# Exponential equivalence for misanthrope processes in contact with weak
reservoirs and applications to totally asymmetric exclusion processes

Julian Kern  0000-0002-8231-0736

###### Abstract

We provide a short proof for the exponential equivalence between misanthrope
processes in contact with weak reservoirs and those with impermeable
boundaries. As a consequence, we can derive both the hydrodynamic limit and
the large deviations of the totally asymmetric exclusion process (TASEP) in
contact with weak reservoirs. This extends a recent result which proved the
hydrodynamic behaviour of a vanishing viscocity approximation of the TASEP in
contact with weak reservoirs. Furthermore, applications to a class of
asymmetric exclusion processes with long jumps is discussed.

##  1 Introduction

The recent work [11] makes use of the method of compensated compactness to
investigate the hydrodynamic behaviour of the totally asymmetric exclusion
process (TASEP) in contact with weak boundaries. An immediate drawback of this
approach is that it can only be applied to a vanishing viscocity approximation
of the model. In [5], a variant of the model with long jumps from [9] in
contact with weak reservoirs has been proposed. In the regime of jumps with
infinite mean, the hydrodynamic limit has been derived therein, but the case
of jumps with finite mean remains open and is expected to be tightly linked to
the TASEP studied in [11],

In this paper, we provide a short proof of the exponential equivalence between
processes in contact with weak reservoirs and with impermeable boundaries
respectively for a large class of models. As an immediate consequence, we can
extend the result from [11] to the TASEP without vanishing viscocity and
derive the hydrodynamic behaviour of the model studied in [5].

The remainder of the paper is divided into three sections: the presentation of
the main result, its applications, and the proof. A generalization is
discussed in Appendix A.

#### Acknowledgement

This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through
grant CRC 1114 Scaling Cascades in Complex Systems, Project Number 235221301,
Project C02 Interface dynamics: Bridging stochastic and hydrodynamic
descriptions. I would like to thank Robert Patterson for valuable discussions.

##  2 Notation and main result

In this section, we will restrict ourselves to the special case of exclusion
processes on $\mathbb{Z}$. For the more general case, see Appendix A.

Write $\Lambda_{N}:=\\{1,\dots,N-1\\}$ for the bulk and
$\Omega_{N}:=\\{0,1\\}^{\Lambda_{N}}$ for the space of configurations. We
denote by $\eta^{x,y}$ the exchange of the sites $x$ and $y$, and by
$\eta^{x}$ the flip of the site $x$ in the sense that

$\eta_{z}^{x,y}:=\begin{cases}\eta_{y}&\text{ if }z=x\\\ \eta_{x}&\text{ if
}z=y\\\ \eta_{z}&\text{ otherwise }\end{cases}\qquad\text{ and
}\qquad\eta_{z}^{x}:=\begin{cases}1-\eta_{x}&\text{ if }z=x\\\ \eta_{z}&\text{
otherwise }\end{cases}.$

Next, define the generators

$\begin{array}[]{rl}\mathcal{L}_{bulk}^{N}f(\eta)&:=\displaystyle\sum_{x,y\in\Lambda_{N}}p(x,y)\cdot\eta_{x}(1-\eta_{y})\cdot\Big{(}f(\eta^{x,y})-f(\eta)\Big{)},\\\
\mathcal{L}_{influx}^{N}f(\eta)&:=\displaystyle\sum_{x\not\in\Lambda_{N}}\sum_{y\in\Lambda_{N}}p(x,y)\cdot\alpha(x)(1-\eta_{y})\cdot\Big{(}f(\eta^{x})-f(\eta)\Big{)},\\\
\mathcal{L}_{outflux}^{N}f(\eta)&:=\displaystyle\sum_{x\in\Lambda_{N}}\sum_{y\not\in\Lambda_{N}}p(x,y)\cdot\eta_{x}\beta(y)\cdot\Big{(}f(\eta^{x})-f(\eta)\Big{)},\end{array}$

where $p:\mathbb{Z}^{2}\rightarrow[0,+\infty)$ is a jump kernel and
$\alpha,\beta:\mathbb{Z}\rightarrow[0,+\infty)$ are bounded. Here and in the
following, we will use the shortcut $x\not\in\Lambda_{N}$ to mean
$x\in\mathbb{Z}\setminus\Lambda_{N}$.

