# Filling Pattern Dependence of Regenerative Beam Breakup Instability in
Energy Recovery Linacs

S. Setiniyaz ID<EMAIL_ADDRESS>R. Apsimon ID
<EMAIL_ADDRESS>Engineering Department, Lancaster University,
Lancaster, LA1 4YW, UK  Cockcroft Institute, Daresbury Laboratory, Warrington,
WA4 4AD, UK  P. H. Williams ID<EMAIL_ADDRESS>STFC Daresbury
Laboratory & Cockcroft Institute, Warrington, WA4 4AD, UK

###### Abstract

Beam breakup instability is a potential issue for all particle accelerators
and is often the limiting factor for the maximum beam current that can be
achieved. This is particularly relevant for Energy Recovery Linacs with
multiple passes where a relatively small amount of charge can result in a
large beam current. Recent studies have shown that the choice of filling
pattern and recirculation scheme for a multi-pass energy recovery linac can
drastically affect the interactions between the beam and RF system. In this
paper we further explore this topic to study how filling patterns affect the
beam breakup instability and how this can allow us to optimise the design in
order to minimise this effect. We present a theoretical model of the beam-RF
interaction as well as numerical modeling and show that the threshold current
can vary by factors of 2-4, and potentially even more depending on the machine
design parameters. Therefore a judicious choice of filling pattern can greatly
increase the onset of BBU, expanding the utility of future ERLs.

††preprint: APS/123-QED

##  I Introduction

In the 2020 European Strategy for Particle Physics
EuropeanStrategyGroup:2020pow , superconducting Energy Recovery Linacs (ERLs)
Merminga2016  were identified as a key accelerator technology requiring
priority research and development to underpin the future anticipated needs of
the community. Applications in particle physics cover both ERL-based
colliders, such as LHeC AbelleiraFernandez2012 ; LHeCnew , PERLE PERLE , FCC-
ee Litvinenko:2019txu  and beam coolers for hadron colliders, notably EIC
Accardi2016 . ERLs are also seen as a promising option, in both academic and
industrial contexts, for future Free-Electron laser light sources Socol2013 ;
Socol2011  and for nuclear physics through both direct beam internal target
experiments Hug:2017ypc  and through secondary production of narrowband gammas
via inverse Compton scattering Shimada2010 ; Hayakawa2010 .

One common theme in these future applications is the requirement of high
average beam power, with GW being an aspired-for reasonable mid-term goal.
This is three orders of magnitude beyond that achieved to date, namely by the
JLab FEL upgrade Neil:2000zz ; Alarcon:2013yxa , indicating the importance of
addressing this as a priority for ERL research.

A well known limitation on the current one can support in an ERL is a
particular incarnation of so-called beam breakup (BBU) instabilities. BBU has
potential to occur where the beam interacts with a higher order mode (HOM) of
an RF structure that it traverses and becomes deflected. There are two general
classes of BBU, cumulative Wangler2008 ; Chao2013 ; Padamsee1998 ;
Gluckstern:1985yh ; Delayen2003 ; Delayen2005  and regenerative LYNEIS1983269
; Pozdeyev2005 . Cumulative is where the deflection builds up over multiple
structures and is not the subject of this study. Regenerative BBU is that of
primary concern in recirculating and energy recovery linacs where there are
multiple passes of the same bunch through each RF structure. This constitutes
a feedback loop between the beam position offset and the HOM voltage.

Bunches passing through the cavity will excite a transverse HOM whose
amplitude is dependent on the transverse offset of the passing bunches. In
turn, the HOM voltage will deflect the subsequent bunches and increase their
offsets. If the beam current is at a certain threshold, $I_{th}$, then the
magnitude of the HOM voltage and bunch offsets will reach an equilibrium,
below this the HOM voltage will tend to zero, above it will cause the HOM
voltage and bunch offsets will grow exponentially until beam loss occurs. This
defines the BBU instability Altenmueller1966 ; Neil1970 . Whilst studies have
been undertaken to investigate BBU instabilities for ERLs Hoffstaetter2004 ;
Hoffstaetter2019 ; Pozdeyev2006 ; Volkov2018 ; Tennant2005 ;
TennantDissertation2006 ; Merminga2002 , no studies have yet considered the
impact of different beam filling patterns and beam line topologies on the beam
loading transients imparted on the cavities and bunches. This becomes
important when considering multi-pass ERLs which, having now been demonstrated
experimentally Bartnik:2020pos , feature in many designs for future ERL
facilities due to the obvious advantage of providing higher beam energies
without concomitant increase in the number of accelerating structures. The
effect of the filling pattern, which describes the order in which bunches are
injected into the ERL over subsequent turns, has in a previous publication
Setiniyaz2020  been shown to have a major impact on the stability of the RF
system. Just as the fundamental mode and RF power are affected differently by
our choice of beam line topology and filling pattern, the threshold current of
regenerative BBU is also dependent on these parameters. In the following
section, we show that the impact on BBU is somewhat more complex than it is
for the fundamental mode due to the asynchronous nature of the HOM mode
relative to the beam, as well as the transcendental relationship between the
HOM voltage and the bunch offsets.

For regenerative BBU, the HOM mode has a sufficiently high Q-factor that the
mode persists for a relatively long timescale. As the bunches pass through the
structure they are given a transverse kick, whilst also contributing to the
excitation of the HOM due to their off-axis trajectories. On recirculation
they pass back through the structure with a larger offset, further exciting
the mode. This feedback loop can grow exponentially until beam loss occurs.

Filling patterns are used to describe the order in which bunches are injected
into a ring on subsequent turns, here we use the concept (introduced in
Setiniyaz2020 ) of intra-packet blocks to describe the position they occupy on
each turn. For example, for a 6-turn ERL (3 accelerating passes and 3
decelerating), the filling pattern [1 2 3 4 5 6] indicates that the first
bunch goes to the first intra-packet block, the second bunch goes to the
second block, and so on. The intra-packet block also fixes how many RF cycles
are between bunches, as in the general case bunches are not necessarily
injected into every RF cycle, illustrated in Fig. 1. These bunches forms a
packet and multiples of such packets fill up the ring as shown in the Fig. 2.
Similarly, filling pattern [1 3 2 4 5 6] indicates that the first bunch goes
to the first block, second bunch goes to the third block, and so on. These
filling schemes are called FIFO (first in first out) as the bunches maintain
their order in the packet, however, their turn number changes turn by turn.
One can also generated a packet where turn number in the packet doesn’t change
turn by turn by and consequently the cavity always see same packets passing
thorough. Such a scheme is called sequence preserving (SP) scheme and denoted
with curly brackets to be differentiated from square brackets of FIFO schemes.
For example, SP pattern $\\{1~{}4~{}2~{}5~{}3~{}6\\}$ describes the first
bucket is always occupied by bunch at the first turn, the second bucket is
always occupied by the bunch at fourth turn, and so on. The key differences
are that in FIFO schemes, bunches remain in the same RF block and bunch turn
numbers change, whilst in SP schemes the arc length for each turn of the ERL
is specifically designed such that the bunches transition between RF blocks on
each turn such that the sequence of bunch turn number in a packet remains
constant on each turn; hence the name sequence preserving.

Figure 1: Intra-packet blocks. RF cycles are shown within the linac. Each
block is colored differently. The red/blue bunches are on the peak/trough and
being accelerated/decelerated.

Figure 2: Bunch packets in the ERL. Bunches at the third turn go through a
transition arc where there is at least an extra half RF cycle delay.

The number of possible filling patterns increases as $\left(N-1\right)!$,
where $N$ is the number of recirculation passes of the ERL. The filling
patterns are dependent on the bunch injection and recombination schemes. FIFO
and SP patterns are merely the most simple subsets of all possible schemes
that can be generated, this will be further studied in a future work employing
a group theoretic classification technique. In this work we focus on SP
schemes to show that BBU is affected by the filling pattern. There are several
factors in multi-turn ERLs that can affect BBU, which are not present in
circular machines: bunches at different turns have different energies so under
the same HOM voltage higher energy bunches are deflected less than low energy
ones; bunches transitioning from accelerating phase to decelerating phase need
to be delayed by at least half an RF cycle and thus have a changed revolution
time; bunches accumulate offset and kicks within the recirculating process
which become amplified as the bunch energy decreases, this is termed adiabatic
anti-damping. All these effects must be taken into account when determining
filling pattern dependent BBU threshold currents.

In section II, we present an analytical derivation for the threshold current
and its dependence on filling patterns and beam line topology and consider how
it drives BBU. Due to the transcendental relationship between HOM voltage and
beam trajectories, the analytical approach is unable to provide a simple
calculation of the threshold current, however we are able to exploit it to
provide insight into the behaviour of BBU under certain conditions. In Section
III, we provide the results of numerical simulations from Matlab in order to
determine the threshold current for different filling patterns for the SP
schemes. These numerical simulations allow us to look at the frequency
dependence of the threshold current for different filling patterns as well as
to determine the threshold current for all filling patterns as a frequency-
averaged threshold current. From this, we are able to show that with the
correct choice of filling pattern, the threshold current can be increased
significantly with appropriate optimisation.

##  II Analytical Model

In this section, we derive an analytical expression for the threshold current
to allow us to understand how the filling pattern and recirculation scheme of
the ERL affects the threshold current. We start by following the standard
derivation described in other works, such as Pozdeyev2005 ;
TennantDissertation2006 ; Pozdeyev2006 . Then we augment this initial
expression for the threshold current, casting it in terms of a generic beam
filling pattern.

