) 1994.13• diff. in collab.-multidisc. effects (α3) 1994.17N avg. publications/author 1994.30• avg. no. collaborations (〈κ〉) 1995.04� avg. SV-clustering coeff. (〈c〉) 1995.26� avg. authors/publication (〈a〉) 1995.65� no. publications 1996.16� no. collab. pairs (m) 1996.17� no. authors (n) 1996.19N correlation by no. publications (rpub) 1996.30

• correlation by no. coauthors (rcol) 1997.37• diff. in collab.-multidisc. effects (α3) 1998.35N correlation by no. publications (rpub) 2000.76N global exclusive clustering coeff. (C×) 2001.08N avg. collab. publications/coauthor 2001.65

� global SV-clustering coeff. (C) 2001.74N avg. publications/author 2001.74N avg. collab. weight (〈w〉) 2002.06� avg. SV-clustering coeff. (〈c〉) 2002.22• scale-freeness (S) 2002.40� avg. authors/publication (〈a〉) 2002.51• avg. no. collaborations (〈κ〉) 2002.59� no. collab. pairs (m) 2002.72� no. authors (n) 2003.65

We also contrasted the aggregate network witha “few-author” network constructed from publi-cations of 6 authors or fewer, which would beunaffected by the reversals of trends exhibited inFig. 2 (c). For uniformity in our change pointanalysis we drew all statistics from largest com-ponents, so time series for many statistics differfrom those presented earlier. (In Table 2, themul-tidegree of a node vi in a weighted graph is the sumκi of the weights of its links; we then say that au-thor vi has engaged in κi “collaborations”.) Thefew-author network exhibited fluctuations similarto, but not always simultaneous with, those of theaggregate. By several metrics it experienced thefirst event later than the aggregate but the secondevent at essentially the same time (Fig. 6 (c,d)).This suggests that highly collaborative projectswere inceptive to the first event while not neces-sarily to the second.

The chronologies of both events, as arrangedin Table 2, suggest “top-down” narratives, withshifts in hierarchical metrics sensitive to highlycentral or prolific authors preceding shifts in met-rics of local connectivity, and shifts in network-wide averages and totals manifesting last.

7

6. Discussion

Over 25 years the mathematics collaborationnetwork grew steadily larger, more collaborative,and better-connected both locally and globally.While the applied network was better connectedlocally (〈a〉, 〈k〉, 〈c〉) and exhibited more hierar-chical structure (rcol, rpub, S), the pure networkwas better connected globally (〈ℓ〉) and exhibitedstronger local connections (〈w〉, 〈s〉, α3). In par-ticular, while the small-world properties of lowmean separation and high clustering have been re-produced together by a variety of real-world andmodel networks [ASBS00] [WS98, Jac08, BM10],neither of our major subnetworks is clearly thesuperior “small world” of the two.The mid-90s event was characterized by prolif-

erated and strengthened collaboration (Fig. 2 (e),〈k〉, Fig. 4 (g,h)), a weakening relationshipbetween collaboration and multidisciplinarity(Fig. 3 (g–i), and moderately increased assorta-tive mixing (Fig. 4 (a,c)). The rise in several-author publications explains the stabilization ofclustering; exclusive clustering had already beenrising (Fig. 4 (g,h)). However, increases in clus-tering and hierarchical metrics were still evidentin the network constructed from 6- or fewer-author publications. The delay in shifts fromthe aggregate to this few-author network indicatesthat highly collaborative projects were inceptiveto the event (Fig. 6 (c)), a proposition supportedby the “top-down” progression of change ponits.These qualities of the event, the similar be-

havior of the pure and applied disciplines, andtiming suggest a possible factor: the rise ofe-communications and the World Wide Web.Among academic Internet milestones are the in-troductions of the arXiv in 1991, which went on-line in 1993 [Gin09], and of MathSciNet in 1996,which made theMR publishing database availablethrough a graphical web interface [Jac97]. Weshould expect researchers in more applied sub-disciplines, who historically made greater use ofcomputing resources, to have made quicker use ofthese tools, and indeed the applied network andits subdisciplines exhibited the above trends moreclearly (Fig. 4 (a–f)).The early-2000s event tells a dissimilar story.

