# Mid-career pitfall of consecutive success in science

Noriyuki Higashide  Department of Technology Management for Innovation, The
University of Tokyo, Tokyo 113-8656, Japan  Takahiro Miura  Department of
Technology Management for Innovation, The University of Tokyo, Tokyo 113-8656,
Japan  these authors contributed equally to this work  Yuta Tomokiyo
Department of Technology Management for Innovation, The University of Tokyo,
Tokyo 113-8656, Japan  Kimitaka Asatani  Department of Technology Management
for Innovation, The University of Tokyo, Tokyo 113-8656, Japan  Ichiro Sakata
Department of Technology Management for Innovation, The University of Tokyo,
Tokyo 113-8656, Japan

###### Abstract

The creativity of scientists often manifests as localized hot streaks of
significant success. Understanding the underlying mechanisms of these
influential phases can enhance the effectiveness of support systems and
funding allocation, fostering groundbreaking discoveries worthy of accolades.
Historically, analyses have suggested that hot streaks occur randomly over
time. However, our research, through meticulous examination, reveals that
these phases are not flatly distributed but are more frequent at the early and
late stages of scientists’ careers. Notably, both early and late hot streaks
are marked by dense tie collaborations, with the former typically involving
close partnerships with particular authors and the latter being characterized
by involvement in large-scale projects compared with single-top or ordinary
papers. This pattern indicates that mid-career researchers lack both intimate
relations and resources to keep big projects, leading to “mid-career pitfall”
of consecutive success. This insight holds profound implications for the
development of policies and initiatives aimed at bolstering innovative
research and discovery.

## Introduction

Over the past half-century, the role of scientists has expanded beyond simply
writing quality papers, making it more difficult to focus solely on academic
interests due to the growing need for broader engagement in academia.
University faculty members are increasingly busy with committee meetings for
selecting other faculty members and Ph.D. candidates, grant reports and
presentations, teaching, meetings with lab team members, trips to discuss with
external collaborators, and attending a kick-off event for a new research
consortium[1]. This diversification of a scientist’s skill set dilutes the
resources available for accelerating sciences[2], even emerging the
possibility that the labor advantage in scientific fields could be becoming a
bottleneck in driving productivity[3]. Given the escalating challenges in
achieving remarkable results with limited time and resources, it is important
to understand the mechanisms that enable scientists to be actively engaged in
their work from a more systematic and ecosystem-based perspective. This
understanding is crucial for research support, including those involved in
research management initiatives, funding, and science and technology policy,
as it poses a significant challenge that needs to be clearly addressed.

The phenomenon known as hot streaks[4, 5] is one of the distinguishing markers
of a researcher’s engagement. Einstein’s “miracle year,” where he made four
revolutionary discoveries, are typical example of a hot streak. New ideas
often do not arise in isolation but are part of a chain where one discovery
sparks another. Previous research shows that scientists’ top three papers are
typically produced in a consecutive sequence[4], so scientific success
somewhat manifests with a curious continuity. Triggers for consecutive success
may be more valuable than triggers for a singular success because single
success may merely represent the fortunate extraction of a good idea from a
universal distribution of ideas[6], whereas the hot streak can be seen as a
‘bonus time’ that elevates the impact of every output over a certain
period[4]. Thus, uncovering the universal patterns of hot streaks can help
efficiently boost researchers’ intellectual creativity, potentially
revolutionizing how we stimulate and sustain scientific innovation.

In investigating the mechanisms that could trigger hot streaks, a type of
career success, the career ages should be considered[7]. Hot streaks are
supposed to occur with a certain probability throughout any career ages[4],
but if we assume this legitimacy, consistently publishing consecutive high-
impact works during the mid-career stage seems quite challenging due to the
large number of involved projects, heavy workloads, funding issues, and other
factors that demand attention. These challenges are explained in Nature
Career’s special podcast series “Muddle of the Middle”[8]. In their mid-
career, having recently earned their Ph.D. and within the first few years of
obtaining tenure, face a unique set of challenges compared to their younger
counterparts who often enjoy more funding and support, or even harder than
senior scientists who have settled assets and experienced skills to manage
various activities. These mid-career researchers are required to juggle an
array of responsibilities, including managing their first teams with limited
budgets, preparing new lectures, and consistently publishing papers to secure
their next positions. Consequently, mid-career researchers may find it
difficult to allocate enough time for both exploration and exploitation which
often marks the onsets of hot streaks[5].

It’s valuable to examine how hot streaks, i.e. consecutive successes, of
scientists differ by career stages. Scientists’ academic career successes are
related to their collaboration patterns such as team formations and co-
authoring[9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Hot streaks are more
likely to be produced by larger teams[5], but team composition approaches vary
among career stages. Early-career researchers might be likely to focus on
building fresh connections while working closely with their supervisors[12,
13, 14]. In contrast, senior researchers might leverage their established
networks with previous collaborators and effectively manage super-ties,
ensuring long-term collaboration[9, 18]. What kind of collaboration produces
consecutive success?

In this study, we investigated when top papers are produced in a researcher’s
career and what factors are behind their creation depending on career stages,
using a large-scale literature database. Contrary to previous studies, our
results revealed a U-shaped pattern, indicating that hot streaks are more
common in the early and later stages of a career. For the early and later
stages, we clarified the differences in supporting factors from the
perspectives of team collaboration.

## Consecutive successes concentrate on a career’s early and late stages

A researcher’s career history of publication impact is used to identify
consecutive success. The career sequence dataset was crafted from over 100,000
scientists with careers spanning more than 20 years and 30 publications
(Supplementary information 1). We defined a consecutive success as a duration
when scientists’ top-$k$% most impactful papers are concentrated $X$ times
within $N$ publications (see Methods for mathematical descriptions). In Figure
1a, for example, the top $10\%$ high-impact works indicated by blue circles,
are concentrated with $X=3$ out of $N=5$ works in the period from 12 to 16
marked in orange which represents a consecutive success period. If these
periods are detected consecutively within overlapping windows, these are
consolidated and considered as a single consecutive success with a longer
length. Various parameters can be set to detect consecutive successes. A
larger $N$ indicates longer consecutive successes, while a smaller $N$ means
shorter ones. Instances detected with parameters like $X/N=5/9$ are regarded
as strong hot streaks, and $X/N=1/1$ captures the weakest form of ones, which
is a single success. The occurence timing of consecutive success is the
starting position of consecutive $N$ papars. The timing is normalized for
different career lengths among researchers. For example, the relative timing
of the $i$-th paper in a career is calculated as $i/(N_{T}-N)$, where $N_{T}$
is the total number of published papers.