BBU can be a long and/or short-range effect and is caused by transverse
offsets of the bunch centroids exciting deflecting modes, which are usually
dipole fields, which then give a transverse kick to the bunch on the next
turn. For short-range BBU, the HOM mode excited has a small loaded Q-factor
($Q_{L}$) and as such the mode is sufficiently attenuated before the bunches
return in order to prevent a significant transverse kick to the bunch after it
has completed one revolution, but it can have a significant effect on bunches
directly behind the driving bunch. From the Panofsky-Wenzel theorem pw , we
relate the transverse voltage of the HOM mode in terms of the transverse
variation in longitudinal voltage as:

$V_{\perp}=-\frac{ic}{\omega}\frac{V_{z}\left(x\right)}{x}e^{i\phi}.$ (1)

It can also be shown that a bunch of charge $q_{bunch}$ excites this HOM mode
and increases the transverse voltage as

$\delta{V}_{\perp}=-i\frac{q_{bunch}\omega}{2}\left(\frac{\omega{x}}{c}\right)\left(\frac{R}{Q}\right)_{\perp}e^{i\phi}$
(2)

where $x$ is the transverse offset from the electrical centre of the cavity,
$c$ is the speed of light, $\omega$ is the angular frequency of the HOM mode,
$\left(\frac{R}{Q}\right)_{\perp}$ is the transverse $R/Q$ and $\phi$ is the
phase of the beam with respect to the peak of the HOM voltage. A derivation of
this beam loading can be found in apsimon .

For a recirculating ERL, the energy of the bunches will vary from one turn to
the next and therefore, we need to consider the one-turn map for each turn,
taking into account the energy variation of the bunch during the turn.
Furthermore, we also need to factor in the deflection from the HOM on each
turn, which will depend on the bunch energy. In general, we can write a simple
first-order one-turn map as

$\begin{array}[]{l}\begin{pmatrix}x_{n}\\\
x^{\prime}_{n}\end{pmatrix}=\begin{pmatrix}R_{11}^{\left(n\right)}&R_{12}^{\left(n\right)}\\\
R_{21}^{\left(n\right)}&R_{22}^{\left(n\right)}\end{pmatrix}\begin{pmatrix}x_{n-1}\\\
x^{\prime}_{n-1}\end{pmatrix}+\begin{pmatrix}\delta{x}_{HOM}^{\left(n\right)}\\\
\delta{x^{\prime}}_{HOM}^{\left(n\right)}\end{pmatrix}\end{array}$ (3)

where $R_{ij}^{\left(n\right)}$ is the $i,j$ matrix element for the one-turn
map for turn $n$, $x_{n}$ and $x^{\prime}_{n}$ are the position and angle
trajectory of the bunch on turn $n$ and $\delta{x}_{HOM}^{\left(n\right)}$ and
$\delta{x^{\prime}}_{HOM}^{\left(n\right)}$ are the position and angle
deflection from the HOM mode. From Eq. 3, we have a means of determining all
the bunch positions if the transverse HOM voltage is known. In addition, from
Eq. 2, we can write an expression for the collective beam loading of the HOM
mode due to $k$ bunches, taking into account the $Q_{L}$ of the mode and the
phase shift between bunches as

$\delta{V}_{\perp,k}=-i\frac{\omega^{2}}{2c}\left(\frac{R}{Q}\right)_{\perp}e^{\left(-\frac{\omega{}T_{k+1}}{2Q_{L}}+i\phi_{0}\right)}\sum_{j=1}^{k}{q_{j}x_{j}e^{\omega{T_{j}}\left(\frac{1}{2Q_{L}}+i\right)}}.$
(4)

In Eq. 4, we define without loss of generality that $T_{k+1}=0$, hence
$e^{-\frac{\omega{T}_{k+1}}{2Q_{L}}}=1$. The stored energy in a cavity is
given as

$U=\frac{\left|V_{\perp}\right|^{2}}{\omega\left(\frac{R}{Q}\right)_{\perp}},$
(5)

from Eq. 5, we can determine the RF power transferred by the beam as

$\frac{\delta{U}}{\delta{t}}=\frac{2\left|V_{\perp}\right|}{\omega\left(\frac{R}{Q}\right)_{\perp}}\frac{\left|\delta{V}_{\perp}\right|}{\delta{t}}.$
(6)

By substituting Eq. 4 into Eq. 6 and assuming that the bunch charge is either
constant for all bunches or periodic such that the mean bunch charge is
bounded, then
$\sum_{j=1}^{k\rightarrow\infty}{\frac{q_{j}}{\delta{t}}}=I_{beam}$, then we
obtain:

$\frac{dU_{beam}}{dt}=I_{beam}\frac{\omega\left|V_{\perp}\right|}{c}\left|\sum_{j=1}^{k}{x_{j}e^{\omega{T_{j}}\left(\frac{1}{2Q_{L}}+i\right)}}\right|.$
(7)

We also know that when the cavity HOM voltage is at equilibrium,

$\frac{dU_{cav}}{dt}=\frac{dU_{beam}}{dt}+P_{c}=0,$ (8)

where $P_{c}$ is the ohmic power dissipation in the cavity, given as

$P_{c}=\frac{\left|V_{\perp,k-1}\right|^{2}}{2\left(\frac{R}{Q}\right)_{\perp}Q_{L}}.$
(9)

The threshold current is defined as the maximum allowed beam current before
the HOM voltage grows exponentially, thus it is the limit of stability for the
accelerator. At the threshold current, the HOM voltage is at steady state at
some value, below it, the HOM voltage will decay exponentially. Thus the only
solutions to Eq. 8 are the trivial case where $\left|V_{\perp}\right|=0$, or
when $I_{beam}=I_{th}$, where $I_{th}$ is the threshold current. We can obtain
the threshold current from Eqs. 8 and 9 as

$I_{th}=I_{beam}=-\frac{\left|V_{\perp}\right|c}{2\left(\frac{R}{Q}\right)_{\perp}\omega{Q}_{L}\left|\sum_{j=1}^{k}{x_{j}e^{\omega{T_{j}}\left(\frac{1}{2Q_{L}}+i\right)}}\right|}.$
(10)

From hereon, it is convenient to introduce some definitions as the beam
loading will be periodic over several timescales. We define a bunch packet as
an ensemble of $N$ consecutive bunches for an $N$-turn ERL, such that each
bunch has completed a different number of recirculating passes. Furthermore,
it is assumed that on a given turn, each bunch packet is equivalent. Finally,
we know that from Eq. 3 that if the bunch offsets are dominated by the kicks
from the HOM, then $x_{j}$ will depend on the HOM voltage, $V_{\perp}$, which
in turn will depend on the bunch charge, $q_{j}$. If we assume that the
variation in $q_{j}$ is small, then it can be approximated as a constant. Thus
if $V_{\perp}$ and $x_{j}$ both depend on the bunch charge, then the overall
charge dependence is cancelled in the threshold current. As we shall see
further on in this section, the threshold current will have a charge in the
equation, although as we know that the charge dependence cancels, we can take
$q=1$ C for convenience without loss of generality.

From Setiniyaz2020  it is known that for any choice of bunch filling pattern
and recirculation scheme, the filling pattern will repeat every $N$ turns, or
in some cases an integer divisor of $N$. Additionally, in Setiniyaz2020  the
concept of a sequence preserving (SP) scheme is discussed, whereby for a given
filling pattern, a recirculation scheme can be chosen such that the sequence
of bunch turn numbers in a packet is the same on each turn. For SP schemes, we
see that the bunch packets are equivalent for all turns and so we can also
trivially see that this will indeed be periodic over $N$ turns. To aid in the
following derivation, we shall define a form factor for a single bunch packet
($f_{n,k}$), a single turn ($F_{n,k}$), and $N$-turns
($\langle{F_{k}\rangle}$) respectively as

$\begin{array}[]{l}f_{n,k}=e^{-\frac{\omega{T}_{N+k}}{2Q_{L}}}\left|\sum_{j=k}^{N+k-1}{x_{j}e^{\omega{T}_{j}\left(\frac{1}{2Q_{L}}+i\right)}}\right|\\\
F_{n,k}=e^{-\frac{\omega{T}_{MN+k}}{2Q_{L}}}\left|\sum_{j=k}^{MN+k-1}{x_{j}e^{\omega{T}_{j}\left(\frac{1}{2Q_{L}}+i\right)}}\right|\\\
F_{k}=e^{-\frac{\omega{T}_{MN^{2}+k}}{2Q_{L}}}\left|\sum_{j=k}^{MN^{2}+k-1}{x_{j}e^{\omega{T}_{j}\left(\frac{1}{2Q_{L}}+i\right)}}\right|\end{array},$
(11)

where $k$ is the bunch number in the packet, $n$ is the turn number. These
form factors are a convenient means of quantifying the effect on HOM voltage
from the bunches over different timescales. These form factors depend
explicitly on the filling pattern and recirculation scheme.

