                                                                                                                       Vol 465 | 3 June 2010 | doi:10.1038/nature09040




LETTERS
The role of mentorship in protégé performance
R. Dean Malmgren1,2, Julio M. Ottino1,3 & Luı́s A. Nunes Amaral1,3,4


The role of mentorship in protégé performance is a matter of import-                   remains an open question. Indeed, we are unaware of any studies that
ance to academic, business and governmental organizations.                               systematically track mentorship success over the entire career of a
Although the benefits of mentorship for protégés, mentors and their                    mentor, so the validity of the rising-star hypothesis has yet to be fully
organizations are apparent1–9, the extent to which protégés mimic                      explored. Here we investigate whether protégés acquire the mentor-
their mentors’ career choices and acquire their mentorship skills is                     ship skills of their mentors, by studying mentorship fecundity, that is,
unclear10–16. The importance of a science, technology, engineering                       the number of protégés that a mentor trains over the course of their
and mathematics workforce to economic growth and the role of                             career. This measure is advantageous as it directly measures an out-
effective mentorship in maintaining a ‘healthy’ such workforce                           come of the mentorship process that is relevant to sustained mentor-
demand the study of the role of mentorship in academia. Here we                          ship, allowing us to quantify the degree to which mentor fecundity
investigate one aspect of mentor emulation by studying mentorship                        determines protégé fecundity.
fecundity—the number of protégés a mentor trains—using data                                Scientific mentorship offers a unique opportunity to study this
from the Mathematics Genealogy Project17, which tracks the mentor-                       question because there is a structured mentorship environment
ship record of thousands of mathematicians over several centuries.                       between advisor and student that is, in principle, readily accessible18,19.
We demonstrate that fecundity among academic mathematicians is                           We study a prototypical mentorship network collected from the
correlated with other measures of academic success. We also find                         Mathematics Genealogy Project17, which aggregates the graduation
that the average fecundity of mentors remains stable over 60 years of                    date, mentor and protégés of 114,666 mathematicians from as early
recorded mentorship. We further discover three significant correla-                      as 1637. This database is unique in its scope and coverage, tracking the
tions in mentorship fecundity. First, mentors with low mentorship                        career-long mentorship record of a large population of mentors in a
fecundities train protégés that go on to have mentorship fecundities                   single discipline (see the MPACT Project (http://ils.unc.edu/mpact/)
37% higher than expected. Second, in the first third of their careers,                   for a smaller database of theses on information and library sciences
mentors with high fecundities train protégés that go on to have                        and references therein). From this information, we construct a net-
fecundities 29% higher than expected. Finally, in the last third of                      work in which links are formed from a mentor to each of his k pro-
their careers, mentors with high fecundities train protégés that go on                 tégés, where k denotes mentorship fecundity. We focus here on the
to have fecundities 31% lower than expected.                                             7,259 mathematicians who graduated between 1900 and 1960, because
    A large literature supports the hypothesis that protégés and mentors               their mentorship record is the most reliable (Methods).
benefit from the mentoring relationship1,2. Protégés that receive career                   Although the mentorship records gathered from the Mathematics
coaching and social support, for instance, are reportedly more likely to                 Genealogy Project provide the most comprehensive data source avail-
have high performance ratings, a higher salary and receive promo-                        able for the study of academic performance throughout a mathemati-
tions1,3. In return, mentors receive fulfilment not only by altruistically               cian’s career, there are obviously other plausible metrics for evaluating
improving the welfare of their protégés, but also by improving their own               academic performance20–22. We have also compared the mentorship
welfare4,5,10. Organizations benefit as well, because protégés are more                data against a list of publications for 4,447 mathematicians and a list of
likely to be committed to their organization6,7 and to exhibit organiza-                 269 inductees into the US National Academy of Sciences (NAS;
tional citizenship behaviour6. These benefits are not obtained only                      Methods). We find that mentorship fecundity is much larger for
through the traditional dyadic mentor–protégé relationship, but also                   NAS members than for non-NAS members (Fig. 1a). We further find
through peer relationships that supplement protégé development8,9.                     that the number of publications is strongly correlated with fecundity,
    The benefits of mentorship underscore the importance of under-                       regardless of whether or not a mathematician is an NAS member
standing how mentors were in turn trained to foster the development                      (Fig. 1b). These results demonstrate that although fecundity is not a
of outstanding mentors. It might be suspected that protégés learn                      typical measure of academic performance, it is closely related to other
managerial approaches and motivational techniques from their men-                        measures of academic success. Thus, even though our investigation
tors and, as a result, emulate their mentorship methodologies; this                      concerns how fecundity is correlated between mentor and protégé, our
suggests that outstanding mentors are trained by other outstanding                       results also address questions in the academic evaluation literature
mentors. This possibility is sometimes formalized as the rising-star                     concerning the success of a mathematician.
hypothesis11,12; it postulates that mentors select up-and-coming pro-                        We first investigate whether it is possible to predict the fecundity of
tégés on the basic of their perceived ability and potential and past                   a mathematician by modelling the empirical fecundity distribution,
performance10,13,14, including promotion history and proactive career                    p(kjt), as a function of graduation year, t. Considering that some
behaviours12. Rising-star protégés are reportedly more likely to                       mathematicians remain in academia throughout their careers whereas
intend to mentor, resulting in a ‘perpetual cycle’ of rising-star pro-                   others spend only a portion of their careers in academia, it might be
tégés that emulate their mentors by seeking other rising stars as their                expected that there are two types of individual when it comes to
protégés15.                                                                            academic mentorship fecundity—‘haves’ and ‘have-nots’—in the
    However, there is conflicting evidence concerning the rising-star                    sense that mathematicians from these types respectively have or have
hypothesis16, so the extent to which protégés mimic their mentors                      not had the opportunity to mentor students throughout their career.
1
  Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, USA. 2Datascope Analytics, Evanston, Illinois 60201, USA. 3Northwestern
Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA. 4Howard Hughes Medical Institute, Northwestern University, Evanston, Illinois 60208, USA.