######  Definition 2.1.

  1. 1.

An _exclusion process in contact with impermeable boundaries_ is an
$\Omega_{N}$-valued Markov process with generator $\mathcal{L}_{bulk}^{N}$.

  2. 2.

An _exclusion process in contact with weak reservoirs_ is an
$\Omega_{N}$-valued Markov process with generator
$\mathcal{L}^{N}:=\mathcal{L}_{bulk}^{N}+\theta(N)\left(\mathcal{L}_{influx}^{N}+\mathcal{L}_{outflux}^{N}\right)$
for some $\theta(N)=o(1)$.

In the following, we will identify $\Omega_{N}$-valued processes with the
corresponding measure-valued process via the map

$\pi^{N}:\Omega_{N}\rightarrow\mathcal{M}_{F}([0,1]),\qquad\eta\mapsto\dfrac{1}{N-1}\sum_{x\in\Lambda_{N}}\eta_{x}\delta_{x/N},$

where $\mathcal{M}_{F}([0,1])$ denotes the space of finite measures on
$[0,1]$. To compare the different processes, we endow the space of measure-
valued càdlàg processes
$\mathbb{D}_{[0,T]}\big{(}\mathcal{M}_{F}([0,1])\big{)}$ with the Skorokhod
topology induced by the weak topology on $\mathcal{M}_{F}([0,1])$. In the
following, we will write $d_{J_{1}}$ for any complete metric inducing this
topology.

######  Definition 2.2 (Exponential equivalence).

We say that two sequences of probability measures $\mathbb{Q}_{1}^{N}$ and
$\mathbb{Q}_{2}^{N}$ on
$\mathbb{D}_{[0,T]}\big{(}\mathcal{M}_{F}([0,1])\big{)}$ are _exponentially
equivalent_ if there exists

  1. i)

a sequence of (abstract) probability spaces
$(\mathcal{X}^{N},\mathcal{F}^{N},\mathcal{Q}^{N})$ and

  2. ii)

a sequence of random variables
$\pi_{1}^{N},\pi_{2}^{N}:\mathcal{X}^{N}\rightarrow\mathbb{D}_{[0,T]}\big{(}\mathcal{M}_{F}([0,1])\big{)}$
with respective laws $\mathbb{Q}_{1}^{N}$ and $\mathbb{Q}_{2}^{N}$

such that the sets $\\{d_{J_{1}}(\pi_{1}^{N},\pi_{2}^{N})>\varepsilon\\}$ are
$\mathcal{F}^{N}$-measurable and

$\limsup_{N}\dfrac{1}{N}\ln\mathcal{Q}^{N}\left\\{\Delta(\pi_{1}^{N},\pi_{2}^{N})>\varepsilon\right\\}=-\infty$

for all $\varepsilon>0$.

In order to state the main result, we introduce the notion of a time change.
We say that we _speed up_ a process $(\pi_{t})_{t\geq 0}$ by a factor
$\vartheta$ if we consider the process $(\pi_{\vartheta t})_{t\geq 0}$
instead. In the case of a Markov process with generator $\mathcal{L}$, this is
equivalent to considering the Markov process with generator
$\vartheta\mathcal{L}$.

######  Theorem 2.3.

If both process are sped up by some factor $\vartheta(N)$, then the exclusion
processes in contact with impermeable boundaries is exponentially equivalent
to the exclusion process in contact with weak boundaries provided

$\vartheta(N)\theta(N)\sum_{x\in\Lambda_{N},y\not\in\Lambda_{N}}\Big{(}p(x,y)+p(y,x)\Big{)}=o(N).$

Under the assumptions of the theorem, the two processes are indistinguishable
at the level of large deviations. In particular, both have the same
hydrodynamic behaviour.

