# Transition to synchrony in chiral active particles

Arkady Pikovsky1,2 1 Institute of Physics and Astronomy, University of
Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany  2 Control
Theory Department, Institute of Information Technologies, Mathematics and
Mechanics, Lobachevsky University, 603950 Nizhny Novgorod, Russia

###### Abstract

I study deterministic dynamics of chiral active particles in two dimensions.
Particles are considered as discs interacting with elastic repulsive forces.
An ensemble of particles, started from random initial conditions, demonstrates
chaotic collisions resulting in their normal diffusion. This chaos is
transient, as rather abruptly a synchronous collisionless state establishes.
The life time of chaos grows exponentially with the number of particles.
External forcing (periodic or chaotic) is shown to facilitate the
synchronization transition.

##  1 Introduction

Active or self-propelled particles are a subject of active current research
(see introductory review [1]). For microscopic particles, random interactions
with an environment are essential, and one speaks about active Brownian
particles, subject to noisy forces. For macroscopic particles, existence of
random forces is not so obvious, but here one also often introduces them to
model observed fluctuations in the motions of animals and birds (cf. famous
Vicsek model [2]). A large class of active particles constitute chiral active
particles, natural trajectories of which are not straight lines, but circles.
The origin of chirality can be asymmetry in the particle shape or in the
propulsion mechanism; also external magnetic field may lead to circular motion
(see discussion and examples in [1]).

There are different models for interaction of active particles. In many cases,
inspired by the Vicsek model [2], one assumes an aligning interaction:
neighboring particles “prefer” to align their velocities. In the context of
chiral Brownian (i.e. noise-driven) particles, such an aligning interaction
can lead to an appearance of synchronized rotating clusters (see, e.g., Refs.
[3, 4, 5, 6]). In this paper, I consider deterministic particles without
alignment forces. The main finding is that this system demonstrates a
transition from supertransient chaos to synchronous clusters.

I introduce the basic model in Section 2. In Section 3 I demonstrate that
while for small times, circling colliding active particles demonstrate chaos
leading to their diffusion, at large times a transition to a synchronous state
without collisions occur. The life time of chaos grows exponentially with the
number of particles, what allows to speak about supertransients, typical for
spatio-temporal chaos. In section 4 I demonstrate that external periodic or
random forcing induces a transition to a collisionless state already at small
times. I conclude with discussion in Section 5.

##  2 Model formulation

I consider active particles in two dimensions, with circular natural
trajectories. The two components of velocity $(v=\dot{x},u=\dot{y})$ for one
particle obey following equations:

$\eqalign{\dot{v}&=-\omega u+\mu(W^{2}-u^{2}-v^{2})+m^{-1}F_{x}\;,\\\
\dot{u}&=\omega v+\mu(W^{2}-u^{2}-v^{2})+m^{-1}F_{y}\;.}$ (1)

Here $W$ is a steady speed, which a particle attends if the other forces
$F_{x,y}$ vanish; parameter $\mu$ describes the rate of the relaxation to this
steady speed; $m$ is the particle’s mass. The velocity field rotates with
frequency $\omega$. In a steady state, an isolated particle rotates on a
circle of radius $W/\omega$ with frequency $\omega$.

It is worth noting that Eq. (1) is widely used in synchronization and coupled
oscillators studies as the Stuart-Landau model (see, e.g., [7]). In this
interpretation two variables $u,v$ describe a state of an autonomous
oscillating system close to the Hopf bifurcation point. It is also well-known
that under a periodic or random force this oscillator synchronizes [7], this
effect will be explored in Section 4.

Below I consider two types of forces acting on particles. The first force is
the interaction between the particles. I assume, following Ref. [8], a
conservative repulsing interaction governed by a truncated Lennard-Jones
potential, which depends on the distance $R$ between the particles:

$V(R)=\left\\{\eqalign{\epsilon\left[\left(\frac{\sigma}{R}\right)^{12}-2\left(\frac{\sigma}{R}\right)^{6}+1\right]&\;\;\rm{for}\;\;R\leq\sigma\;,\\\
0&\;\;\rm{for}\;\;R>\sigma\;.}\right.$ (2)

This potential takes from the full Lennard-Jones potential only its repulsing
part, and the attracting part is absent. Thus, this potential mimics not-so-
hard discs with radius $\sigma/2$, which repulse each other when collide, and
do not interact aside of collisions. I stress here that the interactions have
no any alignment action (the latter is often assumed in models of Vicseck
type). If there where no activity and chirality (i.e. $\omega=\mu=0$ in (1)),
then the model reduces to a Hamiltonian one of elastically colliding disks. In
the case of hard disks it is believed that the dynamics is fully chaotic (the
Boltzmann-Sinai hypothesis), although the proof [9] has some restrictions. For
smooth potentials, stable periodic orbits in the Hamiltonian dynamics may
appear [10]. Much less is known for colliding active particles, but our
simulations in Section 3 indicate for chaos.

In addition to interaction forces described by (2), I will consider external
forces specified in Section 4.

Below I study numerically an ensemble of active particles with circular orbits
(1),(2) in a periodic geometry, i.e., on a torus $L\times L$. I fix $\mu=2$,
$W=\omega=1$, $\sigma=0.2$, $m=1$, and $\epsilon=0.1$ throughout the paper.
The main parameters to explore is the number of the particles $N$ and the
density $\rho=NL^{-2}$.

##  3 Chaotic state and spontaneous transition to synchrony

###  3.1 Quasistationary chaotic state

Figure 1: Positions of particles vs time for $N=20$, $\rho=3$. Synchronization
transition occurs at $t\approx 27000$.  Figure 2: Trajectories of particles
for the same data as Fig. 1.

Because collisions of hard disk have a scattering property with an essential
degree of instability (like dispersive billiards), one can expect chaos in
colliding active particle described by Eqs. (1),(2). Due to multiple
collisions, velocity of a particle is random, so its motion is a diffusion in
two dimensions. I illustrate this with figures 1,2. They show trajectories of
$N=20$ particles in a particular run. Up to time $T_{st}\approx 27000$ (I
measure time in periods of rotations) motion of all particles is irregular.

Quantitatively, a good characteristics of irregularity is the mean diffusion
constant of the particles. It can be calculated from the mean squared
displacement after a large time interval

$D=\frac{\langle(x(T)-x(0))^{2}+(y(T)-y(0))^{2}\rangle}{T}\;.$

Diffusion in the system is normal, as the graph of
$\langle(x(T)-x(0))^{2}+(y(T)-y(0))^{2}\rangle$ versus time interval $T$ (Fig.
3(a)) shows. Because $D$ is an intensive quantity, one can expect that it
depends on the intensive parameter - density of particles $\rho$, but not on
the number of particles $N$. However, calculations of the diffusion constant
Fig. 3(b) show, that for a small number of particles a significant depletion
of the diffusion constant is observed. I attribute this to correlations which
appear in small communities where the same particles collide many times. In
large ensembles, a particle has again and again new neighbors, so that
correlations effectively disappear (see nearly coinciding values for $N=30$
and $N=40$ in Fig. 3(b)).

Figure 3: Calculation of diffusion constant. Panel (a): mean squared
displacements vs. time for $N=20$ and different $\rho$ are nearly perfect
straight lines confirming normal diffusion of particles. Panel (b): diffusion
constants for different $N$ and $\rho$.