Figure 1: (a) Definition of consecutive success. Top $k$% impactful papers are
shown as blue circles and they appear $X$ times within $N$ publications with
parameters $X=3,N=5,k=10\%$. We use the number of publications in sequence as
the length of the researchers’ career instead of years. (b-d) Probability
distribution when consecutive success occurs in the scientists’ careers. The
x-axis, relative timing, represents normalized career length. The numbers in
parentheses represent the percentages of researchers, out of 100,000, who have
experienced at least one hot streak under that parameter. As consecutive
success becomes stronger, the number of researchers experiencing consecutive
success decreases. (b) The distributions of consecutive success with different
length-related parameters $X$ and $N$ ($k=10\%$). A single success ($X/N=1/1$)
occurs with a constant probability but long continuous successes ($X/N=5/9$)
occur more frequently in the early and late throughout the career. (c) The
distributions of consecutive success for different $k$ ($X/N=3/5$). Big
success occurs more in the early and late stages. (d) compares the
distribution between the raw and shuffled career sequences with parameters
$X/N=3/5$ and $k=10\%$.

The previous study captured consecutive success by using the hot-streak
model[4]. The hot-streak model assumes a temporal rise in a researcher’s
potential and thus is applied for the moving average of career sequence data
with the window size of $\Delta N=max(5,0.1N_{T})$. The size is ceiled by 10%
of a career length to smooth out influence by random factors. However, we find
that the window size could determine the duration of hot streaks
(Supplementary information 3.3). The average duration of 3.7 years derived by
the hot-streak model may be an artifact based on single successes stretched by
the window size, suggesting it may not accurately capture the true sequence of
consecutive successes in a career. Contrarily, our approach more directly
characterizes consecutive successes by examining raw career sequence data with
fewer artificial assumptions.

Detected consecutive successes with several parameters show intriguing
characteristics. Firstly, we found that these are less likely to occur in the
mid-stage of a career but are more likely to appear in the early and late
stages (Fig. 1b-d). Especially, single success are spread flat in the career,
but consecutive successes concentrate on career beginnings and ends (Fig. 1b).
Further, moderate successes, $k=50\%$ for example, happen randomly, but
intensive successes like $k=5\%$ are more likely to happen in the early and
later phases (Fig. 1c). Shuffling the order of career sequence data eliminates
the U-shaped distribution, which means that the pattern of early and late
career successes is not a coincidental phenomenon (Fig. 1d). These findings
indicate that consecutive successes do not occur at a constant probability
throughout a career. Instead, the likelihood of their occurrence is
concentrated on the early and late stages, especially for successes of greater
length and intensity.

Looking at the starting year distribution of our consecutive success
extraction, the observed years of consecutive success tend to increase as
careers progress into their later stages. However, compared to instances of
single hits, the mid-career periods, from 7 years after the career begins to
5-10 years before the career ends, show lower likelihoods of achieving
consecutive hits (Supplementary information 2.1). This illustrates the
dynamics whereby true consecutive successes manifest initially during the
Ph.D. and early postdoctoral phases, subsequently subside during the mid-
career period, and become less likely until the final stages of one’s career.
These U-shaped patterns demonstrate robustness over time (Supplementary
information 2.2) and, even when considering variations in career dynamics of
success across different fields, mostly hold true (Supplementary information
2.3). Our identification method shows U-shaped distributions over the career
but the hot-streak model demonstrates that consecutive successes occur with
consistent probability throughout careers[4]. We estimated that this
discrepancy mainly comes from preprocessing for career sequence data and false
positive detection of the previous method (Supplementary information 3).

## Patterns behind consecutive successes

Figure 2: Characteristics of consecutive success. (a) A typical example of a
‘Hot’ career sequence. The period filled in orange is detected as a hot streak
of five consecutive papers. Blue dots represent top 10% papers, appearing
three times within the five papers. (b) An example of ‘Top’ sequence,
represents a period of single success, not consecutive. A green dot indicates
a top 10% paper, appearing once among five papers. (c) ‘Ordinary’ sequence
example as comparisons, five consecutive non-top 10% papers, representing a
period without significant hits in careers. (d-i) Comparison of the 3
sequences across 6 metrics. The relative career timing is divided into ten
bins, and metrics are calculated for sequences with the same start timing in
each bin. This allows observation of how metrics vary in early, mid, and late
career stages. (d) Team size, focused on author lists of fewer than 10 people
(Supplementary information 4). (e) Proportion of teams with 10+ members. (f)
Proportion of sequences with dense ties, appearing 3 or more times in the co-
author list of the 5 consecutive papers. (g) Betweenness centrality of the
focal author in the co-authorship network of the 5 papers. (h) The number of
topics tackled in the 5 papers, ranging from 1 to 5. (i) Proportion of new
topics not previously tackled. In (d) and (f-i), calculations are performed on
sequences where the team size for all 5 papers is less than 10. The bars in
(d,g,h) represent the 95% confidence interval that includes the estimated
population mean, assuming a normal distribution. See Methods for details of
definitions and calculations of each metric.

To understand factors related to the occurrence of consecutive successes, we
characterize the career-stage dependency of indicators related to team
activities. Science and collaborations are closely intertwined, with factors
such as team size[11], the strength of connections between authors[9, 16], and
the topics[10, 18] particularly influencing scientific outcomes. Based on
these indicators, our findings revealed unique patterns associated with
consecutive successes.

We defined three types of career sequences: ‘Hot’, ‘Top’, and ‘Ordinary’ (Fig.
2a-c). ‘Hot’ represents consecutive success with parameters of $X=3$, $N=5$,
$k=10\%$, that is five consecutive papers by the same author including at
least three highly-cited ones. ‘Top’ denotes single success, including five
consecutive works starting with a top 10% paper but fewer than three high-
impact papers. ‘Ordinary’ works comprise five non-top 10% papers. Our analysis
covered a sufficient number of each across all career stages: 47,356 ‘Hot’,
25,227 ‘Top’, and 39,278 ‘Ordinary’ sequences (see Methods). This comparison
helps us to uncover several patterns of ‘Hot’ career sequences.

1\. Large teams: We found that larger teams are more common in later career
stages and for ‘Hot’ sequences, suggesting that consecutive successes are more
likely to arise from larger teams than ‘Top’, and ‘Ordinary’ (Fig. 2d), which
is consistent with the previous study[5]. This trend is measured in teams of
nine or fewer members.

2\. Big projects: We defined teams of 10 or more as ’big projects’ and
calculated the probability of such projects. If at least one paper in a
consecutive period is a big project, the sequence is counted as a big project.
The later career stages have a higher probability of big projects, with ‘Hot’
sequences notably more prevalent than the other two types (Fig. 2e),
indicating that big projects are likely to be in a series of high-impact works
of an individual’s career. The probability of ‘Hot’ sequences having papers
with more than 100 authors is about ten times higher than the other two types,
no matter the career stage (S15).