##  3 Applications

In this section, we discuss applications of the exponential equivalence to
prove the hydrodynamic behaviour and the large deviations for asymmetric
exclusion processes.

###  3.1 The TASEP in contact with weak reservoirs

The recent paper [11] considers the nearest-neighbour TASEP in contact with
weak reservoirs given by the above through the choice
$p(x,y)=\mathds{1}_{y=x+1}$, and $\alpha,\beta$ some (possibly time-dependent,
but locally) bounded rates. Due to the asymmetry, the process evolves on the
timescale $\vartheta(N)=N$. This means in particular that

$\theta(N)\vartheta(N)\sum_{x\in\Lambda_{N},y\not\in\Lambda_{N}}\Big{(}p(x,y)+p(y,x)\Big{)}=\mathcal{O}\big{(}N\theta(N)\big{)}=o(N),$

verifying the condition of 2.3. To include time-dependent rates, we make use
of the more general result discussed at the end of Appendix A.

2.3, then, ensures that the process is exponentially equivalent to the
(nearest-neighbour) TASEP with impermeable boundaries. We can extend the
latter to the left by zeros and to the right by ones without changing the
dynamics, transforming it into the TASEP on $\mathbb{Z}$. Using [8] (see also
[10, Theorem 2.1]), we obtain the hydrodynamic behaviour. The large deviations
are considered in [10] and completed in [6, Main Theorem]. Through the
exponential equivalence, this translate directly to the TASEP in contact with
weak reservoirs.

In contrast to [11], this result does not necessitate the addition of a
vanishing viscocity to the model in order to prove the hydrodynamic behaviour.

###  3.2 The ALJEP in contact with weak reservoirs

Similarly, we may consider the jump kernel

$p(x,y)=\dfrac{\mathds{1}_{y>x}}{|y-x|^{1+\gamma}}$

for some $\gamma>0$. As a process on $\mathbb{Z}$, this is a special case of
the model from [9]. This Asymmetric Long Jump Exclusion Process (ALJEP) in
contact with reservoirs has first been introduced in [5].

The ALJEP on the whole of $\mathbb{Z}$ undergoes a phase transition at
$\gamma=1$: for $\gamma\in(0,1)$, the mean jump size is infinite and the long
range effects remain at the macroscopic level; for $\gamma>1$, the mean
becomes finite and the model behaves like the TASEP. The phase transition can
also be read off of the correct time scales given by

$\vartheta(N)=\begin{cases}N^{\gamma}&\text{ if }\gamma<1\\\ \frac{N}{\ln
N}&\text{ if }\gamma=1\\\ N&\text{ if }\gamma>1\end{cases},$

see [9] for details. In the setting with reservoirs, more phase transitions
(depending on the reservoir strength) are expected, see [5, Figure 1].

In the case of weak boundaries, we may check that for every $\gamma>0$ and the
above choice of $\vartheta(N)$, one has

$\vartheta(N)\theta(N)\sum_{x\in\Lambda_{N},y\not\in\Lambda_{N}}\Big{(}p(x,y)+p(y,x)\Big{)}=o(N),$

so that 2.3 is applicable. As in the case of the TASEP, we can extend the
ALJEP with impermeable boundaries to the left with zeros and the right with
ones to recover the ALJEP on $\mathbb{Z}$. However, this does not yet allow us
to deduce the hydrodynamical behaviour of the ALJEP with weak boundaries.
Indeed, the proof in [9] relies on the fact that the initial profile has the
same asymptotic density $\rho^{*}\in(0,1)$ in both directions. This is
violated here in multiple ways as the asymptotic density to the left is 0 and
the asymptotic density to the right is 1.