This event was characterized by weakening aver-age publishing rates and collaboration strength(Fig. 3 (a–c) and 4 (h)) due in part to an influxof less prolific authors (Fig. 1 (d) and 3 (d)) anddramatically disassortative mixing (Fig. 4 (a–c)).While disassortativity was ubiquitous, lower pub-lishing rates were more evident in applied disci-plines. Increased clustering was largely explainedby a further acceleration in several-author pub-

lications (Fig. 4 (g,h)). Highly collaborativeprojects were not so inceptive (Fig. 6 (d)).The growth in the research community and in-

terconnections within it, simultaneous with weak-ening average publishing rates and collaborationrates, may be largely explained by the surge intransient authors. This surge may reflect an in-creasing trend toward interdisciplinary researchinvolving many researchers outside mathematicswho publish seldom but in larger teams. Thisis consistent with the absence of a specializa-tion trend during this event, which distinguishesit from the earlier event (Fig. 2 (f)). A pos-sible contributing factor to such a trend wouldhave been an increased emphasis on interdisci-plinary projects at funding agencies such the Na-tional Science Foundation, the largest funder ofU.S. mathematics research. We note that theevent was concurrent with increased funding bythe NSF for its Division of Mathematical Sciences[nsf11] recommended by a 1998 report [Odo98].(See the supplementary text for detailed discus-sion.) A change point fit to 5-year funding aver-ages places the surge at 2001.33, toward the be-ginning of the event (Table 2). NSF funding af-fects almost exclusively U.S.-based research, how-ever, while the MR database covers worldwideoutput.

7. Conclusions

The community of researchers in mathemat-ical sciences has grown at an increasing ratesince 1985, and their research output has accel-erated. Amidst this growth the literature has be-come increasingly multidisciplinary and the net-work of researchers has grown better-connectedand individual researchers more collaborative. In-creased collaboration has been due in large partto highly collaborative teams of researchers, manyof whose members have short mathematical pub-lishing histories. Such disassortative authorshiphas been more prevalent in applied disciplines,which nonetheless exhibit more hierarchical or-ganization, while researchers in more pure disci-plines maintain longer collaborations and are lessseparated by degrees of coauthorship. The net-work drastically reorganized twice between 1985and 2009, in different ways that suggest dissimilarcauses and consequences.TheMR network is huge and admits much more

analysis than we have performed. Data collectedsince 1940 are being processed and will be releasedsoon, which will allow investigators to treat thedatabase from conception. We omitted discus-sion of linking mechanisms, and of a range of toolsfor detecting community structure, for which the

8

MR network holds great potential. We suggestedpossible partial explanations for the two eventswe described, but it is beyond the scope of thispaper to consider these hypotheses thoroughly.More detailed information on mathematical pub-lishing and its funding may be obtained from theMR database and from government agencies, andfollow-up investigations may provide deeper in-sights into these possible connections.

Acknowledgments. The authors are grateful tothe American Mathematical Society for provid-ing access to the MR database and for agreeingto make the data publicly available (by request tothe Executive Director). The authors thank Sas-try Pantula and Philippe Tondeur for providinginformation on NSF funding for mathematics, andSid Redner, Ritchie C. Vaughan, Betsy Williams,and fellow participants of the Summer 2010 REUin Modeling and Simulation in Systems Biologyfor helpful conversations and support.

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8. Supporting Information

We performed calculations in R, includinggraph-theoretic calculations using the igraph

package and some original code. All code is avail-able upon request from the first author.

8.1. Publishing rates and connectivity

As discussed in the main text, one-time au-thors increasingly worked in large research teams(Fig. S1 (a)). The also comprised an increas-ing proportion of the community after 2000(Fig. S1 (d)), while the proportion consisting ofmore prolific authors (3 or more publications) de-creased (Fig. S1 (c)).Network density (Fig. S1 (b)) is directly re-

lated to average degree. The relevance of thepure–applied split is evident in the higher densityin both, indicating stronger connectivity amongpure researchers and applied researchers sepa-rately than among all mathematics researchers.The fluctuations, especially in the applied net-work, were similar to those in rcol. The aggre-gate, pure, and applied networks exhibited similarcohesion with respect to largest connected com-ponents (Fig. S1 (c)), and the “S”-shape of thecurves (especially that of the applied) suggeststhat this proportion is approaching its practicallimit.