3\. Dense ties: To understand collaboration patterns in depth, we analyzed co-
authorship networks of five consecutive papers (Methods). Observing the
presence of dense ties, defined as co-authoring three or more of the five
papers with the same authors, we found that ‘Hot’ sequences consistently show
a higher probability of dense connections throughout a career (Fig. 2f). We
also hypothesized that authors with consecutive successes might have higher
betweenness centrality in co-authorship networks, acting as hubs connecting
different author groups. However, no significant overall difference is
observed, suggesting that the network structure of collaborations doesn’t vary
significantly (Fig. 2g). A higher collaboration rate with specific authors
could contribute to experiencing successful periods.

4\. Topic focus: The diversity of research topics in five consecutive papers
is investigated (Methods). ‘Hot’ sequences generally exhibited less topic
diversity, indicating a more focused approach and the pattern shows an inverse
U-shape with a decrease in early stages, an increase in mid-career, and a
decrease in later stages (Fig. 2h). The stronger focus in early and late
stages might relate to the distribution of consecutive successes being more
common in these phases.

5\. New topics: We also examined the probability of new topics tackled during
the sequences, which had not been addressed in the author’s previous career
(Methods). In ‘Hot’ sequences, throughout a career, the likelihood of
venturing into new topics is lower than in the other two types (Fig. 2i). This
indicates that revisiting previously tackled research topics is related to
consecutive success, providing hints for topic selection strategy.

Overall, consecutive successes tend to occur in larger teams, with frequent
collaboration with the same authors, focusing on fewer topics, and these
topics are often previously addressed. These patterns suggest particular
strategies and environments facilitating successive high-impact scientific
works at different career stages.

## Early consecutive success from dense ties, later from large teams

What explains the U-shape distribution of consecutive successes along careers?
In the early stages, we see small teams, dense ties, and a focus on research
topics. These suggest that early-career researchers experience hot streaks
with working intensively with a few co-authors, such as mentors and their
group members. As careers progress to the middle stage, the number of
collaborators increases, the probability of working with specific individuals
decreases, and topic diversity grows. This trend might reflect the expanding
collaborator network over time. In later stages, team size grows further and
researchers tend to focus on specific research topics. The established
reputation and authority of senior scientists might enable them to steer large
teams toward focused topics.

Given the distributions of consecutive successes involving dense ties and big
projects (Fig. 2e, f), we find that these two factors contribute to the
U-shape distribution shown in Figure 1b. Consecutive successes within loose
ties, which do not involve as frequent collaboration as dense ties, occur with
almost constant probability throughout a career, while those involving dense
ties contribute to the spike at the early stages and the slight increase in
the later stages (Fig. 3a). As careers progress into the later stages, the
proportion of consecutive success originating in small teams decreases, while
those arising from larger teams increase. Hereinafter, small teams mean less
than 10 members and large teams mean ten or more (i.e. big projects). These
indicate that both early and late career successes are dominated by dense
ties, while later stages see stronger dominance by larger teams. Thus, by
considering two factors, the strength of connections and team size, we can
categorize types of consecutive successes into four distinct groups. Notably,
about 40% of consecutive successes originate from teams of more than ten
members (Fig. 3b). Each type is intuitively understood by visualizing their
co-authorship networks (Fig. 3c-f).

Figure 3: Four types of consecutive successes. (a) The histogram when
consecutive successes occur with parameters $X/N=3/5$ and $k=10\%$. The four
types of them are displayed in a stacked manner, each represented by a
different color. The top two types, shown in pink and red, represent
consecutive successes with dense ties, more common in early and late stages.
The bottom two, in light blue and blue, depict loose ties occurring
consistently across the career. (b) The four types’ proportions. 65% of them
originated from dense ties and 40% originated from large teams. (c-f) Typical
co-authorship networks for each type. Green nodes represent the focal author,
other nodes are co-authors, and edges indicate the number of collaborations in
five consecutive papers, ranging from one to five. (c) Dense ties in small
teams are shown as thick edges between a few nodes. (d) Thick edges with large
teams composed of many co-authors. (e) The focal author is connected with
fewer authors and thinner edges representing loose ties and small teams. (f)
No thick edges with many co-authors show collaboration in loose ties with
large teams.

Dense ties and large teams can explain why mid-career scientists less
frequently experience consecutive successes. During mid-career, dense
connections are fewer than in the early stages, and the team size is not
sufficiently large. This situation could make consecutive successes more
challenging.

As examples, we consider two physicists in condensed matter physics. The field
has received significant contributions from Japanese scientists, including
some renowned researchers. Tsuneya Ando made significant theoretical
contributions, predicting the quantum Hall effect in semiconductors. Sumio
Iijima is a great experimental researcher who discovered carbon nanotubes.
Both have produced over 350 papers throughout their careers, showing their
high productivity. Ando produced high-impact works early in his career, going
through four consecutive successes (Fig. 4a), with his most significant work
included in his second one. On the other hand, Iijima had three consecutive
successes early in his career and then another later on (Fig. 4b). Tracing
their work trajectories revealed interesting differences. Upon normalizing the
length of their careers and comparing team size, collaboration density, and
topic diversity, it was found that Iijima worked with larger teams later in
his career (Fig. 4c). The maximum co-author frequency and topic diversity
across five papers tend to be higher for Iijima throughout his career, but the
difference is relatively small (Fig. 4d,e). Indeed, in the later stages of his
career, Iijima produced his most significant paper in graphene, outperforming
his early discoveries, with over 10 co-authors. Such a pattern of consecutive
success with larger team sizes in the later career aligns with the overall
trend observed among scientists (Fig. 2d).

Figure 4: Two scientists’ typical career dynamics. (a,b) Career sequences. The
papers with top-10% impact are colored and the orange areas indicate
consecutive success identified by our method. (c) Relation between papers’
publication timing in normalized career length and their team size for two
scientists. (d) Frequency of co-authors in five consecutive papers; higher
values mean repeated collaborations with the same authors, reflecting dense
ties. (e) Topic diversity dependency on career stages, measured by the variety
of topics in five consecutive papers. For clarity, (c-e) display moving
averages with a window size of 5% career length.

## Discussions

In this study, we hypothesized that the mechanisms underlying consecutive
success vary depending on career stages and thus focused on analysis from the
perspective of collaboration patterns. The hot-streak model[4] may not always
capture continuous success because, in the process of applying moving
averages, there’s a possibility that single successes may be mistakenly
identified as consecutive successes (Supplementary information 3.3).
Extracting consecutive successes by more directly observing sequences of
highly-cited works, we found that a series of successes is more likely to
happen in the early and later stages of a career, with a dip during mid-
career. Early-career hot streaks are more likely among researchers with dense
ties to specific colleagues, whereas late-career hot streaks are often
supported by big projects on familiar research topics.