In the long-range regime $\gamma\in(0,1)$, we can instead use the reciprocal
direction and deduce the hydrodynamical behaviour of the ALJEP on $\mathbb{Z}$
for any initial profiles $\rho_{0}$ with compact support. Indeed, this follows
from the study of the ALJEP in contact with weak reservoirs in [5]. For the
short-range regime $\gamma>1$, no results are available for the case with weak
reservoirs. Instead, we will make use of the finite speed of propagation of
mass in the Burger’s equation to extend the result from [9] to suitable
initial profiles. This is enough to deduce the hydrodynamic behaviour of the
ALJEP in contact with weak reservoirs at least when a) the initial profile
$\rho_{0}$ is continuous on $[0,1]$ and satisfies $\rho_{0}(0)=0$ and
$\rho_{0}(1)=1$, and b) the initial configuration is distributed as a product
measure with profile $\rho_{0}$.

######  Lemma 3.1.

Let $\rho_{0}\in C(\mathbb{R};[0,1])$ be a continuous profile satisfying
$\rho_{0}(x)=0$ (resp. $\rho_{0}(x)=1$) for $x$ small (resp. large) enough,
and let $\mu_{0}^{N}$ be the product measure on $\mathbb{Z}$ with marginals
$\mu_{0}^{N}(\eta_{x}=1)=\rho_{0}(x/N)$. Then, the ALJEP on $\mathbb{Z}$ with
initial configuration distributed as $\mu_{0}^{N}$ satisfies a law of large
numbers with hydrodynamic limit given by the solution $\rho$ to Burger’s
equation [9, Equation (3.2)] with initial value $\rho_{0}$.

###### Proof.

See Appendix B. ∎

##  4 Proof of the main result

For every $N\in\mathbb{N}$, let $\mu_{N}$ be a measure on $\Omega_{N}$. An
adaptation of the usual misanthrope (or: attractive) coupling constructs
processes $\eta^{N},\tilde{\eta}^{N}$ and $\hat{\eta}^{N}$ with initial
distribution $\mu_{N}$ and respective generators

$\vartheta(N)\mathcal{L}_{bulk}^{N},\quad\vartheta(N)\mathcal{L}_{bulk}^{N}+\theta(N)\vartheta(N)\mathcal{L}_{influx}^{N}\quad\text{and}\quad\vartheta(N)\mathcal{L}^{N}$

satisfying

$\eta^{N}_{t}\leq\tilde{\eta}^{N}_{t}\qquad\text{ and
}\qquad\hat{\eta}^{N}_{t}\leq\tilde{\eta}^{N}_{t}$

for all $t\geq 0$. See [3] for details of this coupling, or _e.g._ [9, Section
9] for an English version.

Write $\pi^{N}_{t}:=\pi^{N}(\eta^{N}_{t})$,
$\tilde{\pi}^{N}_{t}:=\pi^{N}(\tilde{\eta}^{N}_{t})$ and
$\hat{\pi}^{N}_{t}:=\pi^{N}(\hat{\eta}^{N}_{t})$ for the corresponding
measure-valued processes. Since they are all atomic, the ordering implies

$N\|\tilde{\pi}^{N}_{t}-\pi^{N}_{t}\|_{TV}=|\tilde{\eta}^{N}_{t}|-|\eta^{N}_{t}|\quad\text{
and }\quad
N\|\tilde{\pi}^{N}_{t}-\hat{\pi}^{N}_{t}\|_{TV}=|\tilde{\eta}^{N}_{t}|-|\hat{\eta}^{N}_{t}|,$

where we write $|\eta|:=\sum_{x\in\Lambda_{N}}\eta_{x}$ for the total mass of
a configuration $\eta\in\Omega_{N}$.

Next, note that the Lévy-Prokhorov metric induces the topology of weak
convergence and can be bounded from above by the distance in total variation,
see _e.g._ [4, Figure 1]. In particular, using the fact that $\tilde{\pi}$
takes into account the influx of particles, we have

$\left\\{d_{J_{1}}(\tilde{\pi}^{N},{\pi}^{N})>\varepsilon\right\\}\subseteq\left\\{\sup_{t\in[0,T]}|\tilde{\eta}_{t}^{N}|-|\eta_{t}^{N}|>\varepsilon
N\right\\}=\left\\{|\tilde{\eta}_{T}^{N}|-|\eta_{T}^{N}|>\varepsilon
N\right\\}$

and similarly for
$\\{d_{J_{1}}(\tilde{\pi}^{N},\hat{\pi}^{N})>\varepsilon\\}$. Hence, the proof
reduces to the following lemma.