As we assume that the bunch structure is periodic over a bunch packet and also
periodic over $N$ turns, then it is also convenient to define the duration of
a bunch packet as $T_{packet}$ and the mean revolution period of the ring as
$T_{rev}=MT_{packet}$. Based on this, we can express the 1-turn and $N$-turn
form factors in terms of the bunch packet form factor as

$\begin{array}[]{l}f_{n,k}=e^{-\frac{\omega\left(T_{packet}+T_{k}\right)}{2Q_{L}}}\left|\sum_{j=k}^{N+k-1}{x_{j}e^{\omega{T}_{j}\left(\frac{1}{2Q_{L}}+i\right)}}\right|\\\
F_{n,k}=\left|\sum_{m=0}^{M-1}{e^{m\omega{T}_{packet}\left(\frac{1}{2Q_{L}}+i\right)}}\right|f_{n,k}\\\
F_{k}=\left|\sum_{m=0}^{M-1}{e^{m\omega{T}_{packet}\left(\frac{1}{2Q_{L}}+i\right)}}\right|\sum_{n=1}^{N}{f_{n,k}e^{\frac{\left(n-N\right)\omega{T}_{rev}}{2Q_{L}}}}\end{array}.$
(12)

We note in Eq. 12 that the summations for the 1- and $N$-turn form factors are
finite geometric sums which can easily be evaluated. From Eq. 10 it is
convenient to convert the summation in the denominator into a more appropriate
form and we shall do so by exploiting the $N$-turn form factor from Eq. 12 and
will consider the summation in Eq. 10 to tend to infinity.

$\begin{aligned}
\left|\sum_{j=k}^{\infty}{x_{j}e^{\omega{T_{j}}\left(\frac{1}{2Q_{L}}+i\right)}}\right|=F_{k}\left|\sum_{n=0}^{\infty}{e^{-n\omega{T}_{rev}\left(\frac{1}{2Q_{L}}+i\right)}}\right|\\\
=\frac{F_{k}e^{\frac{N\omega{T}_{rev}}{4Q_{L}}}}{\sqrt{2}\sqrt{\cosh{\left(\frac{N\omega{T}_{rev}}{2Q_{L}}\right)}-\cos{\left(N\omega{T}_{rev}\right)}}}\end{aligned}.$
(13)

Next we shall evaluate the HOM voltage from Eq. 10 and for this, we shall
assume that the cavity is at steady state, thus over a single turn, the HOM
voltage amplitude will be periodic over each bunch packet, however for
convenience, we shall determine the HOM voltage from all the bunches in a
single turn. Thus we can say that the condition for convergence is given as

$\begin{array}[]{l}V_{\perp}=V_{\perp}e^{-\omega{T}_{rev}\left(\frac{1}{2Q_{L}}-i\right)}\\\
-i\frac{\omega^{2}}{2c}\left(\frac{R}{Q}\right)_{\perp}e^{-\frac{\omega{T}_{rev}}{2Q_{L}}}e^{i\phi_{0}}\sum_{j=k-1}^{MN+k-2}{q_{j}x_{j}e^{\omega{T}_{j}\left(\frac{1}{2Q_{L}}+i\right)}}\end{array}.$
(14)

We can rearrange Eq. 14 and solve for $V_{\perp}$, but we also only need the
magnitude of this. We shall once again assume that the bunch charge is
constant, thus we can consider the mean bunch charge and we can also express
the summation as the 1-turn form factor from Eq. 13 to obtain

$\left|V_{\perp}\right|=\frac{\langle{q}\rangle\frac{\omega^{2}}{2c}\left(\frac{R}{Q}\right)_{\perp}e^{-\frac{\omega{T}_{rev}}{4Q_{L}}}F_{n,k-1}}{\sqrt{2}\sqrt{\cosh{\left(\frac{\omega{T}_{rev}}{2Q_{L}}\right)}-\cos{\left(\omega{T}_{rev}\right)}}}.$
(15)

We can now substitute Eqs. 13 and 15 into Eq. 10 to obtain

$\begin{array}[]{l}I_{th}=\\\
-\frac{\langle{q}\rangle\omega}{4Q_{L}}\sqrt{\frac{\cosh{\left(\frac{N\omega{T}_{rev}}{2Q_{L}}\right)}-\cos{\left(N\omega{T}_{rev}\right)}}{\cosh{\left(\frac{\omega{T}_{rev}}{2Q_{L}}\right)}-\cos{\left(\omega{T}_{rev}\right)}}}\frac{F_{n,k-1}e^{-\frac{\left(N+1\right)\omega{T}_{rev}}{4Q_{L}}}}{F_{k}}\end{array}.$
(16)

It should be noted that in general the 1-turn form factor changes turn by
turn, therefore this implies that the threshold current is time-dependent as a
result of this. We also have that the form factor in the numerator is counting
from the $\left(k-1\right)^{\text{th}}$ bunch, whereas in the denominator it
is counting from the $k^{\text{th}}$ bunch. We can express the 1- and $N$-turn
form factors in terms of the bunch packet form factors and sum the threshold
current over all bunches and turns to give the following general expression
for the average threshold current

$\begin{array}[]{l}\langle{I_{th}}\rangle=-\frac{\alpha}{N^{2}}\frac{\langle{q}\rangle\omega}{4Q_{L}}\sum_{k=1}^{N}{\frac{\sum_{n=1}^{N}{f_{n,k-1}}}{\sum_{n=1}^{N}{f_{n,k}e^{\frac{n\omega{T}_{rev}}{2Q_{L}}}}}}\\\
\\\
\alpha=e^{\frac{\left(N-1\right)\omega{T}_{rev}}{4Q_{L}}}\sqrt{\frac{\cosh{\left(\frac{N\omega{T}_{rev}}{2Q_{L}}\right)}-\cos{\left(N\omega{T}_{rev}\right)}}{\cosh{\left(\frac{\omega{T}_{rev}}{2Q_{L}}\right)}-\cos{\left(\omega{T}_{rev}\right)}}}\end{array}.$
(17)

In Eq. 17, $\frac{\langle{q}\rangle\omega}{4Q_{L}}$ is essentially the most
basic possible estimate for the threshold current, $\alpha$ is an enhancement
factor that depends on the HOM frequency, mean revolution period and number of
recirculation turns in the ERL.

###  II.1 Worked example: SP schemes

As previously mentioned and also discussed in more detail in Setiniyaz2020 ,
SP schemes have the property that the sequence of bunch turn numbers in a
packet does not change from one turn to the next. It can be shown that all
filling patterns will have a unique recirculation scheme to form an SP scheme,
but the converse is not necessarily true. We can use the properties of SP
schemes to find a special case form of Eq. 17 because the filling pattern and
therefore bunch packet form factors are independent of turn number.

$\begin{array}[]{l}\langle{I_{th}}\rangle=-\frac{\alpha^{\prime}}{N}\frac{\langle{q}\rangle\omega}{4Q_{L}}\sum_{k=1}^{N}{\frac{f_{k-1}}{f_{k}}}\\\
\\\
\alpha^{\prime}=\frac{\sinh{\left(\frac{\omega{T}_{rev}}{2Q_{L}}\right)}}{\sinh{\left(\frac{N\omega{T}_{rev}}{2Q_{L}}\right)}}\sqrt{\frac{\cosh{\left(\frac{N\omega{T}_{rev}}{2Q_{L}}\right)}-\cos{\left(N\omega{T}_{rev}\right)}}{\cosh{\left(\frac{\omega{T}_{rev}}{2Q_{L}}\right)}-\cos{\left(\omega{T}_{rev}\right)}}}\end{array}.$
(18)

The ratio of form factors needs to be determined numerically due to the cyclic
dependence between $\left|V_{\perp}\right|$ and the transverse offset of the
beam (Eqs. 3 and 15), resulting in a transcendental equation for the cavity
voltage. Fig. 3 shows $\alpha^{\prime}$ vs $\omega{T}_{rev}$ for different
numbers of recirculation turns ($N$) where we have assumed that $Q_{L}=1000$.
This shows that we expect the mean threshold current to decrease with
increasing $N$, placing a constraint on the feasibility of designing
arbitrarily many turn ERLs. Furthermore, it shows that $\alpha^{\prime}$ tends
to a series of Dirac $\delta$-functions in the limit that
$N\rightarrow\infty$. As expected, $\alpha^{\prime}$ is maximal when
$\omega{T}_{rev}=2n\pi$, where $n$ is an integer, and also
$\alpha^{\prime}\approx 0$ when $\omega{T}_{rev}=\left(2n+1\right)\pi$.

Figure 3: $\alpha^{\prime}$ vs. $\omega{T}_{rev}/2\pi$ for SP schemes for
different numbers of recirculation turns, with $Q_{L}$ assumed to be 1000.