The uncovered empirical regularities highlight the importance of considering a
scientist successes at each career stages. Early in the career, dense
connections are present, but as the career progresses and the academic network
expands forward, larger team sizes emerge, showing a complementary
relationship. The “mid-career pitfall” may be attributed to the loss of these
two factors.

The mid-career, spanning from the 10th to 20th year, is challenging. Increased
independence and responsibility, departures from initial support structures,
and the expansion of administrative tasks requiring multitasking make it
difficult for researchers to focus on specific topics and allocate resources.
Researchers not only move between organizations seeking positions[20] but also
belong to more research communities, frequently switching topics[10]. These
career dynamics increase the cost of adapting to new environments. In addition
to these factors, a decline in the number of publications in mid-career[21],
looks influence mid-career dip of consecutive success. The discontinuous
success in mid-career may be due to limited time, human resources, and the
difficulty of returning to familiar settings. Yet, in the later stages,
forming large teams can contribute to a revival of hot streaks. Taken
together, this suggests that if a stable research foundation is established
earlier in the mid-career, it would lead to greater achievements which could
further accelerate scientific progress.

The reality of mid-career researchers’ support is also reflected in national
research support policies, such as the Japan Society for the Promotion of
Science, which divides funding opportunities into categories open to all and
those exclusively for young researchers[22]. Consequently, mid-career
researchers are forced to compete with more experienced researchers for
funding, potentially influencing the occurrence of hot streaks. Single
successes can occur randomly at any career stage, suggesting that such kind of
success can be achieved even without funding, but sustaining successes and
maximizing output on specific topics might require new funding strategies
targeted at the overlooked mid-career researchers.

The prevalence of hot streaks in the early stages of a career suggests a
potential survivorship bias within the scientific community. Our study,
focusing on researchers with at least 30 publications over a 20-year period,
indicates that those who peak in the first five years of their career are
likely to sustain their research career with ease. However, this approach may
overlook hidden talents not represented in our data. The emergence of hot
streaks among researchers with dense ties, particularly in the early stages,
implies that survival as a scientist may depend not only on individual
potential but also on the chance of forming a hot streak through good
mentorship initially. Prior research on the positive effect of connecting with
top researchers early in one’s career [12] suggests that this not only impacts
the magnitude of later success but also the likelihood of sustaining a
research career.

The divergence of success distribution due to exploration, followed by a re-
concentration towards the end of a career, is highly suggestive. Researchers
in their later career stages shine by forming large teams of more than ten
people and refocusing on familiar topics. This is particularly evident in
experimental fields that can expand the scale of experiments with significant
budgets, such as Health Science and Manufacturing Engineering, where many hot
streaks are experienced in the last 10% of a career (Fig. S4). In contrast,
fields emphasizing theory, like Algebraic Geometry and Network Science,
experience fewer late hot streaks. This disparity reflects the relationship
between funding strategy and fields where costs and outcomes are more or less
directly proportional. In experimental fields, forming large teams can lead to
hot streaks at any age.

Our study has limitations common to this style of analysis. Firstly, the
results may be biased due to the dataset used. Since Scopus is limited to
reputable journals designated by Elsevier, the observed consecutive successes
are confined to works of a certain level of renown. By restricting data to the
years 1970 to 2012, the dataset may not fully capture the beginning or end of
the careers of individuals who started their careers before 1970 or whose
careers continued beyond 2012 (Supplementary information 1.2). Secondly, while
this study focuses on common trends in how collaboration patterns change
during hot streaks across careers, there also is the diversity across
disciplines (Fig. S4). The U-shape pattern does not appear universally across
all fields, indicating that a more detailed analysis, delving into the content
of the disciplines, is necessary in the future to understand what determines
the occurrence of hot streaks. The patterns discovered in this study
specifically target only academic careers and may not directly apply to a
wider range of creative fields, such as artists, film directors, musicians, or
ballet dancers[5, 23, 24]. Lastly, the findings of this study represent
correlations rather than causations. While it is true that dense ties in the
early stages and larger teams in the later stages are commonly observed during
current hot streaks, real experiments are required to determine whether
interventions can lead to the occurrence of hot streaks.

## Methods

Detection of consecutive successes. Given an author $a$, the set of papers
published by this author is denoted as
$P_{a}=\\{p_{1},p_{2},...,p_{N_{T}}\\}$, where $N_{T}$ is the total number of
published papers. The 10-year normalized citation count for a paper $p$ is
represented by $C_{10}^{p}$. The subset of papers $P_{a}^{\text{k}}\subset
P_{a}$ includes papers whose $C_{10}^{p}$ are in the top $k\%$ of all
$C_{10}^{p}$ in $P_{a}$. For a specified window $N$, a time interval from $i$
to $i+N$ is considered as a consequtive success if:

$\left|\\{p\in P_{a}^{\text{k}}\ |\ p\in p_{i},...,p_{i+N}\\}\right|\geq X$
(1)

Here, $|\cdot|$ denotes the cardinality of a set. Eq. (1) indicates that if
the number of papers in $P_{a}^{\text{k}}$ published within the window from
$i$ to $i+N$ is more than $X$, the interval to be considered consequtive
success. This method detects overlapping periods in a career sequence, and we
have incorporated a process to merge such instances. As a result, the length
of consecutive success is usually five papers, but it can be longer. The
number of people experiencing hot streaks and the frequency of these
occurrences vary depending on the definition (Fig. 1b-c). Compared with
previous hot-streak model[4, 5], which capture the potential change of
reseacher’s output, this definition is more data-centric approach focusing on
the actual output of researchers.

Some authors experience more than one hot streak during their careers. Under
the conditions of $X=3,N=5,k=10\%$, this proportion is 13%. We also found that
authors who produce more papers over their careers tend to experience hot
streaks more frequently, and those who have hot streaks generally have a
higher overall career impact. When handling career sequence data, we use the
number of publications instead of years to normalize career patterns that vary
across fields due to different publication frequencies and numbers[6].

Three types of sequences. To compare patterns of consecutive success across
different career stages, we defined ‘Hot’, ‘Top’, and ‘Ordinary’ sequences
(Fig. 2a-c). To determine these, first, we identified 47,356 hot streaks from
35,495 authors with the parameters $X=3$, $N=5$, and $k=10\%$. Some hot
streaks were longer than five papers due to merging processes, but only the
first five papers were selected for ‘Hot’ sequences to align with the length
of ‘Top’ and ‘Ordinary’. Next, for the 64,505 authors without hot streaks, we
randomly chose five consecutive papers from their career data. If one or two
of five papers were in the top 10%, the sequence was classified as ‘Top’,
resulting in 25,227 cases. Otherwise, it was categorized as ‘Ordinary’,
accounting for 39,278 cases. Each sequence was tagged with the relative
publication timing of its first paper, scaled between 0 and 1, to analyze
within the same career stages. Various metrics were then calculated, based on
the information from these five papers, such as author lists and topics.