###  3.2 Spontaneous transition to synchrony

The main observation of this work is that the chaos described above is in fact
a transient state, it evolves eventually into a configuration without
collisions; such a transition can be seen at $T_{st}\approx 27000$ in Fig. 1.
Indeed, a set of particles (1) has an absorbing synchronous state where all
the velocities are equal: $v_{1}=v_{2}=\ldots=v_{N}$,
$u_{1}=u_{2}=\ldots=u_{N}$. In this state the particles rotate synchronously,
the distances between them remain constant, and they do not collide. Thus this
state continues forever. Such a state can exist also in a Hamiltonian setup,
but there it can occur only for specially constructed initial conditions. If
collisions in a set of Hamiltonian discs occur, they cannot disappear, because
of the reversibility of the dynamics. For active particles with a non-
Hamiltonian dynamics, there is no such a restriction.

Figure 4: Synchronous state for the trajectory shown in Fig. 1. Black filled
circles: positions of the particles at a certain time; colored large circles:
trajectories of the particles.

I stress that the state in Fig. 1 at $T>T_{st}$ is not completely synchronous,
as the phases of rotation of different particles do not coincide. It is
sufficient to achieve a state where collisions disappear, such a regime, which
I call absorbing collisionless state, continues forever. I illustrate the
collissionless synchronous state corresponding to $T>T_{st}$ of Fig. 1 in Fig.
4. Fig. 1 shows that the transition to synchrony is quite abrupt, so this is
an example of type-II supertransients according to classification of [11].

Figure 5: Mean time to synchronization in dependence on the number of
particles and on density. The dependencies in the range $20\leq N\leq 30$ are
well fitted with $T_{st}\approx A(\rho)\exp[B(\rho)N]$, where $B(1)=0.304$,
$B(2)=0.311$, $B(3)=0.309$, $B(4)=0.282$.

The time at which a synchronous absorbing state appears depends on the initial
configuration of particles, it fluctuates highly. In Fig. 5 I show dependence
of the mean time to achieve a collisionless state on the number of particles
and on density $\rho$. The main feature is that the dependence on the density
is rather weak, but the life time of a chaotic state grows exponentially with
the number of particles $N$. Thus, this system belongs to a class of extended
systems with chaotic supertransients [12, 13]. Figure 5 also shows that the
law of exponential growth with the number of particles only weakly depends on
the density parameter.

##  4 Driven transition to synchrony

Noninteracting particles are described by a set of effective Stuart-Landau
oscillators (1). Thus, such an ensemble can be synchronized by an external
periodic or random force, as described in the theory of synchronization [7].
One can expect that the same happens also in the presence of the collisions,
although the synchronization onset can be retarded due to them. I Illustrate
the effect of a periodic force $F_{x}=\gamma\cos(\omega t)$ on the ensemble in
Fig. 6. A small force has almost no effect, the mean life time of the chaotic
state is almost the same as for autonomous particles. At large values of
$\gamma$, the transition occurs within a few periods, the mean life time is
almost the same for different $N$ and $\rho$. Noteworthy, at intermediate
force amplitudes $0.01<\gamma<0.03$, the reduction of the mean life time of
chaos is mostly pronounced for systems with low density (see curves with
$\rho=1$ for $N=20$ and $N=30$).

Figure 6: Mean time to synchrony under periodic force vs force amplitude
$\gamma$ for different $\rho$ and $N$.

Common noise is another source of synchrony in ensembles of uncoupled
oscillators [7]. Here I report on numerical experiments where noise was in the
form of a Poissonian sequence of delta-pulses:

$F_{x}=\sum_{n}a_{n}\delta(t-t_{n})\;,$ (3)

where $t_{n}$ are Poissonian time events appearing with rate $\tau^{-1}$, and
$a_{n}$ are independent amplitudes of the pulses taken from a Gaussian
distribution with standard deviation $\sigma$. Mean life times of chaos are
shown in Fig. 7. Remarkably, under noisy force the mean life time of chaos is
nearly a constant, it does depend on the number of particles and on the
density. Furthermore, for small number of particles common noise retards
transition to synchrony compared to the spontaneous one. For large number of
particles the effect is opposite, here for $N=30$ noise reduces the life time
of chaos by a factor larger than $20$.

Figure 7: Mean time to synchrony under random force (3) with $\tau=0.2$ and
$\sigma=0.2$ (filled markers). Also, for comparison, is shown the unperturbed
mean time (open markers).

##  5 Conclusion

In this paper I studied an ensemble of active particles with circular natural
trajectories, subject to elastic collisions. This setup differs from previous
studies in two aspects: (i) the basic dynamics is purely deterministic (except
for the special considered case of a common random force), and (ii) there are
no aligning interactions. I show that the autonomous system demonstrates
supertransient chaos. Starting with random initial conditions, for a long
interval of time, chaotic dynamics is observed, leading to a normal diffusion
of the particles. However, the final state is a regime without collisions,
where neighboring particles effectively synchronize their circular rotations.
I demonstrate that the life time of chaos depends on the density of particles,
and grows exponentially with the number of them. Furthermore, I show that the
transition to synchronous rotations can be induced by an external periodic
force, common for all particles (this effect has been previously reported in
other context in Refs. [14, 15]). Also I demonstrate that a random common
force also leads to a synchronous dynamics without collisions. Forced
transitions have a certain transient time which for large forces does not
depend on the number of particles.

Author thanks H. Chaté and F. Ginelli for fruitful discussions, and the
Russian Science Foundation (studies of Section 4, Grant Number 19-12-00367)
for support.

## Bibliography

## References

  * [1] Clemens Bechinger, Roberto Di Leonardo, Hartmut Löwen, Charles Reichhardt, Giorgio Volpe, and Giovanni Volpe.  Active particles in complex and crowded environments.  Rev. Mod. Phys., 88:045006, 2016.
  * [2] Tamás Vicsek, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet.  Novel type of phase transition in a system of self-driven particles.  Phys. Rev. Lett., 75:1226–1229, 1995.
  * [3] Guo-Jun Liao and Sabine H. L. Klapp.  Clustering and phase separation of circle swimmers dispersed in a monolayer.  Soft Matter, 14:7873–7882, 2018.
  * [4] Levis D and B. Liebchen.  Micro-flock patterns and macro-clusters in chiral active Brownian disks.  J. Phys.: Condens. Matter, 30:084001, 2018.
  * [5] Demian Levis and Benno Liebchen.  Simultaneous phase separation and pattern formation in chiral active mixtures.  Phys. Rev. E, 100:012406, 2019.
  * [6] Nikita Kruk, José A. Carrillo, and Heinz Koeppl.  Traveling bands, clouds, and vortices of chiral active matter.  Phys. Rev. E, 102:022604, 2020.
  * [7] A. Pikovsky, M. Rosenblum, and J. Kurths.  Synchronization. A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge, 2001.
  * [8] M. Rex and H. Löwen.  Lane formation in oppositely charged colloids driven by an electric field: Chaining and two-dimensional crystallization.  Phys. Rev. E, 75:051402, 2007.
  * [9] N. Simányi.  Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems.  Invent. math., 154:123–178, 2003.
  * [10] D. Turaev and V. Rom-Kedar.  Soft billiards with corners.  J. Stat. Phys., 112(3-4):765–813, 2003.
  * [11] Y.-C. Lai and T. Tél.  Transient Chaos.  Springer, New York, 2011.
  * [12] J. P. Crutchfield and K. Kaneko.  Are attractors relevant to turbulence?  Phys. Rev. Lett, 60(26):2715–2718, 1988.
  * [13] Thomas Lilienkamp, Jan Christoph, and Ulrich Parlitz.  Features of chaotic transients in excitable media governed by spiral and scroll waves.  Phys. Rev. Lett., 119:054101, 2017.
  * [14] A. D. Chepelianskii, A. S. Pikovsky, and D. L. Shepelyansky.  Synchronization, zero-resistance states and rotating wigner crystal.  Eur. Phys. J. B, 60(2):225–229, 2007.
  * [15] D. L. Shepelyansky, A. S. Pikovsky, J. Schmidt, and F. Spahn.  Synchronization mechanism of sharp edges in rings of saturn.  Mon. Not. Roy. Astron. Soc., 395(4):1934–1940, 2009.