###  II.2 Properties of the bunch packet form factor, $f_{n,k}$

From Eqs. 17, we note that the threshold current depends on both $f_{n,k}$ and
$f_{n,k-1}$. This is because $\left|V_{\perp}\right|$ at some moment in time
depends on the cumulative beam loading of all previous bunches; which depends
on $f_{n,k-1}$. However, the threshold current also depends on the effect this
voltage has on the next bunch passing through; which depends on $f_{n,k}$. It
would therefore be beneficial to understand the relationship between the form
factor evaluated over different bunches as well as to gain some insight into
the structure of the form factor. Starting from Eq. 12, it is useful to
express the form factor in a non-modular form. Starting with the modular part,
we can rewrite it as

$\left|\sum_{j=k}^{N+k-1}{x_{j}e^{\omega{T}_{j}\left(\frac{1}{2Q_{L}}+i\right)}}\right|=\left|\sum_{j=k}^{N+k-1}{x_{j}e^{\frac{\omega{T}_{j}}{2Q_{L}}}e^{i\omega{T}_{j}}}\right|.$
(19)

We can now remove the modular form since $\left|z\right|^{2}=zz^{*}$, and in
addition, we can write the multiple of two summations as a double-summation
instead to obtain

$\begin{array}[]{l}\left|\sum_{j=k}^{N+k-1}{x_{j}e^{\omega{T}_{j}\left(\frac{1}{2Q_{L}}+i\right)}}\right|^{2}=\\\
\sum_{j=k}^{N+k-1}{\sum_{l=k}^{N+k-1}{x_{j}x_{l}e^{\frac{\omega\left({T}_{j}+T_{l}\right)}{2Q_{L}}}e^{i\omega\left({T}_{j}-T_{l}\right)}}}\end{array}.$
(20)

From Eq. 20, we can split the double summation into two separate parts, the
first where $j=l$ will become a single summation and for the case where
$j\neq{l}$, due to the symmetry of the equation, when know that
$f\left(j,l\right)=f\left(l,j\right)$, thus without loss of generality, we can
take double the summation and impose the constraint that $j>l$ to obtain

$\begin{array}[]{l}\left|\sum_{j=k}^{N+k-1}{x_{j}e^{\omega{T}_{j}\left(\frac{1}{2Q_{L}}+i\right)}}\right|^{2}=\sum_{j=k}^{N+k-1}{x_{j}^{2}e^{\frac{\omega{T}_{j}}{Q_{L}}}}+\\\
2\sum_{j=k+1}^{N+k-1}{\sum_{l=k}^{j-1}{x_{j}x_{l}e^{\frac{\omega\left({T}_{j}+T_{l}\right)}{2Q_{L}}}\cos{\omega\left({T}_{j}-T_{l}\right)}}}\end{array}.$
(21)

Eq. 21 can be substituted back into Eq. 12 to give a convenient means of
calculating the form factors without needing to compute complex numbers.
Furthermore, in this form, we can explore the relationship between $f_{n,k}$
and $f_{n,k-1}$ by understanding how each summation over $k-1$ relates to the
summation over $k$. We shall assume that the ERL is stable and periodic over a
bunch packet, thus $x_{N+k}=x_{k}$ and also $T_{N+k}=T_{packet}+T_{k}$. From
the single summation, we can express it as

$\begin{array}[]{l}\sum_{j=k-1}^{N+k-2}{x_{j}^{2}e^{\frac{\omega{T}_{j}}{Q_{L}}}}=\sum_{j=k}^{N+k-1}{x_{j}^{2}e^{\frac{\omega{T}_{j}}{Q_{L}}}}-\\\
\\\
2x_{k}^{2}e^{\frac{\omega{T}_{k}}{2Q_{L}}}e^{\frac{\omega{T}_{packet}}{2Q_{L}}}\sinh{\frac{\omega{T}_{packet}}{2Q_{L}}}\end{array}.$
(22)

Applying the same to the double summation, we obtain

$\begin{array}[]{l}\sum_{j=k}^{N+k-2}{\sum_{l=k-1}^{j-1}{x_{j}x_{l}e^{\frac{\omega\left({T}_{j}+T_{l}\right)}{2Q_{L}}}\cos{\omega\left({T}_{j}-T_{l}\right)}}}=\\\
\sum_{j=k+1}^{N+k-1}{\sum_{l=k}^{j-1}{x_{j}x_{l}e^{\frac{\omega\left({T}_{j}+T_{l}\right)}{2Q_{L}}}\cos{\omega\left({T}_{j}-T_{l}\right)}}}-\\\
\\\
r\sum_{j=k}^{N+k-2}{x_{j}x_{k}e^{\frac{\omega\left(T_{j}+T_{k}\right)}{2Q_{L}}}}\cos{\left(\omega\left(T_{k}-T_{j}\right)+\psi\right)}\end{array}.$
(23)

where

$\begin{array}[]{l}r=\sqrt{2}e^{\frac{\omega{T}_{packet}}{4Q_{L}}}\sqrt{\cosh{\left(\frac{\omega{T}_{packet}}{2Q_{L}}\right)}-\cos{\omega{T}_{packet}}}\\\
\text{and}\\\
\psi=\tan^{-1}{\left(\frac{\sin{\omega{T}_{packet}}}{\cos{\omega{T}_{packet}}-e^{-\frac{\omega{T}_{packet}}{2Q_{L}}}}\right)}\end{array}.$
(24)

Substituting Eqs. 21-24 into Eq. 12, we obtain

where

$\begin{array}[]{l}f_{n,k-1}^{2}=e^{\frac{\omega\left(T_{k}-T_{k-1}\right)}{Q_{L}}}\left(f_{n,k}^{2}\right.\\\
\\\
\left.-2x_{k}^{2}e^{-\frac{3\omega{T}_{packet}}{2Q_{L}}}\sinh{\left(\frac{\omega{T}_{packet}}{2Q_{L}}\right)}-2rx_{k}e^{-\frac{\omega{T}_{packet}}{Q_{L}}}\times\right.\\\
\\\
\left.\sum_{j=k}^{N+k-2}{x_{j}e^{\frac{\omega\left(T_{j}+T_{k}\right)}{2Q_{L}}}}\cos{\left(\omega\left(T_{k}-T_{j}\right)+\psi\right)}\right).\end{array}$
(25)

Eq. 25 now provides us with an explicit relationship between $f_{n,k-1}$ and
$f_{n,k}$ that can be used to with the computation of
$\langle{I_{th}}\rangle$.

##  III Simulation

###  III.1 ERL BBU code with Filling Pattern

As the concept of filling patterns for ERLs is relatively new, none of the
existing BBU simulation codes are currently able to incorporate this into
their calculations. We have adapted the ERLBBU algorithm Pozdeyev2005 ;
TennantDissertation2006 ; Pozdeyev2006  to include the calculation of BBU with
filling patterns by not assuming a constant bunch spacing. These modifications
describe the arrival time and energy of each individual bunch. Benchmarking
against the experimental results presented in Table 5.1 of Ref.
TennantDissertation2006  provided consistent results when applying the same
bunch pattern assumed in the literature. Our modified script predicted a
threshold current of 2.39 mA, which is slightly closer to the experimental
results than other codes and analytical estimates due to these corrections.
The ERLBBU algorithm starts with a test current and an initial HOM voltage
(which for the following simulations is assumed to be 10 kV transverse
voltage). Bunches are injected with small, Gaussian distributed, transverse
offsets. This simulation estimates if the cavity voltage increases or
decreases over time under the test current, as shown in Fig. 4, and generates
a new test current accordingly and repeats the process until the threshold
current is determined within a user-defined tolerance range. As seen in Fig.
4, when the threshold current of 2.39 mA is reached, the cavity HOM voltage
has converged to an equilibrium value. The assumed parameters are given in
Table 1. The magnitude of the transverse offset of the bunches also increases
or decreases in a similar fashion to the HOM voltage as shown in the Fig. 5.

Figure 4: HOM voltage when test current Itest is above (black), below (red), and at (blue) the threshold current. Table 1: 6-Turn ERL BBU simulation parameters. Parameter |  Unit |  Value
---|---|---
fundamental mode frequency $f$ |  MHz |  1497.0
HOM frequency $f_{HOM}$ |  MHz |  2105.4$-$2106.6
HOM loaded Q-factor $Q_{L,HOM}$ |  |  $6.11\times 10^{6}$
HOM geometric shunt impedance $\left(\frac{R}{Q}\right)_{HOM}$ |  $\Omega$ |  29.9
revolution period for non-transitioning bunches $T_{rev}$ |  ns |  801.67
revolution period for transitioning bunch $T_{rev,t}$ |  ns |  802.01
bunch energies at turn 1$-$6 |  MeV |  7.3, 46.3, 85.3, 124.3, 85.3, 46.3
bunch spacing |  $T_{RF}$ |  5 and 10
injected beam RMS offset $\sigma_{x,offset}/\sigma_{y,offset}$ |  $\mu$m |  10/1

Figure 5: Particle x-position (in the cavity) when test currents Itest is
above (black), below (red), and at (blue) the threshold current.

###  III.2 Simulation Initial Parameters

The scan is performed for a 6-turn ERL with 3 accelerating and 3 decelerating
turns. To start with we used similar simulation parameters to those used in
Ref TennantDissertation2006  as summarized in Table. 1. The transitioning
bunch is that moving from accelerating to decelerating phase. The transition
is achieved by delaying the bunch by least half an RF cycle. Therefore, the
revolution period of the transitioning bunch is half an RF cycle longer. The
injected bunches are 7.3 MeV and they gain/lose 39 MeV in each in
accelerating/decelerating turn. We simulated two injection frequencies, which
are of $\frac{1}{5}^{\text{th}}$ and $\frac{1}{10}^{\text{th}}$ of fundamental
mode frequency $f$, i.e. bunches are injected in every 5 and 10 RF cycles
$T_{RF}$, respectively. The horizontal injection offsets are set 10 times of
the vertical as bunches are bent in the horizontal plane and would be
dominated by horizontal jitters. Here we don’t need to be the exact on the
initial parameters like HOM voltages and offset jitters as we are looking at
the trend (growth or decline) to decide whether the test current is above or
below $I_{th}$.