Team sizes. For the dataset used in this study, which includes 9,476,817
papers, the average number of authors per paper is 28, with a median of 4, a
minimum of 1, and a maximum of 3,220. Notably, 88% of the teams consist of
fewer than ten members. Considering previous study[11] and computation time,
the characterisation in Figure 2 were performed for sequences with all five
papers having fewer than ten authors each. We used a team size of ten as a
benchmark for defining team sizes. ‘Big Project’ is defined as a sequence of
five papers where at least one paper includes more than ten authors (Fig. 2e).
The classification of whether a consecutive success involves large teams is
based on whether at least one of the papers has more than ten authors (Fig.
3a).

Dense ties and loose ties. To evaluate the collaboration patterns of a focal
author’s five consecutive papers, we examined their co-author list. If a co-
author appears in all five papers, it signifies a strong collaboration, giving
a maximum co-author appearance of five. If each paper has a different team
without any overlaps, the value is one. We define sequences with a co-author
in at least three out of five papers as having ‘dense ties’.

Topic diversity. The research topics of papers are calculated from co-citation
network clustering of each author’s publication[10], applying modularity
maximization by Louvain method with resolution $=$ 1\. For each consecutive
paper, a topic diversity score is defined as the number of topics in these
papers. The score is five when all papers are on different topics, while a
score of one indicates all papers share the same topic. This score reflects
the focal author’s focus on a single or multiple research topics during that
period. The topic of each paper is calculated from clustering co-citation
network on each researcher’s publications [10] (Supplementary information
1.1).

New topics. Within the targeted five papers sequence, we identify the most
frequent topic $T_{f}$. If there is a tie for the highest frequency, the first
topic appearing in the sequence is $T_{f}$. If $T_{f}$ does not exist in the
author’s earlier career, the attribute ‘New Topic’ is true; if it was present
before, it is False.

## Acknowledgements

The authors thank D. Wang, Y. Matsuo, and all members of our research group
for their invaluable comments.

## Author contributions statement

N.H. and T.M conceived the project; N.H., T.M., Y.T. and A.K. collected data
and performed analysis; N.H and T.M. wrote the manuscript, I.S supervised the
project, all authors discussed results and edited the manuscript. Higashide
and Miura have agreed that, due to their equal contribution as co-first
authors, the order of their names can be interchanged when citing this paper
or listing it in their CV.

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## Supplementary information

[1,+*]Noriyuki Higashide [1,+]Takahiro Miura [1]Yuta Tomokiyo [1]Kimitala
Asatani [1]Ichiro Sakata

###### Contents

  1. 1 Data description
    1. 1.1 Career sequence data
    2. 1.2 Potential limitation of datasets
  2. 2 Probability of consecutive success
    1. 2.1 Year distribution of U-shape success
    2. 2.2 Time dependency of U-shape success
    3. 2.3 Field dependency of U-shape success
  3. 3 Relation with the existing model
    1. 3.1 Empirical measurements
    2. 3.2 Revisiting hot-streak model
    3. 3.3 Dependency of the hot-streak model on window size of $\Gamma(N)$
    4. 3.4 The hot-streak model and our data-centric detection
  4. 4 Collaboration patterns
    1. 4.1 Team sizes
    2. 4.2 Examining if early and late hot streaks come from the same work
    3. 4.3 Disruption index of hot works and works of who experience hot streak

##  1 Data description

###  1.1 Career sequence data

We prepared a home-built dataset of research papers based on Elsevier’s Scopus
Custom Data, encompassing all documents recorded between 1970 and 2021. As the
most comprehensive database for abstracts and citations of peer-reviewed
articles, Scopus contains over 73 million papers and 1.2 billion citations
across a broad range of research fields and hence frequently utilized by
researchers for bibliometric and citation analysis[25, 26, 27, 28]. We focused
on researchers who had sufficiently long careers to characterize timely-
clustered success in life-long careers. Hence, we randomly extracted 100,000
authors who published more than 30 papers and 20 years of their whole careers
from the dataset. To calculate citations within 10 years of publication,
papers published from 2013 to 2021 are removed from the author information.
The historical order of publications was organized on a monthly basis,
according to their publication dates.

To assess each paper’s impact, we measured its citations within 10 years of
publication, denoted as $C_{10}$ [6]. We then adjusted for field-specific and
annual variations by dividing it by the average citations for that field and
year [29]. To obtain field information at the publication level, this study
clustered the whole citation network in 2021 using the Leiden algorithm based
on the Constants Potts Model ($\gamma=10^{-6}$)[30, 31]. Clusters with a size
of less than $10^{5}$ papers were merged into other clusters with the
strongest citation connections[32], resulting in the identification of 66
distinct fields. This re-scaling accounts for calibrating publication
frequency and impacts across disciplines.

Each paper’s topic is calculated by clustering the co-citation network within
each career[10]. The topics were calculated by partitioning the co-citation
network of papers published by each scientist using the Louvain method with a
resolution parameter of $\gamma=1$.

###  1.2 Potential limitation of datasets

The dataset used in this study has several limitations. Firstly, this research
targets researchers who have published 30 or more papers over a span of more
than 20 years in Scopus, which may introduce survivorship bias and publication
bias. For instance, researchers in fields such as law and philosophy, who may
spend years producing a single book or publication, are less likely to be
included in the dataset. Additionally, a plausible hypothesis that researchers
who do not experience a hot streak may exit their careers earlier cannot be
tested with this study. Moreover, publications in languages other than English
are not recorded, which means that different outputs in other languages during
periods of consecutive success might not be captured. Scopus, by incorporating
not only articles and conference proceedings but also book chapters,
editorials, and letters, necessitates the consideration of diverse patterns of
success that may vary according to document type.

Secondly, in the analysis of teams, the size may appear larger than in reality
due to Elsevier’s consolidation of group authors, which unfolds group
contributions into individual ones. Group authors, such as the “CONSORT
Group," “AHA Heart Failure Taskforce," or “XY Workshop Members," attribute
contributions to a group rather than individuals for outputs resulting from
specific activities. Since the number of collaborators in such groups can be
quite large, often in the dozens, this could potentially affect the perceived
team size and density of ties.

Thirdly, although this analysis categorizes career stages as early-career and
late-career within the dataset, the focus on publications from 1970 to 2012
means that individuals who began their careers before or continued their
careers after this period might not have their actual career lengths
accurately represented.