# Metrics to Detect Small-Scale and Large-Scale Citation Orchestration

Iakovos Evdaimon1 John P. A. Ioannidis2 Giannis Nikolentzos3 Michail
Chatzianastasis1 George Panagopoulos4 and Michalis Vazirgiannis1, 5
1LIX, École Polytechnique, Institut Polytechnique de Paris, Rte de Saclay,
Palaiseau, 91120, France
2Meta-Research Innovation Center at Stanford (METRICS), Stanford University,
Stanford, CA 94305
3Department of Informatics and Telecommunications, University of Peloponnese,
Akadimaikou G. K. Vlachou Street 221 31 Tripoli, Greece
4Department of Computer Science, University of Luxembourg, Maison du Nombre 6,
avenue de la Fonte L-4364 Esch-sur-Alzette, Luxembourg
5Mohamed bin Zayed University of Artificial Intelligence, Masdar City, Abu
Dhabi, United Arab Emirates

###### Abstract

Citation counts and related metrics have pervasive uses and misuses in
academia and research appraisal, serving as scholarly influence and
recognition measures. Hence, comprehending the citation patterns exhibited by
authors is essential for assessing their research impact and contributions
within their respective fields. Although the $h$-index, introduced by Hirsch
in $2005$, has emerged as a popular bibliometric indicator, it fails to
account for the intricate relationships between authors and their citation
patterns. This limitation becomes particularly relevant in cases where
citations are strategically employed to boost the perceived influence of
certain individuals or groups, a phenomenon that we term “orchestration”.
Orchestrated citations can introduce biases in citation rankings and therefore
necessitate the identification of such patterns. Here, we use Scopus data to
investigate orchestration of citations across all scientific disciplines.
Orchestration could be small-scale, when the author him/herself and/or a small
number of other authors use citations strategically to boost citation metrics
like $h$-index; or large-scale, where extensive collaborations among many co-
authors lead to high h-index for many/all of them. We propose three
orchestration indicators: extremely low values in the ratio of citations over
the square of the h-index (indicative of small-scale orchestration); extremely
small number of authors who can explain at least $50\%$ of an author’s total
citations (indicative of either small-scale or large-scale orchestration); and
extremely large number of co-authors with more than $50$ co-authored papers
(indicative of large-scale orchestration). The distributions, potential
thresholds based on $1\%$ (and $5\%$) percentiles, and insights from these
indicators are explored and put into perspective across science.

##  1 Introduction

Citation counts and related metrics are widely used and misused in scholarly
evaluation [Bornmann, 2014]. They reflect the acknowledgment and utilization
of research by the scientific community and play a significant role in
assessing an author’s research impact and contributions within their
respective fields [Costas et al., 2015, Ioannidis et al., 2014]. However, the
conventional popular metrics used to quantify research impact may fall short
in capturing the intricate relationships between authors and their citation
patterns [Waltman and Van Eck, 2012].

For example, the $h$-index, introduced by Hirsch in 2005, has gained
considerable popularity as a bibliometric indicator, measuring an author’s
productivity and citation impact [Hirsch, 2005]. It represents the maximum
number of an author’s papers that have received at least $h$ citations. While
the $h$-index provides a straightforward measure of an author’s research
impact, it overlooks the underlying complexities of citation patterns and the
potential manipulation of citation metrics [Bornmann and Daniel, 2005].
Extensive criticism has been yielded against the $h$-index and its gaming
potential [Kelly and Jennions, 2006, Chapman et al., 2019, Oravec, 2019].

In the scholarly landscape, there is an emerging concern regarding the
strategic use of citations to enhance the influence of certain individuals or
groups. This phenomenon, which can be referred to as “citation orchestration”
occurs when authors deliberately include citations not solely to endorse
relevant research but rather to boost the reputation or visibility of specific
researchers or their affiliations [Didegah and Thelwall, 2013].

The $h$-index, for example, can be manipulated by self-citations placed
strategically to boost it [Bartneck and Kokkelmans, 2011]. Self-citations may
be the most superficial and readily detectable type of manipulation and some
authors reach extremely high levels of self-citations, e.g. self-citations may
account for more than $50\%$ or even $80\%$ of the citations that they receive
[Van Noorden and Chawla, 2019, Ioannidis et al., 2020]. However, orchestration
may also involve more subtle citation cartels: a small number of scientists
may orchestrate to cite each other, regardless of whether they are co-authors
or not [Fister et al., 2016, Perez et al., 2019]. Finally, with the advent of
team science, it is becoming increasingly common for many teams to have a very
large number of authors on each of the massive number of papers that they
publish [Papatheodorou et al., 2008]. Such large-scale orchestration requires
an in-depth analysis of the co-authorship patterns. All these behaviors and
authorships and citation patterns are important to document so as to avoid a
naive adoption of metrics like the $h$-index [Bornmann and Daniel, 2009, Ding
et al., 2020].

Eventually, citation orchestration can distort citation rankings and
compromise the fairness and accuracy of research evaluation processes [Waltman
et al., 2011]. In order to understand the extent of citation orchestration and
characterize its presence across science, here we use the entire Scopus
database, which provides comprehensive coverage of scholarly literature across
diverse disciplines and facilitates a multi-dimensional exploration of
citation patterns [Burnham, 2006, Falagas et al., 2008]. We develop three
readily accessible and interpretable metrics that may provide hints to
possible citation orchestration behavior.

##  2 Methods

###  2.1 Proposed Metrics

The metrics are defined at the authors’ level. We propose three different
metrics that may offer hints for the presence of inflated $h$-indices due to
small-scale or large-scale citation orchestration:

####  $\mathbf{C/h^{2}}$:

The first metric is equal to the number of citations divided by the square of
the author’s $h$-index. A small value of $C/h^{2}$ indicates that the
citations of the author are coming from papers that explicitly increase the
$h$-index. An extremely small value for this metric may suggest that the
citations of the author may have been strategically placed in order to
increase the $h$-index. This strategic placement may have been done either by
the author himself/herself (self-citations) or by a small number of scientists
who act as a coordinated citation cartel.