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NATURE | Vol 465 | 3 June 2010                                                                                                                                                                                                   LETTERS


   a 100                                                  b 103                                                             a                                                       b                          c
                                                                                                                                                    100




                                                                                                                          Cumulative distribution
                                                          Number of publications
Cumulative distribution




                          10–1                                                                                                                      10–1
                                                                                   102

                          10–2                                                                                                                      10–2
                                                                                                                                                            P = 0.626                   P = 0.758                    P = 0.068
                                                                                   101
                          10–3                                                                                                                      10–3
                                                                                                                                                        0     10        20     30   0    10     20 30 40      50 0   10 20 30 40    50
                                                                                                                                                                                              Fecundity, k
                                                                                                                            d
                          10–4                             100 0                                                                                    0.6
                              0   20   40   60   80   100     10                                   101         102                                  0.5
                                                      Fecundity, k                                                                                  0.4




                                                                                                                           πh
                                                                                                                                                    0.3
                                                                                                                                                    0.2
Figure 1 | Relationship between mentorship fecundity and other                                                                                      0.1
performance metrics. a, Cumulative distribution of the mentorship                                                           e                       0.0
fecundity for NAS members (red) and non-NAS members (black). NAS                                                                                     20
members have an average fecundity of ÆkæNAS 5 14, which is far greater than                                                                          15
the average fecundity of non-NAS members, Ækænon-NAS 5 3.1, indicating that




                                                                                                                           κh
                                                                                                                                                     10
fecundity is closely related to academic recognition. Not all mathematicians                                                                          5
in the non-NAS group were eligible for NAS membership, owing to                                                                                       0
citizenship and other circumstances. This fact makes the result in the figure                                               f                       2.0
all the more striking. b, Average number of publications as a function of the                                                                       1.5
mentorship fecundity, for NAS members (red) and non-NAS members