##  2 Probability of consecutive success

###  2.1 Year distribution of U-shape success

Although we have demonstrated the presence of a pitfall of consecutive
successes in the mid-career phase on a relative career basis, it remains to be
determined at what specific year of their career researchers are most
susceptible to this phenomenon. Figure S1 illustrates the timing of
consecutive success occurrences extracted using our model, translated into
periods within a researcher’s career. Researchers were divided into five
groups based on career year length, and the distribution of single and
consecutive successes was shown for each. It was found that consecutive
successes become less likely approximately from seven years after the career
begins to 5-10 years before the career ends. These timings likely correspond
to the end of youth, the later stages of postdoctoral, and the periods heading
towards retirement to complete careers. This indicates that researchers in
their mid-careers may lack sufficient resources for adequate exploration and
exploitation. Note that the continuous increase in the number of single hits,
which remain flat in terms of relative career from the early to the late
stages, suggests that the number of publications tends to increase over time,
possibly due to the utilization of collaborations, among other factors.

Figure S1: The distribution of consecutive success among researchers with
varying career lengths in five-year increments. Regardless of age, the
probability of consecutive success declines from seven years after the career
begins to 5-10 years before the career ends.

###  2.2 Time dependency of U-shape success

The U-shape we observed might not just be about successes at the start or end
of a scientist’s career. Alterations in the dynamics of science can also serve
as a potential confounder. For example, if it is easier to publish high-impact
papers around 2010 because of citation inflation[33], scientists at any time
in their careers could have hot streaks recently. To check this, we split
scientists into five groups based on when they started their careers and how
long their careers were. Then we checked if the U-shape still showed up.

We found that no matter when a researcher started their career (Figure S2) or
how long they’ve been working (Figure S3), their single big successes happen
randomly, but their hot streaks follow a U-shaped pattern. This is especially
true for researchers with longer careers, as they publish more papers and thus
have more chances for these hot streaks. For example, out of 9,537 researchers
who have been publishing for over 40 years, 5,131 (about 53.8%) had a hot
streak where 3 out of 5 of their papers were in the top 10%. This shows that
the U-shaped pattern happens no matter the time period.

Figure S2: The distribution of hot streaks among researchers with varying
initiation periods of career in five years increments. Regardless of the ages,
the pattern of consecutive success delineates a U-shaped curve.

Figure S3: The distribution of hot streaks among researchers with varying
lengths of career in five-year increments. Researchers with careers of any
duration exhibit a U-shaped curve in their pattern of consecutive success.

###  2.3 Field dependency of U-shape success

In scientific contribution, there are crudely two types of studies: empirical
and theoretical. Theoretical successes tend to appear early in a career, while
empirical ones are more common later[7]. To make sure the U-shaped pattern
isn’t just showing these different types of scientific contributions, we
looked at whether the distribution of hot streaks varied across 53 fields when
we identified the most common topic during each hot streak as the topic of
that streak.

We made a scatter plot in Figure S4. It shows if the probability distribution
of hot streaks in each field is more than 1 either early in the career
(relative timing below 0.1) or late (above 0.9). Out of 45 fields where more
than 100 people experienced a hot streak (defined as k=10, X=3, N=5), 21
fields showed a U-shaped pattern with both early and late hot streaks.
However, 18 fields had more early but fewer late hot streaks, like in
Algebraic Geometry or Semiconductor materials and devices, which include both
theoretical and empirical studies. Overall, early career hot streaks are
robust, but whether one experiences late career hot streaks seems to depend on
the field.

Figure S4: The distribution of U-shaped patterns in success across various
fields (defined by k=10%, X/N=3/5). Each hot streak(HS) is categorized by its
most common topic, and the graph plots the proportion of hot streaks occurring
either early (probability P(x)<0.1) or late (probability P(x)>0.9) in a
career. The thick dotted line indicates P(x)=1, with fields located in the
upper right quadrant demonstrating a U-shaped pattern, signifying a higher
occurrence of both early and late career peaks in success. The names of the
fields were assigned based on the highest TF-IDF values of keywords in papers.

##  3 Relation with the existing model

###  3.1 Empirical measurements

Hot streaks are a phenomenon in which high-impact papers are concentrated at a
time throughout a scientist’s career [4]. Although there are minor data
differences, we were able to confirm that the hot streak phenomenon is
reproduced in our Scopus dataset.

We began by investigating the timing of the five most impactful works produced
in each career. As confirmed in the original paper, we confirmed that the
probabilities of when the top-impact works occur to each other are correlated.
The normalized joint probability of the highest and second highest-impact work
is calculated and displayed as a heatmap. Similar calculations were performed
for each of the top 1-5 pairs and 10 combinations, and for all pairs, a
pattern of high joint probability was observed on the diagonal (Fig. S5). This
indicates that the timing of the appearance of the top 1-5 in the careers is
correlated and that they tend to occur consecutively. This diagonal line
disappears when the order of the carrier series data is shuffled, indicating
that this tendency is unique to the actual data (Fig. S6).

Figure S5: Distribution of normalized joint probabilities
$P(x_{1},x_{2})/P(x_{1})P(x_{2})$ for the top-1 and 2 high-impact work, $x1$
and $x2$. Plotted 10 combinations of top 1-5 with the bins size of 20. The
diagonal pattern indicates that hits tend to occur relatively consecutively.
Relative timing is the $N$th paper the researcher has published divided by the
total number $N_{T}$. For example, the timing of the 8th paper published by an
author who has written 80 papers would be 0.1. Figure S6: Distribution of
normalized joint probabilities of the $x1$ and $x2$ for the career sequence
data shuffled in order. The diagonal pattern has disappeared. Figure S7:
Normalized distribution of the difference in the relative timing of the top
1,2,3,4,5 papers. All are divided by the distribution of shuffled data. The
yellow dots show the actual data and the blue ones show the shuffled data.
Figure S8: (Top row) Career sequence data for a scientist. The dotted line is
the threshold, and higher-impact jobs are indicated by blue dots. From left to
right: median, average, and top 10% thresholds. (Bottom row) Of works higher
than the thresholds, the largest consecutive length is $L$, and its
distribution is shown in red. Blue is for shuffled data. The higher the
threshold, the shorter the $L$ goes, but all of them are longer than the
shuffle.

We defined temporal distance ${\Delta}x$ as the difference in the relative
timing of the top 1-5 papers to appear and divided the distribution
$P({\Delta}x)$ by the distribution of the shuffled data $P_{s}({\Delta}x)$, we
observed distribution $R({\Delta}x)=P({\Delta}x)/P_{s}({\Delta}x)$ (Fig. S7).
If $R({\Delta}x)$ is higher than 1, it is more likely to occur than in the
random case. The top 1-5 hits all have high values around 0 confirming that
they are also more likely to occur at near time.

We replicated the original paper not only for timing but also for the length
that high-impact works continue. We set a certain threshold and observed the
distribution of the maximum length $L$, of the consecutive periods of higher-
impact work, which tends to be longer than that of the shuffled data (Fig.
S8). This indicates that the actual data tends to be more likely to be
followed by a series of higher-impact papers than the shuffled data. To this
end, hot streaks where high-impact papers are consecutive in time are also
observed in our Scopus data set.