###  III.3 Frequency dependence

The results show the threshold current is frequency dependent. Firstly, the
simulation results show the threshold current is periodic over HOM frequency
range of 1.24 MHz as shown in the Fig. 6. The blue, red, and green colors each
indicate 1 period of the curve. Such periodicity is expected from the general
form the threshold current equation TennantDissertation2006

$I_{th}=\frac{-2E_{beam}}{e\frac{\omega_{H}}{c}\left(\frac{R}{Q}\right)_{HOM}R_{12}sin(\omega_{H}T_{rev})}$
(26)

where $E_{beam}$ is the beam energy in electron Volt, $e$ is electron charge,
$c$ is speed of light in vacuum, $\omega_{H}$ is the HOM angular frequency,
$\left(\frac{R}{Q}\right)_{HOM}$ is geometric shunt impedance of the HOM,
$R_{12}$ is the transfer matrix element relates the angular kick to the off-
set after one recirculation and $T_{rev}$ is the revolution period. Note that
this equation only applicable for estimating threshold current for simple
2-turn ERL. Nevertheless, we can use it to explain periodicity. The period
arises from periodicity of the term $sin(\omega_{H}t_{tr})$. Given the
revolution period is 802 ns, the period $P_{f,HOM}$ is
${1}/{T_{rev}}=1.24~{}\text{MHz}$. Secondly, we see that the periodicity is
not exact but the the threshold current slowly decreases as the HOM frequency
increases. Such relation can be also seen from the Eq. 26.

Figure 6: Threshold current periodicity over HOM frequency. Different period
colored differently. The filling pattern used is {1 2 3 4 5 6} and the phase
advance $\mu=1.5\pi$.

###  III.4 Filling Pattern Dependence

We performed ERL BBU tracking simulations and estimated threshold current for
different sequence preserving filling patterns, where the cavity will always
see same sequence of bunches passing through. The scan is over frequency range
of 2.1054 and 2.1066 GHz, which covers one full aforementioned
$P_{f,HOM}=1.24$ MHz period. As the ERL is 6 turn, 6 bunches form a packet.
Such packets are repeated multiple times to fill up the circular accelerator
ring.

The scan results for $5T_{RF}$ spacing are as shown in the Fig. 7. The
threshold currents as function of HOM frequencies are given for different
patterns. As can be seen, each pattern indeed drives BBU instabilities
differently and have different threshold currents for same frequency. We do
see similar patterns have close $I_{th}$ at some frequencies, as can be seen
from the sub-figure (a) for patterns No. 1$-$3\. In the case of sub-figure
(b), however, we see patterns No. 59 ($\\{1~{}4~{}3~{}6~{}2~{}5\\}$), 60
($\\{1~{}4~{}3~{}6~{}5~{}2\\}$), and 61 ($\\{1~{}4~{}5~{}2~{}3~{}6\\}$) have
mostly different $I_{th}$.

(a)0.2in0.5in
---
(b)0.2in0.5in
Figure 7: Filling pattern dependence of the threshold current: (a) filling
pattern No. 1$-$3; (b) filling pattern No. 59$-$61\. The bunch spacing is
$5T_{RF}$.

###  III.5 Averaged threshold current

We performed threshold current scans for the 120 SP patterns. The simulated
frequency range is 1 period, $P_{f,HOM}$, for each pattern. The averaged
threshold current over for each pattern is shown in Fig. 8 for the case phase
advance $\mu$ is 1.5 $\pi$. As can be seen, pattern number 12 has the highest
threshold current at 2 A, more than 3 times the lowest threshold current.
Pattern numbers 10, 12, 19, 23, 33, and 36 have high threshold currents above
1.6 A for both $10T_{RF}$ and $5T_{RF}$ spacing cases. Clearly, some patterns
are better than others in the terms of lowering BBU instabilities. The highest
threshold current is 2.028 Amps when filling pattern No. 12 is used. Overall,
the $5T_{RF}$ bunch spacings have larger threshold currents as can be seen
from Fig. 8 and Tab 2. The maximum, minimum, and average of the $5T_{RF}$
bunch spacing are higher than of the $10T_{RF}$.

The relationship between threshold current and filling patterns is complicated
as several parameters (like bunch arrival times, energies, and orders) change
simultaneously when filling pattern is changed, consequently the average
threshold currents appear to be random for different filling patterns. The
analytical model fails to predict the threshold current of such a complicated
system and one has to rely on numerical simulation tools only.

Figure 8: Frequency averaged threshold current 120 SP patterns. The red and blue curves are the results with $5T_{RF}$ and $10T_{RF}$ bunch spacings, respectively. Table 2: Maximum, minimum, and average threshold currents for $5T_{RF}$ and $10T_{RF}$ bunch spacings. Threshold currents | Pattern number | Values (mA)
---|---|---
$I_{th,~{}5T_{RF},~{}max}$ | 12 | 2028
$I_{th,~{}5T_{RF},~{}min}$ | 108 | 727
$I_{th,~{}5T_{RF},~{}ave}$ | — | 1102
$I_{th,~{}10T_{RF},~{}max}$ | 36 | 1914
$I_{th,~{}10T_{RF},~{}min}$ | 112 | 544
$I_{th,~{}10T_{RF},~{}ave}$ | — | 1053

Another observation is that the spacing of the bunches has a big effect on the
threshold currents for some patterns but not others. In pattern number 12 for
example, both $5T_{RF}$ and $10T_{RF}$ spacings have similar threshold
current, while pattern number 43 for example does not. We also compared
frequency scans of these two patterns, as shown in Fig. 9.

(a)0.2in2.52in
---
(b)0.2in2.52in
Figure 9: Threshold currents as function of HOM frequency pattern number 12
and 43 at two bunch spacings. In (a) they similar and in (b) different
threshold currents.

###  III.6 Phase advance dependence of threshold current

A scan of threshold currents with the two different phase advances of 1.5 and
2 $\pi$ with $10T_{RF}$ bunch spacing is shown in Fig. 10. When the phase
advance is integer times $\pi$, bunches enter the cavity with minimum offset
and hence the BBU instability is minimized. Consequently, the threshold
currents are increased greatly. We see when the tune changed, the dependence
of the threshold current on the patterns have changed as well.

Figure 10: Average threshold current 120 SP patterns with different phase
advances with $10T_{RF}$ bunch spacing. Threshold currents are higher when
phase advances are integer times of $\pi$

The BBU threshold current is dependent on many parameters that interact in a
complicated manner, making it difficult to choose the best pattern with the
highest threshold current using the analytical approach. We have however
demonstrated that one can find the best filling pattern for a given set of
parameters by performing pattern scans with numerical simulations.

##  IV Conclusion

We have investigated the impact of filling pattern choice in a multi-pass ERL
on the regenerative BBU instability. We find analytically that the mean
threshold current decreases with increasing ERL turn number, placing
constraints on the feasibility of designing arbitrarily many turns. In
Hoffstaetter2004  states a scaling rule of $I_{th}$ with the number of turns
as $1/\left(N\left(2N-1\right)\right)$, which differs slightly from our own
form given as $\alpha$ and $\alpha^{\prime}$ in Eq. 17 and 18, which provides
a more generalised expression for this scaling law, although it is also clear
that the scaling law will depend on the recirculation scheme. We demonstrated
filling pattern dependence of the regenerative BBU instability threshold
current. For example, we observed a factor of 3 difference in the threshold
current between worst and best patterns for SP schemes. The threshold current
and best filling pattern depend on many parameters interacting in a complex
manner. We constructed a new ERL tracking code to numerically scan for the
best filling pattern to maximise the threshold current. This tool will assist
in the design of future ERL projects.

##  V ACKNOWLEDGEMENTS

The authors would like to thank Dr. Graeme Burt for his useful suggestions and
insights. The studies presented have been funded by STFC Grants No.
ST/P002056/1 under the Cockcroft Institute Core Grant.

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Unveiling Scaling Laws in the Regulatory Functions of Reddit

Shambhobi Bhattacharya1,2, Jisung Yoon2,3,4, and Hyejin Youn2,4,5*

1 McCormick School of Engineering and Applied Science, Northwestern
University, Evanston, IL, USA

2 Northwestern Institute on Complex Systems, Evanston, IL, USA

3 KDI School of Public Policy and Management, Sejong, Republic of Korea

4 Kellogg School of Management, Northwestern University, Evanston, IL, USA

5 Santa Fe Institute, Santa Fe, NM, USA

*<EMAIL_ADDRESS>

## Abstract

Online platforms like Reddit, Wikipedia, and Facebook are integral to modern
life, enabling content creation and sharing through posts, comments, and
discussions. Despite their virtual and often anonymous nature, these platforms
need rules and oversight to maintain a safe and productive environment. As
these communities grow, a key question arises: how does the need for
regulatory functions scale? Do larger groups require more regulatory actions
and oversight per person, or can they manage with less? Our analysis of
Reddit’s regulatory functions reveals robust scaling relationships across
different subreddits, suggesting universal patterns between community size and
the amount of regulation needed. We found that the number of comments and
moderator actions, such as comment removals, grew faster than the community
size, with superlinear exponents of 1.12 and 1.18, respectively. However, bot-
based rule enforcement did not keep pace with community growth, exhibiting a
slightly sublinear exponent of 0.95. Further analysis of the residuals from
these scaling behaviors identified a ’trade-off axis,’ where one-way
coordination mechanisms (bots and moderators) counteract two-way interactions
(comments) and vice versa. Our findings suggest that a more proactive
moderation approach, characterized by increased bot activity and moderator
comment removals, tends to result in less user engagement under the scaling
framework. Understanding these natural scaling patterns and interactions can
help platform administrators and policymakers foster healthy online
communities while mitigating harmful behaviors such as harassment, doxxing,
and misinformation. Without proper regulation, these negative behaviors can
proliferate and cause significant damage. Targeted interventions based on
these insights are key to ensuring online platforms remain safe and beneficial
spaces.

## Author Summary

We examine the regulatory functions within Reddit’s communities. Instead of
focusing on individual regulatory functions, as previous literature has done,
we examine how regulation scales and the interplay between different
regulatory functions. Our scaling analysis reveals distinct patterns: user
interactions and moderator actions grow at a superlinear rate (exponents
around 1.15), while bot-driven rule enforcement grows at a slightly sublinear
rate (0.95). Through Principal Component Analysis, we identify a ’trade-off
axis’ that uncovers a key insight: as moderation efforts increase, users’
engagement tends to decrease.