                                                                                                                           κhn
                                                                                                                                                    1.0
(black). NAS members have nearly twice as many publications on average as
                                                                                                                                                    0.5
non-NAS members for all fecundity levels. Error bars, 1 s.e.
                                                                                                                                                    0.0
                                                                                                                                                     1900               1920            1940           1960             1980         2000
If each mentor chooses to train a new academic protégé with                                                                                                                            Graduation year, t

probability jh or jhn, and stops training academic protégés otherwise,                                                 Figure 2 | Evolution of the fecundity distribution. a–c, Cumulative
depending on whether they are a ‘have’ or, respectively, a ‘have-not’,                                                   distribution of the fecundity of mathematicians that graduated during 1910
then we would expect that the resulting fecundity distribution is a                                                      (a), 1930 (b) and 1950 (c) (symbols), compared with the best-estimate
mixture of two discrete exponential distributions                                                                        predictions of a mixture of two discrete exponentials (lines). Monte Carlo
                                                                                                                         hypothesis testing confirms that this model can not be rejected as a model of
                                       p(kjH)~ph p(kjkh )z(1{ph )p(kjkhn )                                        ð1Þ    the fecundity distribution during every year from 1900–1960, as denoted by
                                                                                                                         the P values above the a 5 0.05 significance level (Methods). d–f, Best-
whereph istheprobabilitythatamathematicianisa‘have’andp(kjkh)and                                                         estimate parameters as functions of time, calculated by maximum likelihood
p(kjkhn) are discrete exponential distributions p(kjk) 5 e2k/k(1 2 e21/k)                                                for a mixture of two discrete exponentials. Dashed lines denote average
with respective average fecundities kh 5 1/ln(jh21) and                                                                  parameter values between 1900 and 1960 and coloured circles indicate the
khn 5 1/ln(jhn21) for ‘haves’ and ‘have-nots’. We estimate the                                                           years displayed in panels a–c. The probability, ph, of being a ‘have’ changes
parameters H 5 {ph, kh, khn} of this distribution from the empirical                                                     over time, generally in relation to historic events (hashed grey shading
data using expectation maximization23. Using Monte Carlo hypo-                                                           indicates the First and Second World Wars). In contrast, the average
thesis testing (Methods), we have found that equation (1) cannot be                                                      fecundities remain stable, with time-average values of k h 5 9.8 6 0.4 and
rejected as a candidate description of the fecundity distribution p(kjt)                                                 k
                                                                                                                         hn 5 0.47 6 0.03, until 1960, the time at which mentorship records become
                                                                                                                         incomplete (Methods), and then steadily decrease (grey shaded region).
(Fig. 2a–c). For an alternative description of p(kjt), see Supplementary
Discussion and Supplementary Fig. 1.
    As might be expected, the probability, ph, that an individual is a                                                   generated from uncorrelated branching processes in our investigation
‘have’ experiences drastic changes over time as a result of historical                                                   of the mathematician genealogy network. Here graduation date is
events, such as the First and Second World Wars, the beginning of the                                                    equivalent to birth date and mentors and protégés are equivalent to
Cold War and considerable increases in academic funding (Fig. 2d).                                                       parents and children, respectively.
In contrast, the average fecundities of ‘haves’ and ‘have-nots’ do not                                                      In a branching process24, a parent p, born at time tp, has kp children.
exhibit systematic historical changes, suggesting that these quantities                                                  Child c of parent p is born at time tc and subsequently has kc children.
offer fundamental insight into the mentorship process among math-                                                        The fecundity, k, of each individual is drawn from the conditional
ematicians (Fig. 2e, f). For the sixty year period considered, we find                                                   fecundity distribution p(kjt) for an individual born at time t.
that k h 5 9.8 6 0.4 and k  hn 5 0.47 6 0.03, where the overbar indi-                                                  Networks generated from this type of branching process are therefore
cates a time average of the respective average fecundity.                                                                defined by the birth date of each individual, t, the fecundity distri-
    The stationarity of kh and khn also provides a simple heuristic for                                                  bution p(kjt), and the chronology of child births, {tc}, for each parent
classifying an individual as a ‘have’ or a ‘have-not’; by maximum                                                        (Fig. 3a).
likelihood, an individual is a ‘have’ if k $ 2 and is a ‘have-not’ other-                                                   We compare the mathematician genealogy network with two
wise. These results raise the possibility that similar features, perhaps                                                 ensembles of randomized genealogies from the branching process
with different characteristic scales of fecundity, may be present in                                                     family. Random networks from ensemble I retain the birth date of each
other mentorship domains.                                                                                                individual, the fecundity of each individual and the chronology of child
    Although our description of the fecundity distribution has high-                                                     births for each parent (Fig. 3b), as above. Random networks from
lighted a fundamental property of mentorship among mathematicians,                                                       ensemble II additionally restrict parent–child pairs to have the same
it is not predictive of the behaviour of individual mathematicians in the                                                age difference, tc 2 tp, as parent–child pairs in the empirical network
sense that fecundity, according to this model, is a random variable                                                      (Fig. 3c). All other attributes of these networks are randomized using a
drawn from the distribution in equation (1). We next test whether                                                        link-switching algorithm25,26 (Methods), so neither of these random-
protégés mimic the mentorship fecundity of their mentors, by com-                                                      network ensembles introduces correlations between parent fecundity
paring protégé fecundity with a suitable null model that does not                                                      and child fecundity or temporal correlations in fecundity. They there-
introduce correlations in fecundity. As in the study of genealogical                                                     fore provide a suitable basis for comparison with the mathematician
trees, we perform comparisons of the empirical data with networks                                                        genealogy network.
                                                                                                                                                                                                                                     623
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LETTERS                                                                                                                                           NATURE | Vol 465 | 3 June 2010