######  Lemma 4.1 (No loss of mass).

The events $\\{|\tilde{\eta}^{N}_{T}|-|\eta^{N}_{T}|\geq\varepsilon N\\}$ and
$\\{|\tilde{\eta}_{T}^{N}|-|\eta^{N}_{T}|\geq\varepsilon N\\}$ are
superexponentially unlikely for every $\varepsilon>0$.

###### Proof.

As both are analogous, we will concentrate on the first set only. Since the
change in mass can come only from the influx of particles, it suffices to
prove that the probability of $\varepsilon N$ particles entering up to time
$T$ is superexponentially small. Note that the number of particles is bounded
from above by a Poisson number $P$ with parameter

$\lambda_{N}:=T\|\alpha\|_{\infty}\cdot\theta(N)\vartheta(N)\sum_{x\in\Lambda_{N},y\not\in\Lambda_{N}}p(y,x)=o(N).$

The usual Chernoff bound provides us with the estimate

$\mathbb{P}(P\geq\lambda_{N}+x)\leq\exp\left(-\dfrac{x^{2}}{\lambda_{N}}\cdot
h\left(\dfrac{x}{\lambda_{N}}\right)\right)$

for some function $h(u)=\frac{(1+u)\ln(1+u)-u}{u^{2}}$, vanishing at infinity
like $\frac{\ln u}{u}$, see _e.g._ [2]. In particular,

$\displaystyle\mathbb{P}\left(\exists
t\in[0,T]\;:\;|\tilde{\eta}_{t}^{N}|-|\eta_{t}^{N}|\geq\varepsilon
N\right)\leq\exp\left(-\Theta\left(\varepsilon
N\cdot\ln\left(\dfrac{\varepsilon N}{o(N)}\right)\right)\right)$

which is superexponentially small. ∎

##  Appendix A Generalization to misanthrope processes

The proof in Section 4 relies heavily on the attractive coupling of the
different processes. From [3], it is known that this coupling can be
constructed for the large class of _misanthrope processes_. These cover many
models of interest, including the exclusion and zero range processes.

Let $k\in\mathbb{N}\cup\\{\infty\\}$ denote the maximal number of particles
allowed at a site and set $S_{k}:=\\{0,\dots,k\\}$ or
$S_{\infty}:=\mathbb{N}_{0}$ accordingly.

For $N\in\mathbb{N}$, $N\geq 2$, define the _bulk_
$\Lambda_{N}:=\\{1,\dots,N-1\\}$ and the space of configurations
$\Omega_{N}:=S_{k}^{\Lambda_{N}}$. For a configuration $\eta\in\Omega_{N}$ and
sites $x,y\in\Lambda_{N}$, we define the three actions
$\eta\mapsto\eta^{x\rightarrow y}$, $\eta\mapsto\eta^{x\uparrow}$ and
$\eta\mapsto\eta^{x\downarrow}$ as follows:

  1. 1.

if $\eta_{x}=0$ or $\eta_{y}=k$, set $\eta^{x\rightarrow y}:=\eta$, otherwise
set $\eta^{x\rightarrow y}:=\eta-\delta_{x}+\delta_{y}$, _i.e._

$\eta^{x\rightarrow y}_{z}:=\begin{cases}\eta_{x}-1&\text{ if }z=x\\\
\eta_{y}+1&\text{ if }z=y\\\ \eta_{z}&\text{ otherwise}\end{cases};$

  2. 2.

if $\eta_{x}=k$, set $\eta^{x\uparrow}:=\eta$, otherwise set
$\eta^{x\uparrow}:=\eta+\delta_{x}$;

  3. 3.

if $\eta_{x}=0$, set $\eta^{x\downarrow}:=\eta$, otherwise set
$\eta^{x\downarrow}:=\eta-\delta_{x}$.