This presents a challenge for a moderator, who must strike a balance between
enforcing rules and policies to maintain the system while also fostering users
participation and involvement, and optimize the health and growth of online
communities. Our scaling analysis offers a powerful quantitative method for
comparing governance models across different platforms, enabling
administrators and policymakers to pinpoint areas for improvement and refine
their regulatory frameworks. Finally, our study lays the groundwork for more
detailed analyses of Reddit’s regulatory mechanisms in future research.

## Introduction

Online communities, such as Reddit, Wikipedia, and Facebook, play a
significant role in shaping how people access information [1, 2, 3], socialize
[4, 5], and even collective behavior in real world [6, 7]. Unlike traditional
communities, these digital platforms offer fully observable communication and
unique insights into the dynamic behaviors of users. These platforms, much
like real-world communities, necessitate regulatory functions to prevent
harmful activities, including online harassment [8], doxing [9], and the
spread of misinformation [10, 11]. Traditionally, regulation in online
communities has been widely studied in the field of the human-computer
interaction focusing on categorizing and understanding the multi- level
platform governance of these platforms, developing frameworks for community
values that constitute effective regulation [12, 13], and highlighting the
importance of robust regulatory frameworks [14].

Among the diverse online communities, we focus on Reddit due to its unique
structure and transparency, which offer a rich dataset for examining
regulatory dynamics [15]. Reddit consists of thousands of smaller communities
called ”subreddits,” each created and moderated by regular users. These
subreddits operate independently with their own regulatory systems, managed by
volunteer moderators, facilitating a decentralized approach to regulation [16,
17, 18]. The platform’s extensive digital trace allows for detailed tracking
of regulatory actions and outcomes, providing a comprehensive view of the
community’s regulatory landscape. Furthermore, Reddit’s transparency and open
data policies enable researchers to access detailed information about user
interactions [19], moderation activities [20], and rule enforcement [21],
making it an invaluable resource for studying the complexities of online
regulation.

Especially, Reddit is particularly interesting to study because it shows a
natural transition from a decentralized, free-spirited community [22] to a
semi-autonomous platform with structured governance. Initially, Reddit
operated with minimal universal rules, allowing moderators to create
subreddits like r/WallStreetBets [23] and r/Politics [24], each establishing
its own guidelines in a decentralized manner. However, as Reddit expanded,
managing its vast and diverse community necessitated more structured, site-
wide policies and automated tools like Automoderator. For example, in 2015,
Reddit closed several subreddits, including r/fatpeoplehate and r/CoonTown,
due to extreme anti-harassment activity [25]. Similarly, the rise of “toxic”
subreddits targeting hate speech prompted Reddit to introduce a new anti-
harassment policy [26] and ban several subreddits [27]. Thus, Reddit’s
regulatory functions illustrate how regulation can naturally emerge out of
necessity in a bottom-up manner, making it a compelling case study for
understanding regulation in online community.

Previous studies on regulatory mechanism in online communities have typically
examined the platform from functional perspectives [28, 29, 30] or focused on
specific events [6, 7], rather than providing a comprehensive view of its
overall regulation and coordination. Therefore, our study aims to fill this
gap by offering an in-depth analysis of the regulatory mechanisms in Reddit,
one of the fastest-growing online communities in the world, and showing how
these mechanisms systematically scale with the platform’s growth and
complexity. For this purpose, we employ scaling analysis, which is a powerful
approach for revealing underlying dynamics and structures in various systems.
This method has been used successfully to study systems like urban cities [31,
32], biological systems [33, 34] and ecological systems [35], online platforms
like Wikipedia [36]. Inspired by these studies, we apply scaling analysis to
Reddit to understand the underlying dynamics of its regulatory functions. By
examining the scaling behavior of regulatory functions, we aim to characterize
and quantify the regulation costs associated with different community sizes.
This scaling can be represented with the exponent $\beta$, in the relationship
$Y\propto N^{\beta}$, where $N$ is the number of contributors in a subreddit,
and $Y$ denotes the regulatory actions. The exponent $\beta$ quantifies the
rate of increase in $Y$ concerning relative increases in $N$, indicating
whether this increase is greater ($\beta>1$) or less ($\beta<1$) than expected
in a linear scenario.

Our analysis reveals several key findings regarding the regulatory functions
on Reddit. Firstly, the scaling framework on Reddit is consistent. Moderator
actions scale superlinearly with community size, with an exponent of 1.18,
indicating that as a community grows, the complexity and frequency of
moderator actions increase disproportionately. Similarly, mutual interactions
among users also exhibit a superlinear scaling pattern. Secondly, our analysis
supports previous qualitative studies on user reactions to moderation, but on
a much larger scale. We systematically demonstrate that increased moderation
leads to decreased user activity across all community sizes, providing more
robust and generalizable conclusions. This large-scale analysis sets our study
apart from previous work, such as Jhaver’s [37], by showing that the trade-off
between moderation and user activity exists regardless of community size.
Thirdly, our scaling framework helps compare different systems, such as cities
and online communities, providing deeper insights into the mechanisms at play.
This framework is more powerful than previous methods as it accounts for
variations in community size, allowing for a more comprehensive understanding
of the dynamics in different regulatory environments.

## Results

### Regulatory Functions in Reddit

Reddit, known as the “front page of the internet”, is a dynamic platform where
users join specialized communities called subreddits. These subreddits cover a
wide range of interests and topics, where members share posts, ask questions,
and participate in discussions, contributing to a shared knowledge base.
Regulation is essential to uphold the quality and standards of each community,
elevating valuable content while enforcing rules to maintain order. We
categorize Reddit’s regulatory activities into three distinct types: Mutual
Interactions, Supervision, and Rule Enforcement, as shown in Figure 1.

To understand these activities better, we need to explain the setting of
Reddit. Each subreddit operates with its unique focus, appearance, and rule
set that governs on-topic posts and expected user behavior. Volunteer
moderators, either founding members or those recruited to help, enforce these
guidelines. They are crucial in managing community dynamics by removing posts
and comments that violate subreddit-specific rules or Reddit’s overarching
content policy. Originally, Reddit was established with minimal regulation,
but the landscape gradually evolved as incidents like doxxing underscored the
necessity of scalable and emergent moderation.

To assist in scalable moderation, moderation bots (mod bots) were introduced
to automate tasks like removing posts, limiting rates, and preventing spam.
Automoderator, a popular mod bot, is widely used across subreddits to ensure
rule compliance. Despite the challenges that come with managing increasingly
complex regulatory functions, both human moderators and bots continue to
adapt, ensuring that Reddit remains a well-regulated and thriving community.

Effective regulation prevents subreddits from being banned or quarantined,
helping maintain well-coordinated and functional communities. In our
regulatory framework, we examine three functions: Mutual Interactions,
Supervision, and Rule Enforcement. These actions stem from two-way
coordination and one-way coordination. Two-way coordination is characterized
by mutual interactions without inherent asymmetry between two or more parties.
In contrast, one-way coordination involves predominantly unilateral processes
due to an inherent power differential or asymmetry between individuals, such
as supervision by human moderators or rule enforcement by bots.

To quantify Mutual Interactions, we measure the number of comments per post,
reflecting the level of engagement within discussions. Cumulative comments
represent a deliberative process, where users contribute and respond to each
other, fostering a consensus. Supervision refers to the actions of moderators
who possess distinct authority and influence. We focus on comment removals, a
critical metric illustrating how moderators maintain community identity and
decorum. Rule Enforcement is carried out by moderator bots, primarily through
automated comments related to moderation activities.

Our analysis explores the interplay between Mutual Interactions, Supervision,
and Rule Enforcement, revealing how human and automated mechanisms collaborate
to govern Reddit. Figure 1 provides an example of these regulatory functions
within the subreddit r/News, illustrating how this categorization is
practically applied.

Fig 1: Regulatory Functions in Reddit In Reddit, subreddits provide spaces for
discussions on shared topics. (a) In the subreddit r/news, users discuss
Biden’s election as the US president, where the engagement is quantified
through the number of comments. (b) Bots ensure adherence to rules through
automated comments. (c) Moderators enforce guidelines by removing comments
that violate rules. (d) Summary of regulatory functions and associated
metrics.

### Scaling Laws in Regulatory Functions

In our examination of regulation and moderation mechanisms, we delve into the
concept of the scaling factor [38], a pivotal determinant shaping the growth
associated with such mechanisms. While it’s established that these mechanisms
consistently scale with organizational size across various systems—from
biological systems [34] to urban systems [31] to Wikipedia [36]—the crucial
inquiry revolves around the rate of this growth. Understanding this rate helps
quantify the regulatory landscape and provides context for its dynamics,
allowing for comparisons and building intuition. Our aim is to discern whether
our regulatory functions escalate at a pace surpassing or lagging behind the
organization’s size, thereby aligning with our proposed framework.

To investigate the relationship between our functions and organizational size,
we adopt a scaling perspective, employing subreddits as organizations of
varying sizes in our case (See Methods). Each subreddit’s unique Redditors are
considered a functioning organization, with the number of unique active users
denoted as N serving as the measure of the community size or the subreddit
size.