            a                                                                                        significance of any differences between the empirical data and the null
                                                                                                     models.
           Empirical network                      K. A. Hirsch                                           We use the partitioning of children into classes to examine the
                                                                                                     relationship between the average child fecundity, Ækcæ, and the age
                                                                                                     difference, tc 2 tp, between parent and child (Fig 4a, b and Supplemen-
                                                  B. A. Griffith
                                                                                                     tary Fig. 4a, b). If the data were consistent with a branching process,
                                                                                                     then we would expect Ækcæ to have no temporal dependence. However,
                                    W. Tollmien                                                      the regressions between the Ækcæ z-score (Methods) and tc 2 tp deviate
                                                                                                     significantly (Fig. 4c and Supplementary Fig. 4c) from this expectation
                                                                                                     for both random ensembles, to reveal three distinct features. First,
                                    H. D. Kloosterman                                                mentors with kp , 3 train protégés that go on to have mentorship
           b                                                                                         fecundities 37% higher than expected throughout their careers.
                                                                                                     Second, in the first third of their careers, mentors with kp $ 10 train
           Generating ensemble I




                                                                                                     protégés that go on to have fecundities 29% higher than expected.
                                                                                                     Finally, in the last third of their careers, mentors with kp $ 10 train
                                                                                                     protégés that go on to have fecundities 31% lower than expected.
                                                                                                         The fact that mentors with k , 3 train protégés with higher-than-
                                                                                                     expected fecundities throughout their careers is somewhat counter-
                                                                                                     intuitive. From the rising-star hypothesis11,12, it might be expected
                                                                                                     that protégés trained by mentors with k , 3 are likely to mimic their
                                                                                                     mentors and therefore have lower-than-expected fecundities. Our
                                                                                                     results demonstrate that this is not the case. One possible explanation
            c
                                                                                                     is that mentors with k , 3 are more aware of the resources they must
                                                                                                     allocate for effective mentorship, leading to a more enriching men-
           Generating ensemble II