8.2. Assortative mixing by publications

Newman [New03] defines a normalized degreecorrelation coefficient by way of “remaining de-gree”: Starting with a pair of nodes (vi, vj), takethe number of neighbors of each excluding theother, (ki − 1, kj − 1). These are their remain-ing degrees. We define rpub analogously using thenotion of remaining prolificity.Consider authors vi and vj who have authored

zi and zj publications, respectively, and have col-laborated on wij of them. zi is then the “pro-lificity” of vi (and zj that of vj) while wij is the“collaboration weight” of vi and vj together. De-fine the remaining prolificity of vi with respect tovj to be zi−wij , the number of publications by vinot coauthored with vj . Since in graph-theoreticlanguage we say that vi is adjacent to the link(vi, vj), we refer to the remaining prolificity ofthe adjacency of node vi to link (vi, vj).Where the network includes nx authors of pro-

lificity x, set px = nx/∑

x′ nx′ , the proportionof nodes in the network of prolificity x. Nowconsider the adjacencies: They number twice asmany as the number of links. If we let px,w be theproportion of adjacencies with author (node) pro-lificity x and collaboration (link) weight w then

10

1.8

2.2

2.6

3.0

(A)

1.5e

−05

2.5e

−05

(B)

0.3

0.4

0.5

0.6 (C)

1990 1995 2000 2005

0.42

0.46

0.50

0.54 (D)

1990 1995 2000 2005

0.16

60.

170

0.17

40.

178

(E)

1990 1995 2000 2005

0.09

00.

100

(F)

Figure S1: Across 5-year sliding windows: (a) Average number of authors on publications by one-time authors. One-timeauthors appeared on increasingly collaborative publications throughout our interval. (b) Network density. The appliednetwork grew dramatically sparser after 1995 while the pure grew denser until 2006. (c) Proportion of nodes in the largestcomponent. There is no difference between the pure and applied subnetworks to be discerned from the proportions ofauthors comprising their largest components, and the proportion comprising the largest component of the aggregate isconsistently larger than both. (d–f) Proportions of authors with 1, 2, and 3 publications.

qr =∑

w≥1 pw+r,w is the proportion of all adja-cencies having remaining prolificity r.Let us have ordered pairs (r, s) range over the

remaining prolificities of linked nodes, so thateach link is counted twice (as (r, s) and as (s, r)).Our statistic of interest is then

rpub =E(rs)− E(r)E(s)√

V ar(r)V ar(s),

the correlation coefficient for the remaining pro-lificities of linked nodes.Define er,s to be the joint probability distribu-

tion of the remaining prolificities at the ends ofa uniformly randomly chosen link. Since r ands are drawn from the same distribution, we maysimplify the numerator as

E(rs) = E(r)E(s) = E(rs) − E(r)2

=∑

r

∑

s

rser,s − (∑

r

rqr)2

and the denominator as√

V ar(r)V ar(s) = V ar(r) = E(r2)− E(r)2

=∑

r

r2qr − (∑

r

rqr)2.

If we index the links by i = 1, . . . ,m and (ar-bitrarily) label the remaining prolificities of theirends ri and si then we may rewrite

∑

r

∑

s

rser,s =1

m

∑

i

risi,

∑

r

rqr =1

2m

∑

i

(ri + si),

∑

r

r2qr =1

2m

∑

i

(ri2 + si

2).

This provides the computational formula

rpub =

1

m

∑

i

risi − (1

m

∑

i

1

2(ri + si))

2

1

m

∑

i

1

2(ri

2 + si2)− (

1

m

∑

i

1

2(ri + si))

2

.

If the remaining prolificities of linked nodes areindependent then er,s = qrqs. If, instead, linkedpairs are perfectly correlated in this respect thenwe get er,s = qrδr,s, where δr,s is the Kroneckerdelta (1 if r = s, 0 otherwise). The authors ofa collaboration network are perfectly correlatedby remaining prolificities r precisely when theyare precisely correlated by prolificity x — that is,when the network consists of connected compo-nents of uniform prolificity.

8.3. Assortative mixing with low-count authorsremoved

To check that the trends and fluctuations weobserved in rcol and in rpub were not artifacts ofthe mixing behavior of authors with only one col-laborator or publication, we ran the calculationson the aggregate with such authors removed fromconsideration. The overall trends were the same(Fig. S2).