####  $\mathbf{A_{50\%C}}$:

This metric is equal to the number of citing authors who cumulatively explain
$50\%$ of the citations of an author. An extremely small value of this metric
may highlight either small-scale or large-scale orchestration. Small-scale
orchestration results in a low $A_{50\%C}$ when a single author either cites
himself/herself or is cited by some specific people within a citation cartel
in order to promote his or her $h$-index. An extremely low $A_{50\%C}$ may be
seen also with large-scale orchestration when a tightly-knit team effort
dominates the scientific discipline and all/most authors are cited in all
papers, a situation typical of nuclear and particle physics work affiliated
with the European Organization of Nuclear Research. To calculate $A_{50\%C}$,
we take into account the citing author who contributes the most citations to
the examined author. Once this citing author $A_{1}$ is identified, we re-sort
the remaining citing authors based on the remaining unexamined citing papers
(i.e. excluding citing papers authored by $A_{1}$). The highest contributor
$A_{2}$ is identified then and similarly removed. The process is iterated
until citing author $A_{i}$, when at least $50\%$ of the citations are
explained. This iterative process allows us to determine the number of citing
authors who collectively explain at least $50\%$ of the examined authors’
citations while avoiding double-counting of citations from the same papers.

####  $\mathbf{A_{50}}$:

The metric is equal to the number of co-authors with shared co-authorship
exceeding $50$ papers. If this number is large, this may highlight large-scale
orchestration, where the authors belong to a big cluster of authors that
heavily cite one another in work that they do jointly. Typically, this
reflects the situation where prolific teams publish papers with massive co-
authorship, and then these papers massively cite papers produced by the same
team. This pattern is overwhelmingly seen in work done by the European
Organization of Nuclear Research but this type of practice may also be
appearing increasingly also in other scientific fields [Cronin, 2001].

###  2.2 Dataset

As part of the project, we were given access to the International Center for
the Study of Research (ICSR) Lab111https://www.elsevier.com/icsr/icsrlab, a
cloud-based computational platform that allows its users to analyze large
structured datasets such as the publication metadata of
Scopus222https://www.elsevier.com/solutions/scopus/how-scopus-works/metrics.
The platform provides access to the publication metadata of $95,242,081$
papers in total as of $12^{th}$ of March 2023. These papers are authored by
$49,255,352$ different author IDs in total. The precision and accuracy of the
author ID files in Scopus have been well characterized [Kawashima and
Tomizawa, 2015]; while some authors have their papers split into more than one
ID file and some specific ID files contain papers by more than one author, the
errors are very small to materially affect the distributions of the proposed
metrics, and may be even less influential for the extreme tails that are
likely to be most informative for orchestration.

###  2.3 Preprocessing

We only consider publications that are full papers, i.e. fall into the Scopus
categories of articles, conference papers, or reviews, in concordance with
previous work [Ioannidis et al., 2018], with a total of $82,694,786$ eligible
papers. We next determine the field and subfield of study for each author
using the Science-Metrix333https://science-metrix.com/ [Archambault et al.,
2011] classification. Each publication is assigned to one of 22 fields and one
of 174 subfields. In case an author has published multiple papers and those
papers belong to different fields, we choose the one where most of the
author’s papers are assigned. In case of ties, we choose the field whose
papers have received the largest number of citations. If there are still ties,
we randomly select one of the fields. $2,889,779$ authors without specified
subfield or field are excluded from all analyses. Table 1 shows the number of
authors eligible for analysis in each field.

Field of study | Number of authors | Percentage | Number of authors analyzed | Percentage
---|---|---|---|---
Agriculture, Fisheries & Forestry | $1,446,173$ | $3.12\%$ | $35,839$ | $2.39\%$
Biology | $1,586,467$ | $3.42\%$ | $64,737$ | $4.33\%$
Biomedical Research | $3,057,457$ | $6.60\%$ | $189,457$ | $12.66\%$
Built Environment & Design | $273,254$ | $0.60\%$ | $3,638$ | $0.24\%$
Chemistry | $2,464,113$ | $5.31\%$ | $91,721$ | $6.13\%$
Clinical Medicine | $15,594,844$ | $33.63\%$ | $579,366$ | $38.71\%$
Communication & Textual Studies | $328,254$ | $0.71\%$ | $1,617$ | $0.11\%$
Earth & Environmental Sciences | $1,345,726$ | $2.90\%$ | $58,067$ | $3.88\%$
Economics & Business | $909,251$ | $2.00\%$ | $20,776$ | $1.39\%$
Enabling & Strategic Technologies | $5,103,262$ | $11.00\%$ | $101,820$ | $6.80\%$
Engineering | $3,679,238$ | $7.94\%$ | $56,200$ | $3.75\%$
General Arts, Humanities & Social Sciences | $5,992$ | $0.01\%$ | $2$ | $0.00\%$
General Science & Technology | $30,121$ | $0.06\%$ | $77$ | $0.01\%$
Historical Studies | $241,666$ | $0.52\%$ | $1,550$ | $0.10\%$
Information & Communication Technologies | $4,094,210$ | $8.83\%$ | $57,163$ | $3.82\%$
Mathematics & Statistics | $410,548$ | $0.89\%$ | $9,605$ | $0.64\%$
Philosophy & Theology | $111,270$ | $0.24\%$ | $433$ | $0.03\%$
Physics & Astronomy | $2,649,145$ | $5.71\%$ | $174,028$ | $11.63\%$
Psychology & Cognitive Sciences | $487,914$ | $1.10\%$ | $18,264$ | $1.22\%$
Public Health & Health Services | $1,221,920$ | $2.64\%$ | $19,928$ | $1.33\%$
Social Sciences | $1,300,068$ | $2.80\%$ | $12,372$ | $0.83\%$
Visual & Performing Arts | $25,800$ | $0.06\%$ | $20$ | $0.00\%$
Table 1: Number and percentage of total Scopus author IDs and of Scopus author
IDs eligible for analysis (those with at least 5 full papers and at least 1000
citations).

We focus on a subset of authors who have made a significant contribution to
the research community and satisfy the following two conditions: (1) they have
published more than $5$ full papers, and (2) their papers have received at
least $1,000$ citations in total. Eventually, $1,496,680$ authors satisfy the
above conditions. Their numbers per scientific field are also shown in Table
1.

###  2.4 Analyses of distributions for proposed indicators

For each of the three proposed indicators, we generate the distribution of
values for the $1,496,680$ eligible authors. We focus on the left-side tails
(very small values) for the first two indicators and the right-side tails for
the third one. There is no absolute cut-off that can separate authors where
orchestration has been done on purpose. The most extreme values are more
likely to reflect orchestration, but orchestration may have happened to a
various extend also for citation profiles of authors that have less extreme
values. Here we focus primarily on the lowest $1\%$ percentile for $C/h^{2}$
and $A_{50\%C}$ and the highest $1\%$ for $A_{50}$ metric. We also present
values for the $5\%$ threshold as well as median and interquartile range
values for each of the three indicators. We also show how extreme $1\%$
percentile authors are distributed across the main fields of science, so as to
demonstrate whether these patterns are enriched in specific fields. For
$A_{50\%C}$ and $A_{50}$ specifically, we provide analyses that exclude the
authors categorized in the field of Physics and Astronomy, since their tails
of interest are otherwise overwhelmed by nuclear and particle physics authors
in work affiliated with the European Organization for Nuclear Research.