                                                                                                     torship experience for their protégés. An alternative hypothesis is that
                                                                                                     mentors with k , 3 select for, or are selected by, protégés that have a
                                                                                                     greater aptitude for mentorship.
                                                                                                         The striking temporal correlations for mentors with kp $ 10 are
                                                                                                     also intriguing. Because mentors with kp $ 10 represent the upper
                                                                                                     echelon of mentors in mathematics, these mentors were probably
                                                                                                     ‘rising stars’ early in their academic careers. The fact that these men-
                                                                                                     tors train protégés with high fecundities early in their careers sup-
                     1920              1930        1940       1950      1960     1970                ports the rising-star hypothesis.
                                                  Graduation year, t                                     By the end of these mentors’ careers, however, their protégés have
                                                                                                     lower-than-expected fecundities. Perhaps mentors, who ultimately
Figure 3 | Branching process null models. a, Subset of the mathematician                             have high fecundities, spend fewer and fewer resources training each
genealogy network. Mentors/parents (black circles) are connected to each of
                                                                                                     of their protégés as their careers progress. Alternatively, protégés with
their protégés/children (white circles). The horizontal positions of
mathematicians represent their graduation/birth dates, t. The bottom two                             high mentorship fecundity aspirations might court prolific mentors
parents were born in 1924, the top two parents were born in 1937, and all                            early in their mentors’ careers whereas protégés with low fecundity
four parents have a child born in 1958. From a parent’s perspective, three                           aspirations might court prolific mentors later in their mentors’ careers.
essential features of the empirical network must be preserved in random                              Our findings therefore reveal interesting nuances to the rising-star
networks generated from the two branching process null models: the birth                             hypothesis.
date, tp, the fecundity, kp, and the chronology of child births, {tc}. b, Random                         It is unclear whether the temporal correlations we discover in men-
networks from ensemble I preserve these three essential features. Solid red                          torship fecundity generalize beyond mathematicians in academia.
lines highlight the links in the empirical network whose end points can be
randomized. Dashed red lines illustrate one of the possible randomization
                                                                                                     Anecdotally, mathematicians are thought to perform their best work
moves after switching the corresponding pair of links. We note that the age                          at a young age27, a perception that may influence how mentors and
difference between parent and child is not preserved. c, Random networks                             protégés choose each other. Perceptions in other domains, however,
from ensemble II preserve the three essential features as well as the age                            may differ and subsequently influence mentor and protégé selection in
difference between parent and child. Solid blue lines of the same colour                             different ways. As data for other academic disciplines18,19, business and
highlight the links in the empirical network whose end points can be                                 the government becomes available, it will be important to determine
randomized. Dashed blue lines illustrate one of the possible randomization                           whether temporal correlations in fecundity are a general consequence
moves after switching the corresponding pair of links. Random networks for                           of mentorship or are a particular consequence of mentorship for
each ensemble are generated by attempting 100 switches per link (Methods).
                                                                                                     mathematicians in academia.
   To explore the influence of mentor fecundity and age difference on                                    Regardless, our results offer another means of judging academic
protégé fecundity, we partition protégés according to the fecundity of                           impact in science as well as the impact of managers on their employ-
their mentors and the age difference between mentor and protégé,                                   ees, both of which are notoriously complicated and risky affairs.
tc 2 tp. Given our findings (Supplementary Discussion and Sup-                                       These assessments are multidimensional, metrics and expectations
plementary Figs 2 and 3), it is clear that age differences affect fecundity                          are domain dependent, and placement of creative output, timescales
in a nonrandom manner for protégés whose mentors have kp , 3. We                                   of impact and recognition vary significantly from field to field.
partition the remaining protégés, whose mentors have kp $ 3, into two                              Ultimately, the assessment of individuals for awards and promotion
groups: protégés whose mentors are below-average ‘haves’                                           is based on painstaking individual analysis by selection committees
(3 # kp , 10) and protégés whose mentors are above-average ‘haves’                                 and peers. Although these committees may have varying goals and
(kp $ 10). We then partition these three groups of protégés according                              incentives, it is important that collective arguments—the kind of
to when they graduated during their mentors’ careers. Specifically,                                  arguments we are making here—be based on sound quantitative
we split each group of protégés into terciles, the most fine-grained                               analysis. Although the extent to which our findings extrapolate to
grouping that still gives us sufficient power to examine the statistical                             other domains may vary, we are confident that the kind of analysis
624
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NATURE | Vol 465 | 3 June 2010                                                                                                                                                                                         LETTERS


                                                                             kp < 3                                 3 ≤ kp < 10                           kp ≥ 10
                                 a 100
                                                                                        〈kE〉 = 2.2                          〈kE〉 = 1.7                                〈kE〉 = 2.5




                                Cumulative distribution
                                                                                       〈kM〉 = 2.6                          〈kM〉 = 0.8                                〈kM〉 = 0.8
                                                          10–1                          〈kL〉 = 1.2                          〈kL〉 = 1.3                                〈kL〉 = 0.3


                                                          10–2                                                                                                                            Ens. I
                                                                                                                                                                                          Early
                                                                                                                                                                                          Middle
                                                          10–3                                                                                                                            Late
                                                              0        10        20         30        40 0     10     20     30          40 0        10        20         30        40
                                                                                                              Child fecundity, kc
                                b
                                                            4
                              〈kc〉 z-score