For functions $f:\Omega_{N}\rightarrow\mathbb{R}$, define the generators

$\mathcal{L}_{bulk}^{N}f(\eta):=\sum_{x,y\in\Lambda_{N}}^{N-1}p(x,y)\cdot
b_{bulk}(\eta_{x},\eta_{y})\cdot\Big{(}f(\eta^{x,y})-f(\eta)\Big{)}$

and

$\displaystyle\mathcal{L}_{influx}^{N}f(\eta)$
$\displaystyle:=\sum_{x\in\Lambda_{N}}\sum_{y\not\in\Lambda_{N}}p(y,x)\cdot
b_{influx}(y,\eta_{x})\cdot\Big{(}f(\eta^{x\uparrow})-f(\eta)\Big{)},$
$\displaystyle\mathcal{L}_{outflux}^{N}f(\eta)$
$\displaystyle:=\sum_{x\in\Lambda_{N}}\sum_{y\not\in\Lambda_{N}}p(x,y)\cdot
b_{outflux}(\eta_{x},y)\cdot\Big{(}f(\eta^{x\downarrow})-f(\eta)\Big{)},$

where

  1. i)

$p$ is a jump kernel,

  2. ii)

$b_{bulk}:S_{k}^{2}\rightarrow[0,+\infty)$ is non decreasing in its first
variable, non increasing in its second variable and satisfies $b(n,m)=0$ if
and only if $n=0$ or $m=k$,

  3. iii)

$b_{influx}:\mathbb{Z}\times S_{k}\rightarrow(0,+\infty)$ and
$b_{outflux}:S_{k}\times\mathbb{Z}\rightarrow[0,+\infty)$ are bounded.

We will assume that $b_{bulk}$ is such that the following Markov processes
exist. This is trivially satisfied when $b_{bulk}$ is bounded, _e.g._ if
$k\neq\infty$.

######  Definition A.1.

A _misanthrope process in contact with impermeable boundaries_ is defined
through the generator $\mathcal{L}_{bulk}^{N}$, whereas a _misanthrope process
in contact with weak reservoirs_ has the generator

$\mathcal{L}^{N}:=\mathcal{L}_{bulk}^{N}+\theta(N)\left(\mathcal{L}_{influx}^{N}+\mathcal{L}_{outflux}^{N}\right)$

for some $\theta(N)\to 0$.

As before, we will identify processes with values in the space of
configurations $\Omega_{N}$ with the corresponding process with values in the
space of measures via the map

$\pi^{N}:\Omega_{N}\longrightarrow\mathcal{M}_{F}([0,1]),\quad\eta\mapsto\dfrac{1}{N}\sum_{x\in\Lambda_{N}}\eta_{x}\delta_{x/N}.$

######  Theorem A.2.

If both processes are sped up by a factor $\vartheta(N)$, then the misanthrope
process in contact with weak reservoirs is exponentially equivalent to the
misanthrope process in contact with impermeable boundaries provided

$\vartheta(N)\theta(N)\sum_{x\in\Lambda_{N},y\not\in\Lambda_{N}}\Big{(}p(x,y)+p(y,x)\Big{)}=o(N).$

###### Proof.

The proof is exactly as in Section 4. The only difference is that we replace
$\|\alpha\|_{\infty}$ by $\|b_{influx}\|_{\infty}$. ∎

Note that the proof does not depend on the underlying space. In particular,
the result can be extended to misanthrope processes on any lattice, including
$\mathbb{Z}^{d}$. Furthermore, it can be generalized to time-dependent
interactions with the reservoirs as long as they are locally $L^{1}$ in time
in the sense that $b_{influx}\in
L^{1}_{loc}([0,+\infty);L^{\infty}(\mathcal{Z}\times S_{k}))$ and similarly
for $b_{outflux}$. In this case $T\|b_{influx}\|_{\infty}$ is to be replaced
by $\int_{0}^{T}\|b_{influx}(t)\|_{\infty}\;\operatorname{d}\\!{t}$.