###  3.2 Revisiting hot-streak model

Figure S9: We generated two types of sequence datasets following the null
model and hot-streak model respectively. As shown in previous research, we set
${\Gamma}_{0}$ to represent the average impact level of each researcher, with
${\sigma})=1.0$, and ${\tau}_{H}$ as 4 years. Figure S10: Characteristics of
reproduced hot-streak model. a, Histogram of the number of hot streaks in a
career. b, The distributions of durations of hot streaks $P(\tau_{H})$. c, The
cumulative distributions of the onset timing $P(N_{\uparrow}/N)$.

To explain the observed temporal regularities in career sequence data, the
previous research has proposed a simple model [4]. First, they introduced a
null model, where each work is randomly selected from a normal distribution
with parameters $\mathcal{N}({\Gamma}_{0},{\sigma})$, where ${\Gamma}_{0}$
represents the individual researcher’s specific parameter of impact level, and
${\sigma}$ represents the variability in impact. With these two parameters and
the total number of papers $N_{T}$, we can generate career sequence data
following the null model. Next, they proposed hot-streak model, where
${\Gamma}_{0}$ increases to ${\Gamma}_{H}={\Gamma}_{0}+1.0$ during a certain
period ${\tau}_{H}$ randomly chosen within the career. By considering this
simple assumption, they could reproduce patterns in actual data.

Using our dataset, we generated 100,000 sequence data following both the null
model and the hot-streak model (Fig. S9). Interestingly, in the null model,
the observed patterns in the simultaneous probability map did not exhibit
diagonal patterns, while in the hot-streak model, diagonal patterns were
observed. This indicates that the simple assumption reproduced the temporal
proximity of success in the actual data, as reported in the prior study.

To apply the hot-streak model to actual data, we fitted a specific piecewise
function to the moving average ${\Gamma}(N)$ of the career sequence data. This
function resembled a square wave pulse and could have up to three rising
edges, characterized by ten parameters:$base$, $hot_{1}$, $N_{1\uparrow}$,
$N_{1\downarrow}$, $hot_{2}$, $N_{2\uparrow}$, $N_{2\downarrow}$, $hot_{3}$,
$N_{3\uparrow}$, $N_{3\downarrow}$. Here, consistent with the previous
research, ${\Gamma}(N)$ denotes the moving average over a window size
${\Delta}N$, determined as 10% of the total number of works created, $N_{T}$
on an individual basis. It is important to note that ${\Gamma}(N)$ does not
reflect the impact of the N-th work on the individual. Instead, it summarizes
the average performance of works created before and after the N-th piece. To
ensure a robust statistical representation, we adopted the definition
${\Delta}N=max(5,0.1N_{T})$ from the previous research, ensuring sufficient
statistics when calculating ${\Gamma}(N)$. Additionally, to prevent
overfitting, we introduced an L1 regularization term on the difference between
$base$ and each $hot_{1}$, $hot_{2}$, $hot_{3}$. 20 random initial values are
prepared for the other parameters, adopting the one with the smallest squared
error.

To examine the characteristics of fitting with the hot-streak model, we
quantified the number of occurrences of hot streaks in a career. Consistent
with previous research, Fig. S10a shows that one occurrence was most common,
followed by two and zero occurrences, with three occurrences being the least
common. Fig. S10b indicates that the distribution of hot streak lengths in
years, remained constant across different career stages, averaging 4.1 years
almost similar to previous studies. Lastly, we aggregated the start timing of
hot streak in a career (Fig. S10c). Hot streaks were slightly more common in
the early stages of the carriers, while the probability was about the same in
the other periods. In previous studies, the probability of hot streak
occurrence was equal for all carriers, which is a slight difference from the
model reproduced here. These results indicate that we have reproduced the hot-
streak model of a certain quality.

###  3.3 Dependency of the hot-streak model on window size of $\Gamma(N)$

In the context of successfully replicating the hot-streak model, we focused
our attention on the window size that serves to smooth out the impact and
found that the shape of ${\Gamma}(N)$ changes depending on the window size of
the moving average ${\Delta}N$. With the definition of
${\Delta}N=max(5,wsr*N_{T})$, in which $wsr$ represents window size ratio
among each career, we then calculated ${\Gamma}(N)$ for career sequence data
at four different $wsr$ values. Looking at the two sequences as examples, the
shape of ${\Gamma}(N)$ becomes smoother as $wsr$ increases (Fig. S11).

To test the effect of $wsr$, we fitted a simple version of hot-streak model
over the several ${\Gamma}(N)$ with four different $wsr$. The simple version
assumes only one raise thus it has four parameters of $hot$, $base$, $N_{u}$
and $N_{d}$. In Fig. S12, the impact elevation $Height$, calculated from $hot-
base$, declines as $wsr$ increases and the length of the hot streak, indicated
by the difference between $N_{d}$ and $N_{u}$, increases. This trend is
confirmed by fitting the parameters for 500 researchers.

Notably, the distribution of the length of hot streaks $realtive\ L$ peaks at
the same length of $wsr$. This can be understood more clearly by examining the
lower row of Figure S11. When career sequence data with a single outstanding
impact are converted to ${\Gamma}(N)$, a shape resembling a plateau around the
maximum impact emerges, whose length almost matches $wsr*N_{T}$, as
intuitively obvious from the definition of the moving average. Fitting such
${\Gamma}(N)$ results in tracing a square-wave-pulse shape, suggesting a
potential correlation between the relative length of the hot streak $L$ and
$wsr$. In such cases, the model detects a single significant success rather
than consecutive successes. Given its current definition, the hot-streak model
cannot prevent such false-positive detections. Therefore, there may be room
for improvement in accurately detecting consecutive successes using the hot-
streak model.

Indeed, there are cases where consecutive successes are detected, indicated by
periods where the overall impact increases as shown in the upper row of Figure
S11. The proposed hot-streak model undeniably contributes by capturing a
common feature even among film directors and painters, which is the occurrence
of consecutive hits. However, the main claims such as 1) 90% of researchers
experience a hot streak, 2) hot streaks can occur at any time throughout a
career, and 3) the average length of a hot streak is about four years, does
not fully capture general trends applicable in all cases. This is because
these are based on the distribution of fitting parameters when the window size
for the moving average is arbitrarily assumed to be 10% of the career length.

Figure S11: For two researchers, the moving average ${\Gamma}(N)$ (green) was
calculated from the sequence data (blue), and a function assuming a single hot
streak was fitted (orange). It can be observed that for the same data, as the
window size increases, ${\Gamma}(N)$ becomes smoother and the fit goes lower
and longer. The fitting was done by exploring all possible regions of four
parameters under certain constraints, with the least squared error. Figure
S12: The distribution of parameters for the simple version of hot-streak model
fitted to 500 researchers. As the window size increases, the height of the hot
streak decreases, and its length increases. The height is defined as $hot-
base$ and the length $realtiveL$ is the normalized length
$(N_{d}-N_{u})/N_{T}$.