8.4. Clustering coefficients

The time series for C (Fig. S3(a)) and rcol aresimilar. The dependence between these statisticsreflects the reduced number of possible trianglesamong nodes of different degrees, which admitsless clustering in disassortative graphs [SV05]. Inthe main paper we accounted for this interaction

11

1990 1995 2000 2005

0.12

0.16

0.20

0.24

(A)

1990 1995 2000 2005

0.45

0.50

0.55

(B)

Figure S2: Across 5-year sliding windows on the aggregatenetwork: (a) rcol calculated after removing authors withonly one collaborator from consideration. (b) rpub cal-culated after removing authors with only one publicationfrom consideration.

1990 1995 2000 2005

0.22

0.26

0.30

(A)

1990 1995 2000 2005

8000

1200

016

000 (B)

Figure S3: Across 5-year sliding windows: (a) Global clus-tering coefficient C: the proportion of connected triplesof authors who are in fact pairwise linked. The time se-ries closely resembles that of degree assortativity, partic-ularly before 2001. (b) Clustering normalized by density(Fig. S1(d)), its expected value in a uniformly randomgraph, C · n(n − 1)/2m; the relative probability that twoauthors collaborated provided they had a common coau-thor. The rises in density and in assortative mixing from1995 to 2000 were largely to credit for the perceived rise inclustering during this period, while clustering after 2001becomes more pronounced when corrected for these phe-nomena. Plots (b) and (c) use information from the entiregraph.

using the correction C introduced by Soffer andVazquez [SV05].

Under uniformly random linking, a higher pro-portion of connected triples will form triangles ina denser graph, increasing clustering. To accountfor this, we normalized C by density (Fig. S3(d))to get the relative probability that two authorshave collaborated provided that they have a com-mon coauthor [New01a] (Fig. S3(c)). Becausedensity decreased substantially due to propor-tional growth in the largest component, this nor-malization increased monotonically across largestcomponents; we took the normalization over en-tire graphs instead.

8.5. Change point fits

We fit change point models to time series datathat were not clearly piecewise linear, but thatexhibited one or two major changes in behavioramid smaller perturbations that the models inter-pret as normally-distributed error. Here we dis-cuss change point models in more detail, and we

display all aggregate residual plots, together withchange point fits, used for the analysis.The principle behind change point models is the

same as that behind linear models. A traditionallinear fit takes the form

yi = β0 + β1xi + ǫi, ǫiiid∼ N(0, σ),

while our change point model takes the form

yi = β0+β1xi+β2(xi−c)δxi>c+ǫi, ǫiiid∼ N(0, σ).

We caution that some basic statistical assump-tions for change point models are not met by thisdata: Particularly because adjacent sliding win-dows share 4 out of their 5 years but also becausemost authors publish in multiple years, measure-ments performed on these windows cannot be con-sidered independent. Because each window con-tains a different (increasing) number of publica-tions, they cannot be considered identically dis-tributed. By performing change point analysiswe do not intend to make predictions of futurebehavior but only to take advantage of an effec-tive method for identifying shifts in behavior oth-erwise well-modeled linearly.What follows is a simplification of the code we

used to perform change point analysis in R. Werequired a guess c at the change point and cal-culated estimators for the coefficients by fittinga linear model lm1 to the data below c (provid-

ing β0 and β1) and a linear model lm2 with fixedintercept at (c, lm1(c)) to the data above c (pro-

viding β2).

# FUNCTION: Change point analysis on a

# collection of ordered pairs

changepoint.model <- function(

x, # points (independent), sorted

y, # values (dependent)

c # number in range(x)

) {

len <- length(x)

stopifnot(len == length(y))

# Linear model to estimate b0 and b1

m <- max(which(x < c))

lm1 <- lm(y[1:m] ~ x[1:m])

b0 <- lm1$coeff[1]

b1 <- lm1$coeff[2]

# Scaling model to estimate b2

# y-value at x = c

int <- lm1$coeff[1] + lm1$coeff[2] * c

# x-values with origin (c,int)

x2 <- x[(m + 1):len] - c

# y-values with origin (c,int)

y2 <- y[(m + 1):len] - int

lm2 <- lm(y2 ~ x2 + 0)

b2 <- lm2$coeff[1] - lm1$coeff[2]

12

−0.15 0.00

(A)

−0.03 0.00 0.03

(B)

−0.02 0.02

(C)

−0.02 0.01

(D)

−0.05 0.05

(E)

−0.010 0.005

(F)

−0.10 0.05

(G)

−0.10 0.00

(H)

−0.010 0.010

(I)

−0.02 0.02

(J)

−5000 5000

(K)

−20000 20000

(L)

19901995

20002005

−5000 5000

(M)

19901995

20002005

−0.04 0.00 0.04

(N)

Figure

S4:Resid

uals

from

best

linearfits

overla

idwith

ach

angepointfitaboutthemid-90seven

t(5-yearslid

ingwindow

s).