Although pathological counter-examples can be constructed, the statement of
A.2 is sharp in most situations. This includes also the symmetric case, see
for example [7] (and particularly [1] therein) for the treatment of the
symmetric exclusion process with long jumps in contact with reservoirs.
Unfortunately, it is generally equally hard to derive the hydrodynamic
behaviour of the process in contact with weak reservoirs or in contact with
impermeable boundaries, so that the result only shortens proofs by providing a
general argument for ignoring boundary terms.

In the context of totally asymmetric processes, however, A.2 provides a
shortcut for proving the hydrodynamic behaviour (and even higher order
behaviour as the fluctuations or the large deviations) as shown in Section 3.

##  Appendix B Proof of 3.1

For simplicity, assume that $\rho_{0}(x)=0$ for $x\leq 0$ and $\rho_{0}(x)=1$
for $x\geq 1$. Let $(\rho_{0}^{m,\ell})_{m,\ell\in\mathbb{N}}$ be a family of
continuous functions satisfying
$\rho_{0}^{m,\ell}|_{[-m,\ell]}=\rho_{0}|_{[-m,\ell]}$ and
$\rho_{0}^{m,\ell}(x)=\frac{1}{2}$ on $\mathbb{R}\setminus[-2m,2\ell]$. We may
choose the family such that it is pointwise non increasing in $m$ and
pointwise non decreasing in $\ell$. Write $\mu_{0}^{N,m,\ell}$ for the
corresponding product measure on $\\{0,1\\}^{\mathbb{Z}}$. Furthermore, denote
by $\mu_{0}^{N,\ell}$ the measures obtained from the pointwise limit
$\lim_{m}\rho_{0}^{m,\ell}$ which is vanishes to the left of $0$.

Using the attractive coupling, we may construct the ALJEPs $\eta^{N,m,\ell}$
and $\eta^{N,\ell}$ on $\mathbb{Z}$ started from $\mu_{0}^{N,m,\ell}$ on a
common probability space such that

$\eta_{t}^{N,\ell}\leq\eta_{t}^{N,m,\ell}\leq\eta_{t}^{N,m^{\prime},\ell}\qquad\text{
and }\qquad\eta_{t}^{N,\ell}\leq\eta_{t}^{N,\ell^{\prime}}\leq\eta_{t}^{N},$

for any $m^{\prime}\leq m$, $\ell\leq\ell^{\prime}$ and $t\geq 0$, a.s.

Similarly to the proof of 2.3, it is enough to show that both

$\lim_{m}\sup_{\ell}\limsup_{N}\mathbb{P}^{N}\left(\sum_{x\geq
1}\eta_{T}^{N,m,\ell}(x)-\eta_{T}^{N,\ell}(x)>\varepsilon N\right)=0$

and

$\lim_{\ell}\limsup_{N}\mathbb{P}^{N}\left(\sum_{x=1}^{N-1}\eta_{T}^{N}(x)-\eta_{T}^{N,\ell}(x)>\varepsilon
N\right)=0.$

As both quantities are similar, we will concentrate on the former. By
construction, $\sum_{x\geq 1}\eta_{t}^{N,\ell}(x)=\sum_{x\geq
1}\eta_{0}^{N,\ell}(x)=\sum_{x\geq 1}\eta_{0}^{N,m,\ell}(x)$. In particular,
it is enough to bound