                                                            2
                                                                                                                                                                                         1900s
                                                            0                                                                                                                            1910s
                                                                                                                                                                                         1920s
                                                           –2                                                                                                                            1930s
                                                                  Slope, 0.023                               Slope, 0.026                       Slope, –0.10                             1940s
                                                           –4     Intercept, 0.18                            Intercept, –0.42                   Intercept, 2.1                           1950s
                                                                  10        20         30        40       10     20    30       40              10        20         30        40
                                                                                                         Age difference, tc – tp (yr)
                                 c                          3                                                                                                                              102




                                                                                                                                                                                                 Probability density
                                                            2
                                Intercept




                                                            1

                                                            0                                                                                                                              101
                                                           –1
                                                           –2      P = 0.009                                  P = 0.366                          P < 0.001
                                                           –3                                                                                                                              100
                                                                  –0.1           0.0         0.1             –0.1      0.0         0.1          –0.1           0.0         0.1
                                                                                                                      Slope

Figure 4 | Effect of age difference between mentor and protégé, tc 2 tp, on                                                 black line; slope and intercept as shown). The regression lines for networks
protégé fecundity. a, Fecundity distribution of children born during the                                                    from our null model (grey lines) vary around the expectation of our null
1910s (for which the average fecundity was 1.4) to parents with kp , 3,                                                       model (dashed black line). c, Significance of linear regressions in b. We
3 # kp , 10 and kp $ 10, compared with the expectation from ensemble I                                                        compare the slope and intercept of the empirical regression (black circle)
(grey line). We separate children into terciles (early, middle, late) according                                               with the distribution of the slope and intercept of the same quantities
to tc 2 tp, and denote the average fecundities of the children born early,                                                    computed from the null model. Because these quantities are approximately
middle and late in their parents’ lives as ÆkEæ, ÆkMæ and ÆkLæ, respectively. The                                             distributed as a multivariate Gaussian, we compute the equivalent of a two-
average fecundity of children born to parents with kp , 3 is higher than                                                      tailed P value by finding the fraction of synthetically generated
expected, regardless of whether they were born during the early, middle or                                                    slope–intercept pairs that lie outside the equiprobability surface of the
later part of their parents’ lives. We also note that the average fecundity of                                                multivariate Gaussian (dashed ellipse). The slopes and intercepts of the
children born to parents with kp $ 10 decreases throughout their parents’                                                     regressions for children of parents with low (P 5 0.009) and high (P , 0.001)
lives. b, We quantify the significance of these trends during each decade                                                     fecundities are significantly different from the expectations for the null
(coloured symbols) by computing the z-score of the average child fecundity,                                                   model, consistent with the data displayed in a. Comparisons with
Ækcæ, compared with the average child fecundity in networks from ensemble I.                                                  expectations from random networks from ensemble II yield the same
This information is summarized by identifying the linear regression (solid                                                    conclusions (Supplementary Fig. 4).