−0.10 0.00

(A)

−0.02 0.00 0.02

(B)

−0.010 0.010

(C)

−0.02 0.01

(D)

−0.04 0.00

(E)

−0.015 0.005

(F)

−0.10 0.05

(G)

−0.05 0.05

(H)

−0.010 0.005

(I)

−0.02 0.01 0.04

(J)

−5000 5000

(K)

−20000 0 20000

(L)

19901995

20002005

−5000 5000

(M)

19901995

20002005

−0.04 0.00 0.04

(N)

Figure

S5:Resid

uals

from

best

linearfits

overla

idwith

ach

angepointfitaboutthemid-90seven

t(5-yearslid

ingwindow

s,few

-authornetw

ork).

13

# Change point model using estimators for

# c (given), b0, b1, and b2

return(summary(nls(

as.formula(

’y ~ B0 + B1 * x +

B2 * (x - C) * (x >= C)’

),

start = list(

C = c, B0 = b0, B1 = b1, B2 = b2

)

)))

}

Fig. S4–S7 depict the change point fits we usedto examine the two events in the main paper, withthe exception of Fig. S4(q); the fit, we judged,was too poor to warrant inclusion, and it demon-strates by comparison the superior fits obtainedin other cases. In each plot the dotted verticallines demarcate the intervals used to construct themodel.

8.6. NSF funding for mathematics

The 1998 Odom Report [Odo98] recommendedsteep increases in funding for mathematics re-search, and from 2001 to 2004 annual NSF fund-ing for its Division of Mathematical Sciences rosedramatically (Fig. S8). A change point fit to thesenumbers over 1995–2004 identifies a change yearof 2000.36. Using 5-year averages instead, witheach interval identified by its last year followingthe pattern used for other statistics, a changepoint fit over 1996–2006 identifies the change year2001.33. Both values are toward the beginning ofthe collection of change years identified for net-work statistics, supporting a causal hypothesis,but significantly later than several specific statis-tics, suggesting that increased NSF funding mayhave contributed to, but was not the sole driverof, the second event.

1985 1990 1995 2000 2005

5000

015

0000

Figure S8: Funding by the National Science Foundation’sDivision of Mathematical Sciences, 1985–2007. A surge infunding beginning in 2001 was concurrent with the secondevent, specifically the surge in authorship, and leveled offafter 2005.

14

−0.03 0.00 0.03

(A)

−0.015 0.000

(B)

−0.04 0.02

(C)

−4e−04 2e−04

(D)

−0.10 0.05

(E)

−0.02 0.02

(F)

−0.10 0.05

(G)

−0.03 0.00

(H)

−0.010 0.010

(I)

−0.02 0.00

(J)

−0.02 0.02

(K)

−0.10 0.00

(L)

19901995

20002005

−20000 10000

(M)

19901995

20002005

−5000 5000

(N)

Figure

S6:Resid

ualsfro

mbest

linearfits

overla

idwith

ach

angepointfitabouttheearly

-00seven

t(5-yearslid

ingwindow

s).

−0.02 0.00

(A)

−0.015 0.005

(B)

−0.04 0.02

(C)

−6e−04 0e+00

(D)

−0.10 0.00

(E)

−0.010 0.010

(F)

−0.10 0.05

(G)

−0.015 0.005

(H)

−0.010 0.005

(I)

−0.02 0.01

(J)

−0.02 0.01 0.04

(K)

−0.05 0.05

(L)

19901995

20002005

−20000 0 20000

(M)

19901995

20002005

−5000 5000

(N)

Figure

S7:Resid

uals

from

best

linearfits

overla

idwith

ach

angepointfitabouttheearly

-00seven

t(5-yearslid

ingwindow

s,few

-authornetw

ork).

15