##  3 Results

###  3.1 Distributions of indicators

The distribution of the three proposed indicators appears in Figure 1 and
Figure 2 zooms out specifically on the lower $5\%$ percentile of the first two
indicators and the upper $5\%$ percentile of the third orchestration
indicator. The distributions are right-skewed. For the $A_{50\%C}$ and
$A_{50}$ metrics, we provide also distributions excluding the field of Physics
and Astronomy so as to allow revealing better the patterns for all other
fields.

###  3.2 Metric $C/h^{2}$

For the metric $C/h^{2}$, the $1^{st}$ percentile value is $2.45$ and the
$5^{th}$ percentile value is $2.76$. Comparatively, the median is $4.11$ and
the interquartile range is $3.36$ to $6.11$. Here, we focus on the 1%
percentile ($14,967$ authors) that may have the strongest hint for small-scale
orchestration. Table 2 shows that the scientific field allocation of these
$14,967$ authors is largely similar to the allocation of all examined authors,
with some exceptions. Specifically, there are proportionally more than
$1.5$-fold higher representation of authors with extremely low $C/h^{2}$
values compared to all examined authors in 3 fields: Agriculture, Fisheries
and Forestry, Chemistry, and Earth and Environmental Sciences.

(a) The distribution of the metric $C/h^{2}$ (authors with values above 20 are
not shown)

(b) The distribution of the metric $A_{50\%C}$ (authors with values above 200
are not shown)

(c) The distribution of the metric $A_{50}$ (authors with values of 0 and
above 40 are not shown).

(d) The distribution of the metric $A_{50\%C}$ (excluded authors with values
exceeding 200 and those in the field of Physics and Astronomy).

(e) The distribution of the metric $A_{50}$ (excluded authors with values
above 40 and those in the field of Physics and Astronomy).

Figure 1: The distribution of all the proposed metrics

(a) The distribution of the metric $C/h^{2}$ on the lowest 5%

(b) The distribution of the metric $A_{50\%C}$ on the lowest 5%

(c) The distribution of the metric $A_{50}$ on the upper 5% (authors with
values above 100 are not shown)

Figure 2: The distribution of all the proposed metrics on the lowest or upper
5%

Field of study |  Percentage of examined authors |  Percentage of lowest 1% authors of $C/h^{2}$ metric |  Percentage of lowest 1% authors of metric $A_{50\%C}$ (excluding Physics & Astronomy) |  Percentage of upper 1% authors of metric $A_{50}$ (excluding Physics & Astronomy)
---|---|---|---|---
Agriculture, Fisheries & Forestry |  $2.39\%$ |  $5.74\%$ |  $2.19\%$ |  $1.43\%$
Biology |  $4.33\%$ |  $5.99\%$ |  $4.34\%$ |  $0.75\%$
Biomedical Research |  $12.66\%$ |  $12.19\%$ |  $4.76\%$ |  $9.03\%$
Built Environment & Design |  $0.24\%$ |  $0.25\%$ |  $0.44\%$ |  $0.02\%$
Chemistry |  $6.13\%$ |  $13.80\%$ |  $15.47\%$ |  $3.07\%$
Clinical Medicine |  $38.71\%$ |  $29.20\%$ |  $17.09\%$ |  $73.29\%$
Communication & Textual Studies |  $0.11\%$ |  $0.03\%$ |  $0.26\%$ |  $0.00\%$
Earth & Environmental Sciences |  $3.88\%$ |  $5.93\%$ |  $4.82\%$ |  $1.34\%$
Economics & Business |  $1.39\%$ |  $0.62\%$ |  $2.13\%$ |  $0.00\%$
Enabling & Strategic Technologies |  $6.80\%$ |  $7.96\%$ |  $10.84\%$ |  $7.31\%$
Engineering |  $3.75\%$ |  $4.64\%$ |  $12.18\%$ |  $1.17\%$
General Arts, Humanities & Social Sciences |  $0.00\%$ |  $0.00\%$ |  $0.00\%$ |  $0.00\%$
General Science & Technology |  $0.01\%$ |  $0.00\%$ |  $0.00\%$ |  $0.00\%$
Historical Studies |  $0.10\%$ |  $0.08\%$ |  $0.24\%$ |  $0.00\%$
Information & Communication Technologies |  $3.82\%$ |  $1.06\%$ |  $11.00\%$ |  $1.59\%$
Mathematics & Statistics |  $0.64\%$ |  $0.48\%$ |  $9.44\%$ |  $0.04\%$
Philosophy & Theology |  $0.03\%$ |  $0.00\%$ |  $0.16\%$ |  $0.00\%$
Physics & Astronomy |  $11.63\%$ |  $9.77\%$ |  $-$ |  $-$
Psychology & Cognitive Sciences |  $1.22\%$ |  $0.84\%$ |  $1.99\%$ |  $0.25\%$
Public Health & Health Services |  $1.33\%$ |  $0.92\%$ |  $0.96\%$ |  $0.67\%$
Social Sciences |  $0.83\%$ |  $0.46\%$ |  $1.68\%$ |  $0.03\%$
Visual & Performing Arts |  $0.00\%$ |  $0.00\%$ |  $0.02\%$ |  $0.00\%$

Table 2: Authors with extreme metrics: Distribution across the 22 main fields
of science.

(a) Distribution of the citations, considering only whoever author has an
$C/h^{2}$ value less than $2.45$ (authors with citations above $5,000$ are not
shown).

(b) The distribution of the number of papers published by authors with an
$C/h^{2}$ value on the lowest 1% (authors with more than 200 papers are not
shown).

(c) Distribution of the $h$-index of the authors that have an $C/h^{2}$ value
less than $2.45$ (authors with $h$-index above 60 are not shown).

Figure 3: The distribution of citations, number of papers and $h$-index of the
authors that belong to the lowest 1% of metric $C/h^{2}$.

Figure 3 depicts the distribution of citations, the number of published
papers, and the $h$-index, respectively, for this group of $14,717$ authors
with the lowest $C/h^{2}$ metric. These are highly productive authors (median
$63$ papers, the interquartile range is $50$ to $81$ publications) and they
all have $h$-indices exceeding $21$, while some of them have even very large
$h$-indices.

###  3.3 Metric $A_{50\%C}$

For the $A_{50\%C}$ metric, we observed that $73\%$ ($4,339$ out of $5,881$)
of authors with the lowest $1\%$ percentile values of $A_{50\%C}$ belong to
Physics & Astronomy (of those, $3,808$ belong to the subfield of nuclear and
particle physics). Therefore, we excluded Physics & Astronomy authors in order
to examine better the distribution for the rest of science. All results that
follow retain analyses excluding Physics & Astronomy. For the $A_{50\%C}$
metric, the $1^{st}$ percentile is $5$ while the median is $36$, and the
interquartile range is $21$ to $60$. The $11,277$ authors who belong to this
lowest $1\%$ percentile (metric value less than $5$) are more likely to engage
in self-citation or be part of networks that highly cite each other. Table 2
shows the disciplines to which these $11,277$ authors belong. There is a more
than 1.5-fold higher representation of authors with exceptionally low
$A_{50\%C}$ values compared to all examined authors in $10$ specific fields:
Chemistry, Enabling & Strategic Technologies, Engineering, Historical Studies,
Information & Communication Technologies, Mathematics & Statistics, Philosophy
and Theology, Psychology and Cognitive Sciences, Social Sciences, and Visual
and Performing Arts.