presented here will serve to elevate the discourse on scientific and                                                          Average-fecundity z-score. By the central limit theorem, the average of variates
managerial impact.                                                                                                            drawn from p(kcjtc) is normally distributed because p(kcjtc) is well described by a
                                                                                                                              mixture of discrete exponential distributions that has finite variance. Given a set
METHODS SUMMARY                                                                                                               of child fecundities, Kc 5 {kc}, we quantify how significantly a subset of these
Data acquisition. We use data from the Mathematics Genealogy Project17 to                                                     child fecundities, Kc* , Kc, deviates from Kc by measuring the z-score of Ækcæ, the
identify the 7,259 protégé mathematicians that are in the giant component28 and                                             average child fecundity of all nodes within the subset Kc*, compared with Ækcæs,
graduated between 1900 and 1960, of which 4,447 have linked publication                                                       the average child fecundity computed for children within a subset equivalent to
records through the American Mathematical Society’s research database                                                         Kc* in the synthetic networks. That is, we compute z 5 (Ækcæ 2 m)/s, where m is
MathSciNet. We use a text-matching algorithm29 to semi-automatically match                                                    the ensemble average of {Ækcæs} and s is the standard deviation of the ensemble
members of the NAS with mathematicians from the Mathematics Genealogy                                                         {Ækcæs} over the 1,000 realizations generated for our null models.
Project.                                                                                                                      Full Methods and any associated references are available in the online version of
Monte Carlo hypothesis testing for p(kjt). We use Monte Carlo hypothesis                                                      the paper at www.nature.com/nature.
testing30 to determine whether equation (1) with maximum-likelihood23 para-
meters H can be rejected as a candidate model for p(kjt) at the a 5 0.05 signifi-                                             Received 21 December 2009; accepted 19 March 2010.
cance level.
Random-network generation. We use a variation of the Markov chain Monte                                                       1.     Kram, K. E. Mentoring at Work: Developmental Relationships in Organizational Life
                                                                                                                                     (Scott Foresman, 1985).
Carlo algorithm25,26 to construct each of the 1,000 random networks in ensembles
                                                                                                                              2.     Chao, G. T., Walz, P. M. & Gardner, P. D. Formal and informal mentorships: a
I and II. Specifically, we restrict the switching of end points of links p R c that
                                                                                                                                     comparison on mentoring functions and contrast with nonmentored
belong to the same link class L, where the link classes are defined as                                                               counterparts. Person. Psychol. 45, 619–635 (1992).
LI(t) 5 {p R cjtc 5 t} and LII(s, t) 5 {p R cjtp 5 s, tc 5 t} for networks from                                               3.     Scandura, T. A. Mentorship and career mobility: an empirical investigation. J.
ensembles I and II, respectively. Each link class can be thought of as a subgraph,                                                   Organ. Behav. 13, 169–174 (1992).
which can then be randomized in the usual way by attempting 100 switches per                                                  4.     Aryee, S., Chay, Y. W. & Chew, J. The motivation to mentor among managerial
link in the class25,26.                                                                                                              employees. Group Organ. Manage. 21, 261–277 (1996).
                                                                                                                                                                                                                           625
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LETTERS                                                                                                                                              NATURE | Vol 465 | 3 June 2010


5.    Allen, T. D., Poteet, M. L., Russell, J. E. A. & Dobbins, G. H. A field study of factors   21. Moed, H. F. Citation Analysis in Research Evaluation (Springer, 2005).
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doi:10.1038/nature09040