$\displaystyle Y=Y_{0}\cdot N^{\beta}$ (1)

Here, the variable $Y$ denotes the regulatory actions, and $Y_{0}$ serves as a
normalization constant. The exponent $\beta$ quantifies the rate of increase
in $Y$ concerning relative increases in $N$, indicating whether this increase
is greater ($\beta$ $>$ 1) or less ($\beta$ $<$ 1) than expected in a linear
scenario. This approach allows us to systematically analyze and comprehend the
scaling factors affecting regulatory actions in online communities of
different sizes.

We use Pushshift data to carry out our analysis. Refer to data sources for
more info on this archive dataset of posts and comments on Reddit. Our
empirical analysis initially focused on subreddits with at least 100 active
unique users from 2010 to 2020, considering variables like the number of
subreddits, unique active users, comments, removed comments, posts, and
removed posts. Plotting growth curves and analyzing descriptive statistics, we
observed consistent upward trends in subreddits, users, posts, and comments,
indicating the growth of Reddit over the years. Notably, moderation and bot
actions significantly increased in 2015. Consequently, we focused on post-2015
data for our final scaling analysis to understand Reddit’s stabilized
regulation system (Refer to S3 in SI for more details).

Our empirical analysis spans from 2015 to 2020. We utilized a dataset that
includes 2,828 labeled subreddits across 88 categories from
r/ListofSubreddits. We sourced our data from the Reddit Pushshift archive. The
choice of this timeframe is validated by the observed stability in regulatory
practices post-2015, which is crucial for ensuring the robustness of our
findings. A parallel analysis of all active subreddits, totaling 107,006
subreddits from 2015 to 2020, confirmed the consistency of our results,
further attesting to the reliability of our research timeframe and subreddit
sampling.

Fig 2: Scaling Behavior of Regulatory Functions in Reddit (a) Descriptive
statistics of our sampled Subreddits (b) Mutual interactions (# of comments),
(c) Supervision (# of removed comments), and (d) Rule Enforcement (# of bot
comments) exhibit scaling behavior with unique users. Each subreddit is log-
binned (error bars in blue), and the $\beta$ in $N^{\beta}$ is estimated,
depicted by the orange line. The gray line shows a linear trend for reference.
Our findings indicate that mutual interaction and supervision demonstrate
superlinear scaling, growing faster than the number of users, while rule
enforcement exhibits slightly sublinear, very close to linear scaling,
increasing proportionally to user numbers.

Scaling Analysis Results and Intuition:

Figure 2 shows remarkably consistent scaling laws of regulatory functions in
Reddit. Notably, all three plots consistently form a straight line (orange)
across seven orders of magnitude, suggesting a unified power law equation for
Reddit’s regulatory functions. Despite this, the exponents governing the power
law vary across regulatory measures. Figure 2 indicates that mutual
interaction on Reddit, measured by the number of comments, exhibits
superlinear scaling (comment = 1.12). This finding suggests that Reddit
discussions and disagreements escalate at a rate surpassing user growth,
indicative of a descaled economy of scale. Such patterns in mutual interaction
align with the coordination costs observed in societal interactions,
reminiscent of urban features displaying superlinear scaling. The supervision
aspect on Reddit, as reflected in the number of removed comments and posts,
reveals intriguing scaling patterns. The removed comments exhibit superlinear
scaling (mod.act. = 1.18), indicating a rapid increase that outpaces user
growth, pointing to a diseconomy of scale.

In our discussion, we explore the implications of these results. We compare
the exponents with those from Wikipedia to understand why Reddit requires more
moderation per person. Why does Reddit overall need more moderation than
Wikipedia per person? We wanted to get more meaning out of understanding these
exponents. For example, given that our exponent for the number of removed
comments is 1.18, why do larger subreddits need more removals per person than
smaller ones? Additionally, while supervision (comment removals) shows
superlinear scaling, rule enforcement (bot comments) is close to linear. There
must be some interplay between these regulatory functions, and they don’t seem
to replace each other. All of these questions probe the relationship between
these regulatory functions and what might be the driving or influencing forces
behind their interaction.

### Interplay within Regulatory Functions

Residual Analysis

To provide insight into the factors influencing regulatory costs beyond the
average behavior predicted by the scaling framework (the orange line), we
quantify the residuals of each regulatory function [39]. The residual is the
difference between the observed and predicted values, denoted as

$\xi_{i,q}\equiv\log\frac{Y_{i,q}}{Y_{q}(N_{i})}=\log\frac{Y_{i,q}}{Y_{0,q}N_{i}^{\beta_{q}}},$
(2)

where $i$ indicates each subreddit, and $q$ indicates each regulatory
function.

Table S3 shows the correlation among the residuals, illustrating the
relationships between the regulatory functions after removing the average
behavior of each function. The positive correlation among all activities
suggests that different methods of coordination do not seamlessly replace each
other, contrary to previous assumptions. This raises several questions: What
is the underlying reason for this overall positive correlation? Does an
increase in interaction always lead to an increase in rule enforcement by bots
and supervision by moderators? Is this pattern consistent across all types of
subreddits?

Principal Component Analysis on the Residuals To delve deeper into these
questions, we employ Principal Component Analysis (PCA) to investigate
potential overlooked confounding factors. PCA allows us to reduce the
dimensionality of the data by identifying a few linear combinations of the
original variables that explain most of the variance (See Methods). This
approach helps us break down the regulatory functions into two principal axes,
providing a clearer understanding of their interplay and identifying the main
driving forces behind regulation costs on Reddit.

Fig 3: Principal Component Analysis of Subreddit Categories PC1 represents the
”Controversy Axis,” and PC2 signifies the ”Trade-off Axis,” exploring a trade-
off among regulatory functions. Grey dots denote average residual values for
each subreddit category, with bolded categories identified. Arrows illustrate
variable loadings on the principal components, with positive loadings on PC1
and a mix of positive and negative loadings on PC2, indicating an interplay
between functions.

Trade-off axis Our second finding uncovers a trade-off between the regulatory
functions, as moderation efforts increase, users’ engagement tends to
decrease. Figure 3 shows the resulting two primary orthogonal principle axes
with the arrows in the plot indicating the loading of each regulatory
function, $q$. In PCA analysis, loadings are the correlation between the
original variables and the principal components. We label the first axis
“Controversy Axis”. Contentious topics like News/Politics or Sports are
positioned on the right side of the figure $(PC1>0)$, while less contentious
topics like Gardening and Art tend to be on the left side $(PC1<0)$. An
increase in PC1 values correlates with heightened contention, resulting in
increased subreddit discussions, more comment removal by moderators, and an
uptick in bot comments. Conversely, PC2 reveals a trade-off mechanism, with
positive loadings for moderator and bot activity (red and yellow arrows) and
negative loadings (green arrow) for discussion, termed the “Trade-off Axis”.
PC2 suggests that a stricter moderation approach, marked by increased bot
activity and moderator comment removals, tends to result in fewer user
comments. This finding is consistent with earlier research by Jhaver et al.
(2019) [30]. Jhaver et al. mention that with the increase in Automoderator
adoption, there has been an increase in comment removals, which has led to
less engagement from users post-removal. Their qualitative research also
indicates user frustration with these removals. Our framework, by illustrating
the dynamic interactions between regulatory aspects on Reddit, highlights the
need for policy intervention and further research to balance effective
regulation with user engagement. As Reddit grows, it is crucial to maintain
robust regulatory mechanisms while educating users to mitigate frustration and
increase engagement. This comprehensive understanding can guide future
regulatory strategies.

## Discussion

In this discussion, we explore Reddit’s regulatory functions through a scaling
framework to understand the unique dynamics of managing large online
communities and the potential for broader application of these methodologies
to other platforms.

Scaling laws provide a powerful framework for understanding these dynamics.
Philosophically, scaling laws suggest that size becomes an independent driving
factor rather than a controlling one. This perspective helps us see how
properties within a system disproportionately increase or decrease with its
size, such as headcount becoming an independent factor rather than a control
factor. For example, in urban systems, communication costs per person can be
analyzed similarly to interactions within a subreddit, offering insights into
why behaviors differ across contexts like Slack, Reddit, or a city like
Chicago. As a tool, scaling laws offer quantifiable measures across different
sizes, allowing for side-by-side comparisons of varied systems. This approach
helps us understand which properties disproportionately increase or decrease
with system size. By controlling for the size effect, we gain insights into
how and why certain dynamics change as systems grow. This aggregated
perspective enables comparisons of communication costs in a city versus a
subreddit, helping us understand why user behaviors might differ between
platforms like Slack and Reddit.

Our analysis reveals unique scaling patterns that correlate with the size and
activity of individual subreddits. For instance, mutual interactions on Reddit
follow a superlinear scaling pattern with an exponent of 1.12, indicating that
the volume of interactions grows disproportionately as subreddit membership
increases. This pattern contrasts with Wikipedia’s stronger superlinear
scaling exponent of 1.3 [36], reflecting Wikipedia’s well-established norms
and structures guiding contributor behavior. Reddit’s high degree of task
uncertainty, anonymity, and fluid user base differentiate its scaling patterns
from those seen in Wikipedia’s more structured environment. Wikipedia’s mutual
adjustment scaling (1.3) is higher than Reddit’s, suggesting stronger
compliance with rules and guidelines due to a clear editorial framework. On
Reddit, the elevated growth rate of moderator actions (1.18) compared to
administrative actions (0.91) and bot actions (0.69) on Wikipedia reflects the
increasing complexity faced by Reddit’s moderators in regulating community
norms as subreddits grow. This complexity is exacerbated by Reddit’s
transient, often temporary accounts and the lack of persistent user histories,
leading to unpredictable regulatory challenges. The transient nature of Reddit
interactions, characterized by anonymity and immediate, ephemeral exchanges,
makes community regulation uniquely challenging. Unlike Wikipedia, which
fosters long-term engagement and encourages the development of expertise,
Reddit prioritizes conversation over structured learning. As a result, users
may not internalize community norms effectively, contributing to a lower
overall rate of rule adherence. This necessitates an increasing reliance on
moderators, whose actions scale superlinearly due to the nature of the
platform’s rapid growth.