Figure 4 depicts the distribution of citations, the number of published
papers, and the h-index, respectively, for this group of $11,277$ authors with
the lowest $1\%$ percentile for the $A_{50\%C}$ metric. These are mostly very
productive authors (median $104$ papers, interquartile range $71$ to $153$
publications) and they mostly ($59\%$ of these authors) have $h$-indices
exceeding $20$, while some of them have even large $h$-indices.

(a) Distribution of the citations, considering only whoever author has an
$A_{50\%C}$ value less than $5$ (authors with citations above $10,000$ are not
shown).

(b) The distribution of the number of papers published by authors with an
$A_{50\%C}$ value on the lowest 1% (authors with more than $500$ papers are
not shown).

(c) Distribution of the $h$-index of the authors that have an $A_{50\%C}$
value less than $5$ (authors with $h$-index above $60$ are not shown).

Figure 4: The distribution of citations, number of papers and $h$-index of the
authors that belong to the lowest 1% of metric $A_{50\%C}$.

###  3.4 Metric $A_{50}$

Metric $A_{50}$ aims to identify authors that heavily collaborate with each
other, essentially functioning as a metric for detecting large-scale
orchestration. Among the $1,496,680$ considered authors, we exclude the
authors that belong to the field of ”Physics & Astronomy”, as it is widely
recognized for its consistent involvement in large-scale collaborative
endeavors, leading to a potential bias in the analysis. In particular, we
observed that around $99\%$ of authors with the largest $1\%$ percentile
values of $A_{50}$ belong to Physics & Astronomy. By excluding those authors
from our analysis, we aim to mitigate this bias and gain a clearer
understanding of metric $A_{50}$ in a broader context. Thus, we end up
examining $1,322,652$ authors. Among them, $242,281$ have a value greater than
zero for the $A_{50}$ metric, they have at least $1$ co-author with whom they
have co-authored more than $50$ papers.

For this metric the $99^{th}$ percentile value is $7$, and the $95^{th}$
percentile is $2$. The median is equal to $0$ as well as the interquartile
range. Here, we focus on the top $1\%$ of authors who possess an $A_{50}$
value exceeding $7$, as we place particular emphasis on exceptional cases
where an author has collaborated with a diverse set of individuals more than
$50$ times, providing the strongest indication for large-scale orchestration.
Table 2 indicates the allocation of these $12,015$ authors across scientific
disciplines. A strong predilection is shown for authors within Clinical
Medicine, as they represent $73.29\%$ of these $12,015$ authors, as opposed to
only $38.71\%$ among all authors. No other field has an enrichment among the
$12,015$ authors. Supplementary table 4 shows the breakdown of Clinical
Medicine authors into subfields. As shown, allergy, arthritis & rheumatism,
cardiovascular system & hematology, gastroenterology & hepatology, general
clinical medicine, geriatrics, and psychiatry subfields are 1.5-fold or more
enriched among those with $A_{50}$ values exceeding the highest $1\%$
percentile.

Figure 5 depicts the distribution of citations, the number of published
papers, and the $h$-index, respectively, for this group of $12,015$ authors
with the largest $A_{50}$ metric. These are highly productive authors (median
$596$ papers, interquartile range $315$ to $1061$) and their $h$-indices are
typically very high or even extraordinarily high (median $56$, interquartile
range $38$ to $80$, $12.8\%$ of these authors with $h$-index¿100).

(a) Distribution of the citations, considering only whoever author has an
$A_{50}$ value greater than $7$ (authors with citations above $100,000$ are
not shown).

(b) The distribution of the number of papers published by authors with an
$A_{50}$ value on the upper 1% (authors with more than $2,000$ papers are not
shown).

(c) Distribution of the $h$-index of the authors that have an $A_{50}$ value
greater than $7$ (authors with $h$-index above $200$ are not shown).

Figure 5: The distribution of citations, number of papers and $h$-index of the
authors that belong to the upper 1% of metric $A_{50}$.

###  3.5 Co-existence of extreme values for the metrics

We examined the co-existence of extreme values for the three metrics (lowest
$1\%$ percentile for the first two, highest $1\%$ percentile for the third
one) among all $1,322,652$ authors excluding Physics and Astronomy. As shown
in Table 3, there was strong co-existence for extremely low values of
$C/h^{2}$ and $A_{50\%C}$ (odds ratio $6.4$, $95\%$ confidence interval
$5.9-6.9$), modest coexistence for extremely low values of $A_{50\%C}$ and
extremely high values of $A_{50}$ (odds ratio $1.5$, 95% confidence interval
$1.3-1.7$), and extremely low values of $C/h^{2}$ very rarely co-occurred with
extremely high values of $A_{50}$ (odds ratio $0.09$, $95\%$ confidence
interval $0.05-0.16$).

|  Lowest percentile of $A_{50\%C}$ |  Highest percentile of $A_{50}$
---|---|---
| YES | NO | YES | NO
Lowest percentile of $C/h^{2}$ | YES | $659$ | $12,566$ | $11$ | $13,214$
NO | $10,618$ | $1,298,809$ | $12,004$ | $1,297,423$
Lowest percentile of $A_{50\%C}$ | YES |  |  | $151$ | $11,126$
NO |  |  | $11,864$ | $1,299,511$

Table 3: Co-existence of small- and large-scale orchestration indicators

##  4 Discussion

The phenomenon of orchestrated citations may markedly enhance the influence of
specific individuals or groups. Using science-wide data from Scopus, we
present three readily accessible metrics that may offer hints to orchestration
patterns. This work may contribute to the ongoing efforts to enhance the
assessment of research impact and recognize the contributions of authors
within their respective fields [Ioannidis and Maniadis, 2023b, Ioannidis and
Maniadis, 2023a]. We found that authors with hints of orchestration based on
these 3 metrics have enriched presence in specific scientific fields.
Importantly, these authors almost always had high or even extremely high
h-indices. These high $h$-indices would be very misleading and would need to
be interpreted in the light of the authorship and citation patterns to place
them in a more proper context.

Authors at the extreme tails of these metrics may or may not engage in
individual or group practices that lead to inflated authorship contributions
and inflated perceived impact. Moreover, most authors who do occasionally
engage in practices that lead to inflated authorship contributions [Masic and
Jankovic, 2021] or inflated perceived impact would not have such extreme
values in these metrics. We propose to use these metrics with a very stringent
threshold, such as 1% percentile, where chance is unlikely to generate such
extremes in the absence of an organized plan or group arrangement regarding
the allocation of authorship and/or citations.

However, any claims of ethical wrong doing should be withheld in the absence
of additional evidence. Each author with hints of extreme orchestration would
need to be examined carefully as to the nature of their collaborations,
teamwork arrangements and standards in their environment, and citation
practices in the networks of scientists who cite them. These metrics should
not be seen as synonymous with unethical practices but as hints for further
appraisal. Many of these scientists may have perfectly legitimate standards
for their field, e.g. massive co-authorship in their teamwork may be the
accepted norm, as in the case of nuclear and particle physics work.
Nevertheless, even in these cases, the extreme nature of orchestration should
be noted and the citation metrics of these authors should be properly adjusted
(e.g. for co-authorship [Ioannidis, 2008] and interpreted. This is essential
in order to avoid unfair comparisons against other scientists in the same or
neighboring fields who have different work and publication arrangements.
Otherwise, comparisons would be grossly unfair.