###  3.4 The hot-streak model and our data-centric detection

To evaluate the correspondence between the replicated hot-streak model and our
method, data-centric consecutive success detection, we examined the extent of
alignment in the onset of hot streak periods. We normalized career timelines
from 0 to 1 and defined the onset of a hot streak within a tolerance range of
±0.1 as a match. The comparison results are presented in a confusion matrix
(Table. S2). Out of the consecutive success periods identified using our data-
centric method, only 47% are also detected by the hot-streak model. This means
our method overlooks roughly half of the periods caught by the model.
Conversely, of the consecutive successes detected by the hot-streak model,
only 19% are identified by the data-centric method. Given this, the hot-streak
model tends to identify a significant portion of the periods as “success
streaks" even when consecutive success of top-10% works is not occurring. Even
in periods of consecutive success, it fails to recognize them about half the
time (Fig. S13). In this study, we demonstrate the validity of our approach.

When employing the hot-streak model on sequences of raw impacts without
applying a moving average, approximately 30% of researchers are found to
experience at least one hot streak, while less than 10% undergo multiple hot
streaks (Fig. S14a). This closely aligns with our model’s ratio of X/N=3/5 and
suggests that the 90% hot streaks identified in prior studies may be artifacts
of high impacts being smoothed out by moving averages. Furthermore, while the
average duration of hot streaks in models with a set window size is around
four years, the variation is significantly greater in hot-streak models fitted
to raw data (Fig. S14b). Comparing our method to the raw data fitting hot-
streak model reveals that the latter does not fit well with non-consecutive
successes. Comparing Tables S2 and S2, the number of hot streaks detected only
by our data-centric method is fewer in the raw data fit. The count of hot
streaks identified solely by the hot-streak model has increased. However,
similar to the moving average, the distribution of the onset of hot streaks
appears nearly flat(Figure S14c).

As limitations, due to computational constraints, the fitting was only
performed on 500 individuals, and the method only considers up to a single hot
streak, thus it is not a complete replication experiment. In the future,
following previous studies, a method to fit up to three hot streaks will be
applied to sufficient data of 100,000 individuals to observe more precise
parameter distribution.

Our data-centric detection approach is simpler and directly engages with raw
data, enabling a more accurate capture of real-world phenomena. The hot-streak
model, on the other hand, works with career sequence data smoothed by moving
averages. This model doesn’t always successfully identify sequences of good
work in a career, specifically the concentration of top-10% achievements, with
only a 50% probability.

Table S1: The number of hot streaks detected
by hot-streak model (n=100,000) | Data-centric detection
---|---
Hot-streak model |  | Positive | Negative
Positive | 24,030 | 104,310
| Negative | 22,293 | -

Table S2: The number of hot streaks detected
by hot-streak model for raw data (n=100,000) | Data-centric detection
---|---
Hot-streak model for raw data |  | Positive | Negative
Positive | 12,667 | 30,531
| Negative | 33,916 | -

Figure S13: Differences in hot streak extraction trends between the hot-streak
model and our data-centric approach. Red circles indicate the top-10% works in
a career. a, When the timing of the hot streak was common. b, Cases in which
the onset timing is not common and more than one hot streak was extracted by
the hot-streak model. c, Cases where the onset timing was not common and one
or more of them were extracted by the data-centric detection. Figure S14:
Characteristics of reproduced hot-streak model with raw-impact arrays. a,
Histogram of the number of hot streaks in a career. b,The distributions of
durations of hot streaks $P(\tau H)$. c, The cumulative distributions the
start of a hot streak.

##  4 Collaboration patterns

###  4.1 Team sizes

In sequences experiencing consecutive successes, ‘Hot’ sequences, team sizes
tend to be larger, with a higher proportion of large projects involving more
than 10 members (Fig. 2d,e). Histograms of team sizes for each sequence show
similar proportions for teams up to 100 members, but ‘Hot’ sequences have a
higher proportion of very large projects with over 100 members (Fig. S15).
This indicates that large-scale projects often publish a cluster of papers at
once, likely contributing to the top 10% high-impact papers in a career.

Figure S15: Histograms of team sizes for each sequence. The dashed line is on
100 of team size. ‘Hot’ sequences have greater proportion of big project.

###  4.2 Examining if early and late hot streaks come from the same work

We hypothesized that consecutive successes in early and late career stages may
originate from the same teams. For instance, a researcher who had multiple
hits as a student or postdoc in their early career might be linked to a mentor
or principal investigator who appears as the last author during the same
period. Therefore, these early-career mentors, often senior researchers later
in their careers, could mirror the later career successes. To test this
hypothesis, 20,000 papers were randomly sampled from early and late career
consecutive success sequences, defined as having a relative career timing less
than 0.2 for early and greater than 0.8 for late stages. If these overlap, it
indicates the authors experienced the consecutive successes at the same time
in their each career stages, early and late. The overlap was measured using
the Jaccard index, which was 0.002. For randomly extracted paper sets,
regardless of career stage, the Jaccard coefficient was 0.1. This suggests
that ‘Hot’ papers in early and late stages are more likely to written by
different teams, strongly supporting the notion that early and late career
successes are qualitatively distinct. Notably, only 1.7% of individuals, 474
in total, experience consecutive successes in both early and late stages of
their career lifecycle.

###  4.3 Disruption index of hot works and works of who experience hot streak

Our findings indicate that scientists belonging to the early hot-streak group
(experience hot streak up to 20% of the career timing) tend to experience less
disruption in their scholarly careers compared to those in the middle (20-60%
of the career timing), late (80% onward of the career timing) group.
Furthermore, even in their hot streak papers, the early group displays low
disruptiveness when compared to the other groups. These results suggest that
early hot streaks are characterized by a limited level of topic diversity (see
Figure S16). Scientists in this group tend to concentrate on developing their
research and maintain a consistent style throughout their careers. Conversely,
the late hot-streak group demonstrates relatively high levels of
disruptiveness, despite also exhibiting low topic diversity (see Figure 2h).
This can be attributed to their involvement in or leadership of large research
teams and the presence of weak ties (see Figures 2d, f). These findings imply
that they engage in research that incorporates diverse ideas and
collaborations, thereby avoiding research with low levels of disruptiveness.

Figure S16: The blue line depicts the disruptiveness rank[11] (D) of hot
streak papers during the early, middle, and late periods. The green line
represents the D rank of all papers authored by scientists experiencing early,
middle, and late hot streaks. The error bar for both series indicates a 95%
confidence interval. The rank of D is determined within groupings that are
classified according to year (in intervals of 5 years), cited count (divided
into 5 percentiles), and the field (calculated in 1.1).