While our findings offer significant insights, our analysis is constrained by
the limitations of available data. Specifically, the Pushshift dataset
provides only net votes (upvotes minus downvotes), hindering our ability to
analyze individual voting behavior and its impact on content regulation.
Future work should address this gap by employing datasets that capture a more
granular voting history. Further research could delve deeper into how
comments, as an indicator of mutual adjustment, can more robustly predict
regulatory behaviors and challenges. Additionally, longitudinal studies
tracking changes in user behavior and the impact of regulatory shifts over
time will shed light on how scaling laws affect community management
strategies.

This structured approach aims to guide readers through our findings
comprehensively, enhancing understanding and fostering further exploration in
the field of online community management. By refining governance approaches,
we can offer a clearer understanding of how to balance user freedom with
effective moderation in rapidly growing online communities.

## Conclusions

Our study of regulatory scaling functions on Reddit provides important
insights into the distinctive dynamics of online community management. We
found that mutual interaction and moderator actions follow superlinear scaling
patterns, with mutual interaction scaling less dramatically than Wikipedia due
to the higher task uncertainty and fluid user base on Reddit. These
differences underscore the challenges faced by moderators in managing Reddit’s
dynamic and often ephemeral environment.

The comparison with Wikipedia highlights the influence of structured editorial
norms and persistent user engagement on regulatory behaviors. Reddit’s unique
design, prioritizing conversation over structured learning, results in a lower
rate of rule adherence, necessitating superlinear growth in moderator actions.

Despite the limitations of available data, our analysis opens avenues for
future research into the role of comments and voting data as mutual adjustment
indicators. Longitudinal studies and causal analyses will further deepen our
understanding of the complexities of managing diverse and rapidly growing
online communities. These insights can guide the development of effective
governance strategies that balance user engagement with regulatory efficacy.

## Materials and Methods

### Data Sources

We utilized publicly accessible archived data from Reddit, specifically
drawing from the Pushshift dataset covering the years 2015 to 2020 [40]. The
rationale behind selecting this timeframe stems from our analysis and
observations of announcements directed at moderators on Reddit. In 2015, there
was a notable shift where moderators were granted enhanced administrative
rights, allowing them to exercise more control over their subreddits,
including the establishment of specific rules tailored to each subreddit’s
unique context.

This timeframe was chosen deliberately to capture a holistic representation of
Reddit activity, considering the pivotal changes in administrative dynamics
that occurred in 2015. By doing so, our dataset spans a significant period,
enabling a comprehensive examination of Reddit’s evolving landscape and
dynamics.

The Pushshift Reddit dataset is bifurcated into two distinct sets of files,
one dedicated to submissions and the other to comments. The submissions
dataset comprises monthly newline-delimited JSON files, with each line
representing a submission in the form of a JSON object. Similarly, the
comments dataset is organized into monthly ndjson files, with each line
corresponding to a comment presented as a JSON object. Both submissions and
comments are uniquely identified through an identifier (id) and encompass a
range of information, including author details, posting time, subreddit name,
subreddit id, the content of the submission or comment, reactions to the
content to name a few.

Of paramount importance to our study is the exploration of aggregated
coordination costs at the subreddit level. Consequently, we concentrated our
analysis by grouping the information from comments and submissions at the
subreddit level. This strategic approach facilitated a focused examination of
coordination dynamics within the targeted subreddit communities.

### Special Subreddits Dataset

For this analysis, we used a curated dataset of 2,828 subreddits categorized
into 88 distinct groups, such as Science, Politics, and Images, obtained from
the r/ListofSubreddits community [41]. The detailed list is available in the
Supplementary materials (refer to S5 in SI). This dataset was sourced from the
collaborative efforts of Reddit users who categorize and organize subreddits,
accessible at r/ListofSubreddits.

The labeled subreddits were essential for our Principal Component Analysis
(PCA) as they provided a structured framework to identify and interpret PCA
axes, especially for distinguishing potentially contentious subreddits. This
categorization enabled a more intuitive analysis of subreddit governance
structures and regulatory patterns [42]. Although the Pushshift Reddit dataset
includes all subreddit data, we filtered out only the labeled 2,828 subreddits
for our results and figures in this paper. The Supplementary materials (refer
to S4 in SI) include results from an analysis of all Reddit subreddits without
labeling, which showed consistent scaling patterns, indicating that our
conclusions are broadly applicable across Reddit.

### Scaling Analysis

Reddit’s varied subreddit sizes, ranging from small communities to those with
millions of users, offer a unique perspective on how regulation changes with
size, community interests, and behaviors. As subreddits evolve, their
regulatory complexities change, leading to new behaviors. This diversity in
size and type, along with vast longitudinal data [40], makes Reddit an
excellent platform for studying regulation. Concepts such as productivity and
crime rate in urban systems [31], as well as metabolic rate in biological
systems [34], have been studied from a systematic, aggregated perspective
using scaling analysis [38]. We aim to extend this scaling framework to online
platforms to capture and comprehend these emergent regulatory behaviors by
examining aggregated regulatory actions. This approach allows us to
empirically examine how regulatory functions scale and interact, enhancing our
understanding of regulation in online communities.

In our exploration of how regulatory actions evolve with the number of
contributors, we conceptualized each subreddit as an operational entity within
the Reddit platform. Our analytical approach involved a comparative study of
subreddits of various sizes, employing a scaling perspective to unravel
regulation requirements.

To gauge the organizational size within each subreddit, we utilized the count
of active unique authors (N). For an in-depth examination of scaling dynamics,
we employed a power-law scaling model:

$\displaystyle Y=Y_{0}\cdot N^{\beta}$ (3)

Here, the variable $Y$ denotes regulation costs, and $Y_{0}$ serves as a
normalization constant. The exponent $\beta$ quantifies the rate of increase
in $Y$ concerning relative increases in $N$, indicating whether this increase
is greater ($\beta$ $>$ 1) or less ($\beta$ $<$ 1) than expected in a linear
scenario.

Our data exhibits a distinct characteristic: the majority of subreddits
feature a relatively small contributor base, typically around 100.
Nevertheless, a few exceptional projects attract a significantly larger
contributor base. If we were to utilize ordinary least squares (OLS)
regression for model fitting, the resulting model would naturally prioritize
explaining the majority of the data. However, our research objective centers
on understanding the growth patterns of subreddits rather than exclusively
modeling the predominant data.

To address this, we adopted a specific strategy for fitting the data into the
scaling law, tailored to the nature of our dataset. Logarithmic binning of the
data preceded model fitting, ensuring that every scale, from small to large
subreddits, received equal consideration in our analysis. This approach allows
us to capture growth patterns across the entire spectrum of subreddit sizes,
providing a comprehensive understanding of how coordination costs scale within
online communities.

### Residual Analysis

The concept of residuals is integral to our understanding of the discrepancies
between predicted and observed regulatory costs in our dataset. Residuals,
denoted as $\xi_{i}$, are calculated using the formula:

$\xi_{i}=\log\left(\frac{Y_{i}}{Y(N_{i})}\right)=\log\left(\frac{Y_{i}}{Y_{0}N_{i}^{\beta}}\right),$

where $Y_{i}$ is the observed regulatory cost for each instance, and
$Y(N_{i})$ is the predicted regulatory cost based on the scaling law, with
$N_{i}$ representing the number of contributors. This calculation helps
identify deviations from the expected behavior defined by the model, offering
insights into factors that influence coordination costs beyond the general
trends modeled by the scaling relationship. We analyze these residuals to
explore the underlying dynamics not captured by the average model behavior.
Tables showing the Pearson correlation coefficients among the observed
regulatory functions and another for the residuals (Table S2 and Table S3,
respectively) provide a quantitative measure of how strongly these costs are
related to the number of contributors and how residuals interrelate once the
mean behavior is adjusted for.

### Principal Component Analysis

Principal Component Analysis (PCA) [43, 44] is a statistical technique used to
reduce the dimensionality of large datasets, enhancing interpretability while
minimizing information loss. By transforming the data into a new coordinate
system, PCA ensures that the greatest variance by any projection lies along
the first principal component, with each subsequent component capturing
progressively less variance.

In this study, we apply PCA to the three-dimensional vector of residuals
$[\xi_{\text{mutual\\_interaction}},\xi_{\text{supervision}},\xi_{\text{bot\\_enforcement}}]$
from our analysis of Reddit’s regulatory functions to uncover significant
patterns and minimize redundancy. The analysis identifies two principal
components that capture the most substantial variation within the residuals:
Principal Component 1 (PC1), which explains 60% of the variance, and Principal
Component 2 (PC2), which explains 21% of the variance.

These components help us identify the key dimensions of variation in Reddit’s
regulatory activities, providing deeper insights into the complexities of
moderation and rule enforcement across different community segments.

## Supporting information

Link to Supporting Information

## Acknowledgments

The authors would like to acknowledge the support of the National Science
Foundation Grant Award Number 2133863. The authors thank Prerna Juneja for
sharing the Modlog Dataset, which, although not directly used in this
research, provided valuable insights into moderator actions. We also extend
our gratitude to our collaborators who provided suggestions and feedback
during this work.

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