We observed that the two metrics of small-scale orchestration tended to
coexist far more common than chance. Authors with double indicators for small-
scale orchestration should be even more carefully scrutinized. The presence of
unjustified extreme self-citations or citation cartels should be assessed in
these cases. This requires an in-depth assessment of the nature, relevance,
and justification of citations. Future work may also explore whether such
behavior is linked to other unethical or spurious publication practices, such
as paper mills [Christopher, 2021], hijacked journals [Dadkhah and Borchardt,
2016, Abalkina, 2021], hijacked citations [Dadkhah, 2016] and other subversive
arrangements.

The two indicators of large-scale orchestration also tended to co-exist more
often than chance. In authors with large-scale orchestration, sometimes
astonishingly high citation counts and $h$-indices can be reached, but they
are largely meaningless. These authors should be assessed for the nature of
their contribution to the massively collaborative teamwork where they
participate. Several unethical behaviors may co-exist in some cases of large-
scale orchestration, e.g. scientists may be placed as authors with gift
authorship [Smith, 1994, Teixeira da Silva and Dobránszki, 2016]. This is a
particular problem with people who acquire power as chairs/heads and exhibit a
massive acceleration of their research productivity (at a time when
administrative duties should probably have cut their productivity). Moreover,
while teamwork is a good research practice, in principle, inflation of
authorship within teams may signal salami-slicing, and a poor sense of
accountability and may reflect badly on the entire team/consortium.

Some limitations of our work should be discussed. First, we only focused on
authors with at least 1000 citations. Orchestration may also affect scientists
who have fewer citations. Scientists with fewer than 1000 citations are the
vast majority of the scientific workforce. However, their influence in the
scientific literature is limited compared with the more influential sample
that we examined. Moreover, it is common for self-citations to be high in the
early career of scientists when they have limited citations and they try to
establish themselves with their work [Seeber et al., 2019]. This would
possibly frequently raise signals of small-scale orchestration in early career
authors that have a very different meaning compared with established authors
with many citations. Second, inaccuracies in Scopus may affect the calculation
of these metrics for these scientists. Inaccuracies are overall low across
Scopus, but in the case of an in-depth assessment of the behavior of specific
authors suspected of unethical orchestration practices, it is important to
verify at the first step that their publication and citation profile is
accurate, so as to avoid mischaracterizations. Third, we did not employ here
more complex methods of community detection, centrality analysis, and network
visualization [Jolly et al., 2020, Fister Jr et al., 2016], but these methods
may also be utilized when unethical orchestration is suspected and in-depth
evaluation of specific author networks is needed.

Overall our work reveals that many authors with high or even exceptionally
high traditional citation metrics such as the h-index have strong hints of
orchestration practices. This distortion should be taken into account in using
citation metrics for evaluation and reward purposes.

## References

  * [Abalkina, 2021] Abalkina, A. (2021).  Detecting a network of hijacked journals by its archive.  Scientometrics, 126(8):7123–7148.
  * [Archambault et al., 2011] Archambault, É., Beauchesne, O. H., and Caruso, J. (2011).  Towards a multilingual, comprehensive and open scientific journal ontology.  In Proceedings of the 13th international conference of the international society for scientometrics and informetrics, volume 13, pages 66–77. Durban, South Africa.
  * [Bartneck and Kokkelmans, 2011] Bartneck, C. and Kokkelmans, S. (2011).  Detecting h-index manipulation through self-citation analysis.  Scientometrics, 87(1):85–98.
  * [Bornmann, 2014] Bornmann, L. (2014).  How are excellent (highly cited) papers defined in bibliometrics? a quantitative analysis of the literature.  Research Evaluation, 23(2):166–173.
  * [Bornmann and Daniel, 2005] Bornmann, L. and Daniel, H.-D. (2005).  Does the h-index for ranking of scientists really work?  Scientometrics, 65:391–392.
  * [Bornmann and Daniel, 2009] Bornmann, L. and Daniel, H.-D. (2009).  The state of h index research: Is the h index the ideal way to measure research performance?  EMBO reports, 10(1):2–6.
  * [Burnham, 2006] Burnham, J. F. (2006).  Scopus database: a review.  Biomedical digital libraries, 3(1):1–8.
  * [Chapman et al., 2019] Chapman, C., Bicca-Marques, J. C., Calvignac-Spencer, S., Fan, P.-F., Fashing, P., Gogarten, J., Guo, S., Hemingway, C., Leendertz, F., Li, B., Matsuda, I., Hou, R., Serio-Silva, J., and Stenseth, N. C. (2019).  Games academics play and their consequences: How authorship, h-index and journal impact factors are shaping the future of academia.  Proceedings of the Royal Society B: Biological Sciences, 286:20192047.
  * [Christopher, 2021] Christopher, J. (2021).  The raw truth about paper mills.
  * [Costas et al., 2015] Costas, R., Zahedi, Z., and Wouters, P. (2015).  Do “altmetrics” correlate with citations? extensive comparison of altmetric indicators with citations from a multidisciplinary perspective.  Journal of the Association for Information Science and Technology, 66(10):2003–2019.
  * [Cronin, 2001] Cronin, B. (2001).  Hyperauthorship: A postmodern perversion or evidence of a structural shift in scholarly communication practices?  Journal of the American Society for Information Science and Technology, 52(7):558–569.
  * [Dadkhah, 2016] Dadkhah, M. (2016).  Types of hijacking in the academic world–our experiment in the scholarly publishing.  Library Hi Tech News, 33(3):1–2.
  * [Dadkhah and Borchardt, 2016] Dadkhah, M. and Borchardt, G. (2016).  Hijacked journals: an emerging challenge for scholarly publishing.
  * [Didegah and Thelwall, 2013] Didegah, F. and Thelwall, M. (2013).  Which factors help authors produce the highest impact research? collaboration, journal and document properties.  Journal of informetrics, 7(4):861–873.
  * [Ding et al., 2020] Ding, J., Liu, C., and Kandonga, G. A. (2020).  Exploring the limitations of the h-index and h-type indexes in measuring the research performance of authors.  Scientometrics, 122:1303–1322.
  * [Falagas et al., 2008] Falagas, M. E., Pitsouni, E. I., Malietzis, G. A., and Pappas, G. (2008).  Comparison of pubmed, scopus, web of science, and google scholar: strengths and weaknesses.  The FASEB journal, 22(2):338–342.
  * [Fister et al., 2016] Fister, I., Fister, I., and Perc, M. (2016).  Toward the discovery of citation cartels in citation networks.  Frontiers in Physics, 4.
  * [Fister Jr et al., 2016] Fister Jr, I., Fister, I., and Perc, M. (2016).  Toward the discovery of citation cartels in citation networks.  Frontiers in Physics, 4:49.
  * [Hirsch, 2005] Hirsch, J. E. (2005).  An index to quantify an individual’s scientific research output.  Proceedings of the National academy of Sciences, 102(46):16569–16572.
  * [Ioannidis, 2008] Ioannidis, J. P. (2008).  Measuring co-authorship and networking-adjusted scientific impact.  PloS one, 3(7):e2778.
  * [Ioannidis et al., 2014] Ioannidis, J. P., Boyack, K. W., Small, H., Sorensen, A. A., and Klavans, R. (2014).  Bibliometrics: Is your most cited work your best?  Nature, 514(7524):561–562.
  * [Ioannidis et al., 2018] Ioannidis, J. P., Klavans, R., and Boyack, K. W. (2018).  Thousands of scientists publish a paper every five days.  Nature, 561(7722):167–169.
  * [Ioannidis and Maniadis, 2023a] Ioannidis, J. P. and Maniadis, Z. (2023a).  Quantitative research assessment: using metrics against gamed metrics.  Internal and Emergency Medicine, pages 1–9.
  * [Ioannidis et al., 2020] Ioannidis, J. P. A., Boyack, K. W., and Baas, J. (2020).  Updated science-wide author databases of standardized citation indicators.  PLOS Biology, 18(10):1–3.
  * [Ioannidis and Maniadis, 2023b] Ioannidis, J. P. A. and Maniadis, Z. (2023b).  In defense of quantitative metrics in researcher assessments.  PLOS Biology, 21(12):1–5.
  * [Jolly et al., 2020] Jolly, B. L. K., Jain, L., Bera, D., and Chakraborty, T. (2020).  Unsupervised anomaly detection in journal-level citation networks.  In Proceedings of the ACM/IEEE Joint Conference on Digital Libraries in 2020, pages 27–36.
  * [Kawashima and Tomizawa, 2015] Kawashima, H. and Tomizawa, H. (2015).  Accuracy evaluation of scopus author id based on the largest funding database in japan.  Scientometrics, 103(3):1061–1071.
  * [Kelly and Jennions, 2006] Kelly, C. D. and Jennions, M. D. (2006).  The h index and career assessment by numbers.  Trends in Ecology & Evolution, 21(4):167–170.
  * [Masic and Jankovic, 2021] Masic, I. and Jankovic, S. M. (2021).  Inflated co-authorship introduces bias to current scientometric indices.  Medical Archives, 75(4):248.
  * [Oravec, 2019] Oravec, J. A. (2019).  The ”dark side” of academics? emerging issues in the gaming and manipulation of metrics in higher education.  Review of Higher Education, 42:859–877.
  * [Papatheodorou et al., 2008] Papatheodorou, S. I., Trikalinos, T. A., and Ioannidis, J. P. (2008).  Inflated numbers of authors over time have not been just due to increasing research complexity.  Journal of clinical epidemiology, 61(6):546–551.
  * [Perez et al., 2019] Perez, O., Bar-Ilan, J., Cohen, R., and Schreiber, N. (2019).  The network of law reviews: Citation cartels, scientific communities, and journal rankings.  The Modern Law Review, 82(2):240–268.
  * [Seeber et al., 2019] Seeber, M., Cattaneo, M., Meoli, M., and Malighetti, P. (2019).  Self-citations as strategic response to the use of metrics for career decisions.  Research Policy, 48(2):478–491.
  * [Smith, 1994] Smith, J. (1994).  Gift authorship: a poisoned chalice?
  * [Teixeira da Silva and Dobránszki, 2016] Teixeira da Silva, J. A. and Dobránszki, J. (2016).  Multiple authorship in scientific manuscripts: ethical challenges, ghost and guest/gift authorship, and the cultural/disciplinary perspective.  Science and engineering ethics, 22:1457–1472.
  * [Van Noorden and Chawla, 2019] Van Noorden, R. and Chawla, D. (2019).  Hundreds of extreme self-citing scientists revealed in new database.  Nature, 572.
  * [Waltman and Van Eck, 2012] Waltman, L. and Van Eck, N. J. (2012).  The inconsistency of the h-index.  Journal of the American Society for Information Science and Technology, 63(2):406–415.
  * [Waltman et al., 2011] Waltman, L., van Eck, N. J., van Leeuwen, T. N., Visser, M. S., and van Raan, A. F. (2011).  Towards a new crown indicator: Some theoretical considerations.  Journal of informetrics, 5(1):37–47.