METHODS                                                                                  using maximum-likelihood estimation23. Second, we compute the test statistic, S
Mathematics Genealogy Project data. We study a prototypical mentorship                   (detailed below), between the model M(Ht) and the empirical fecundity distri-
network collected from the Mathematics Genealogy Project17, which aggregates             bution, p(kjt). Next, we generate a synthetic fecundity distribution, ps(k), from
the graduation dates, mentors and advisees of 114,666 mathematicians from as             model M(Ht) using the best-estimate parameters, ht, and we treat the synthetic
early as 1637. From this information, we construct a mathematician genealogy             data exactly the same as we treated the empirical data: first, we calculate the best-
network in which links are formed from a mentor to each of his or her k protégés.      estimate parameters, Hs, for model M from maximum-likelihood estimation;
   The data collected by the Mathematics Genealogy Project are self-reported, so         second, we compute the test statistic, Ss, between the model M(Hs) and the
there is no guarantee that the observed genealogy network is a complete descrip-         synthetic fecundity distribution, ps(k). We generate synthetic fecundity distribu-
tion of the mentorship network. In fact, 16,147 mathematicians do not have a             tions and their corresponding synthetic test statistics until we accumulate an
recorded mentor and, of these, 8,336 do not have any recorded protégés. To             ensemble of 1,000 Monte Carlo test statistics, {Ss}. Finally, we calculate a two-
avoid having these mathematicians distort our analysis, we restrict our analysis         tailed P value with a precision of 0.001. As is customary in hypothesis testing, we
to the 90,211 mathematicians that comprise the giant component28 of the net-             reject the model M at time t if the P value is less than a threshold value. We select a
work; that is, we restrict our analysis to the largest set of connected mathema-         P-value threshold of 0.05; that is, if less than 5% of the synthetic data sets exhibit
ticians in the mathematician genealogy network.                                          deviations in the test statistic that are larger than those observed empirically, the
   Although the Mathematics Genealogy Project contains information on math-              model is rejected at time t.
ematicians from as early as 1637, this does not necessarily indicate that all of these      Because we are conducting hypothesis tests with the fecundity distribution
records are representative of the evolution of the network. For example, before          p(kjt), which is a distribution with a discrete support, it is important to use a test
1900 the Project records fewer than 52 new graduates per year worldwide.                 statistic S that is appropriate for testing discrete distributions. We use the x2 test
Furthermore, because mathematicians often have mentorship careers lasting                statistic whereby we bin p(kjt) such that each bin has at least one expected
50 years or more, we are not guaranteed to have complete mentorship records              observation according to the model M(Ht). This binning prevents observations
for mathematicians who graduated after 1960. We therefore restrict our analysis          that are exceptionally rare from dominating our statistical test and skewing our
to the 7,259 protégé mathematicians who graduated between 1900 and 1960, for           results.
whom we believe that the graduation and mentorship record is the most reliable.          Random-network generation. We use the Markov chain Monte Carlo algo-
MathSciNet data. Of the 7,259 protégé mathematicians that graduated between            rithm25,26 to build random networks from the mathematician genealogy net-
1900 and 1960, 4,447 of them have linked MathSciNet publication records,                 work. The standard version of this algorithm inherently preserves the
which are used in our analysis.                                                          fecundity of each individual, but it does not preserve the chronology of child
US National Academy of Science data. The US National Academy of Science                  births, {tc}, for each parent. To obtain random networks belonging to ensemble I
maintains two databases of its membership. The first database consists of all            or ensemble II, we restrict the switching of end points of links p that belong to the
deceased members elected to the NAS from as early as 1863. This database                 same link class L, where the link classes are defined as LI(t) 5 {p R cjtc 5 t} and
records the name of the inductee, their election year, their date of death and a         LII(s, t) 5 {p R cjtp 5 s, tc 5 t} for networks from ensembles I and II, respectively.
link to a biographical sketch. The second database consists of all active members        Each link class can be thought of as a subgraph, which can then be randomized
of the NAS. This database records the name of the inductee, their institution,           using the Markov chain Monte Carlo algorithm. Here, we attempt 100 switches
their academic field and their election year.                                            per link in each link class, which sufficiently alters random networks away from
   The challenge to matching this data with the Mathematics Genealogy Project            the original empirical network25,26. We repeat this procedure 1,000 times to
data is that there is no direct link between a member of the NAS and the                 generate a set of 1,000 random networks for each ensemble.
Mathematics Genealogy Project, and vice versa. This is further confounded by             Average-fecundity z-score. The average of variates drawn from p(kcjtc) is normally
the fact that some members of the NAS have the same name. To circumvent these            distributed because p(kcjtc) is well described by a mixture of discrete exponential
problems, we use a text-matching algorithm29 to semi-automatically detect whether        distributions—a distribution with finite variance—and, thus, the central limit
a member of the NAS matches a name in the Mathematics Genealogy Project                  theorem applies. Given a set of child fecundities, Kc 5 {kc}, we quantify how
database. We use this procedure to curate the 269 members of the NAS that                significantly a subset, Kc*, of these child fecundities deviates from Kc, by measuring
definitively match mathematicians in the Mathematics Genealogy Project database.         the z-score of Ækcæ, the average child fecundity of all nodes within the subset Kc*,
Monte Carlo hypothesis testing for p(kjt). Given a model, M, with parameters             compared with Ækcæs, the average child fecundity computed for children within a
Ht for the empirically observed fecundity distribution, p(kjt), we use Monte             subset equivalent to Kc* in the synthetic networks. That is, we compute
Carlo hypothesis testing to determine whether it can be rejected as a candidate          z 5 (Ækcæ 2 m)/s, where m is the ensemble average of {Ækcæs} and s is the standard
model for p(kjt) (ref. 30). The Monte Carlo hypothesis testing procedure is as           deviation of the ensemble {Ækcæs} over the 1,000 realizations generated for our null
follows. First, we calculate the best-estimate parameters, ht, for model M at time t     models.




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