##  Appendix A Supplementary material

Subfield |  Number of eligible authors |  Percentage of eligible authors |  Number of authors that belong to highest 1% percentile |  Percentage over the authors of highest 1% percentile
---|---|---|---|---
allergy |  $2,922$ |  $0.50\%$ |  $93$ |  $1.06\%$
anesthesiology |  $4,562$ |  $0.79\%$ |  $16$ |  $0.18\%$
arthritis & rheumatology |  $6,614$ |  $1.14\%$ |  $168$ |  $1.91\%$
cardiovascular system & hematology |  $61,481$ |  $10.61\%$ |  $1,462$ |  $16.60\%$
dentistry |  $8,488$ |  $1.47\%$ |  $22$ |  $0.25\%$
dermatology & venereal diseases |  $7,218$ |  $1.25\%$ |  $77$ |  $0.87\%$
emergency & critical care medicine |  $4,602$ |  $0.79\%$ |  $35$ |  $0.39\%$
endocrinology & metabolism |  $25,727$ |  $4.44\%$ |  $373$ |  $4.24\%$
environmental & occupational health |  $1,289$ |  $0.22\%$ |  $9$ |  $0.10\%$
gastroenterology & hepatology |  $18,708$ |  $3.23\%$ |  $500$ |  $5.68\%$
general & internal medicine |  $43,032$ |  $7.43\%$ |  $448$ |  $5.09\%$
general clinical medicine |  $287$ |  $0.05\%$ |  $9$ |  $0.10\%$
geriatrics |  $937$ |  $0.16\%$ |  $23$ |  $0.26\%$
immunology |  $46,471$ |  $8.02\%$ |  $470$ |  $5.34\%$
legal & forensic medicine |  $601$ |  $0.10\%$ |  $5$ |  $0.06\%$
neurology & neurosurgery |  $92,605$ |  $15.98\%$ |  $1,096$ |  $12.45\%$
nuclear medicine & medical imaging |  $15,719$ |  $2.71\%$ |  $143$ |  $1.62\%$
obstetrics & reproductive medicine |  $14,860$ |  $2.56\%$ |  $139$ |  $1.58\%$
oncology & carcinogenesis |  $95,554$ |  $16.49\%$ |  $2,041$ |  $23.18\%$
ophthalmology & optometry |  $10,206$ |  $1.76\%$ |  $65$ |  $0.74\%$
orthopedics |  $12,512$ |  $2.16\%$ |  $193$ |  $2.19\%$
otorhinolaryngology |  $4,199$ |  $0.72\%$ |  $19$ |  $0.22\%$
pathology |  $1,130$ |  $0.19\%$ |  $4$ |  $0.05\%$
pediatrics |  $9,012$ |  $1.56\%$ |  $56$ |  $0.64\%$
pharmacology & pharmacy |  $21,038$ |  $3.63\%$ |  $64$ |  $0.73\%$
psychiatry |  $17,236$ |  $2.97\%$ |  $481$ |  $5.46\%$
respiratory system |  $11,667$ |  $2.01\%$ |  $179$ |  $2.03\%$
sport sciences |  $4,521$ |  $0.78\%$ |  $25$ |  $0.28\%$
surgery |  $18,604$ |  $3.21\%$ |  $303$ |  $3.44\%$
tropical medicine |  $3,024$ |  $0.52\%$ |  $25$ |  $0.28\%$
urology & nephrology |  $14,419$ |  $2.49\%$ |  $263$ |  $2.99\%$

Table 4: The break-down of Clinical Medicine authors, who belong to the
highest 1% of metric $A_{50}$, into subfields