$\displaystyle\mathbb{P}^{N}\left(\sum_{x\geq
1}\eta_{T}^{N,m,\ell}(x)-\eta_{0}^{N,m,\ell}(x)>\varepsilon N\right)$
$\displaystyle\quad\leq\dfrac{\mathbb{E}^{N}\left[\sum_{x\geq
1}\eta_{T}^{N,m,\ell}(x)-\eta_{0}^{N,m,\ell}(x)\right]}{\varepsilon N}$
$\displaystyle\quad\leq\dfrac{1}{\varepsilon
N}\int_{0}^{T}\mathbb{E}^{N}\left[N\sum_{x\geq 1}\sum_{y\leq
0}p(y,x)\cdot\eta_{t}^{N,m,\ell}(y)\left(1-\eta_{t}^{N,m,\ell}(x)\right)\right]\;\mathrm{d}{t}$
$\displaystyle\quad=\mathcal{O}\left(\dfrac{1}{\varepsilon}\int_{0}^{T}\sum_{y\leq
0}\dfrac{1}{|y|^{\gamma}}\cdot\mathbb{E}^{N}\left[\eta_{t}^{N,m,\ell}(y)\right]\;\mathrm{d}{t}\right)$
$\displaystyle\quad=\mathcal{O}\left(\dfrac{T}{\varepsilon}\left(\dfrac{mN}{2}\right)^{1-\gamma}+\dfrac{mN}{2\varepsilon}\int_{0}^{T}\int_{-m/2}^{0}\rho_{t}^{m,\ell}(u)\;\mathrm{d}{u}\;\mathrm{d}{t}\right),$

where $\rho^{m,\ell}$ is the hydrodynamic limit of the family
$(\eta_{t}^{N,m,\ell})_{N\geq 2}$, which is a solution to Burger’s equation
with initial profile $\rho_{0}^{m,\ell}$, see [9]. The first term vanishes in
$N$, whereas the latter becomes constantly $0$ independently of $N$ and $\ell$
for $m$ large enough depending only on $T$.

## References

  * [1] Cédric Bernardin, Pedro Cardoso, Patrícia Gonçalves and Stefano Scotta  “Hydrodynamic limit for a boundary driven super-diffusive symmetric exclusion” working paper or preprint, 2022  URL: https://hal.science/hal-02884549
  * [2] Clément Canonne Accessed: August 18, 2023, 2019  URL: http://www.cs.columbia.edu/~ccanonne/files/misc/2017-poissonconcentration.pdf
  * [3] Christiane Cocozza-Thivent  “Processus des misanthropes”  In _Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete_ 70.4, 1985, pp. 509–523  DOI: 10.1007/BF00531864
  * [4] Alison L. Gibbs and Francis Edward Su  “On choosing and bounding probability metrics”, 2002  arXiv:math/0209021 [math.PR]
  * [5] Patrícia Gonçalves, Julian Kern and Lu Xu  “An Asymmetric Exclusion Process with Long Jumps in Contact with Reservoirs”, Forthcoming
  * [6] Jeremy Quastel and Li-Cheng Tsai  “Hydrodynamic large deviations of TASEP”, 2022  arXiv:2104.04444 [math.PR]
  * [7] Stefano Scotta  “Asymptotic behavior of the boundary driven exclusion with long jumps”, 2021
  * [8] Timo Seppäläinen  “Coupling the totally asymmetric simple exclusion process with a moving interface”  In _Markov Processes and Related Fields_ 4, 1998
  * [9] Sunder Sethuraman and Doron Shahar  “Hydrodynamic limits for long-range asymmetric interacting particle systems”  In _Electronic Journal of Probability_ 23.none Institute of Mathematical StatisticsBernoulli Society, 2018, pp. 1–54  DOI: 10.1214/18-EJP237
  * [10] Srinivasa R.. Varadhan  “Large deviations for the asymmetric simple exclusion process”  In _Stochastic analysis on large scale interacting systems_ 39, Adv. Stud. Pure Math.  Math. Soc. Japan, Tokyo, 2004, pp. 1–27  DOI: 10.2969/aspm/03910001
  * [11] Lu Xu  “Hydrodynamics for One-Dimensional ASEP in Contact with a Class of Reservoirs”  In _Journal of Statistical Physics_ 189.1, 2022, pp. 1  DOI: 10.1007/s10955-022-02963-x


# Improving Cyberbullying Detection with User Interaction

Suyu Ge1, Lu Cheng2, Huan Liu2 1Department of Electronic Engineering, Tsinghua
University
2School of Computer Science and Engineering, Arizona State University
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(2021)

###### Abstract.

Cyberbullying, identified as intended and repeated online bullying behavior,
has become increasingly prevalent in the past few decades. Despite the
significant progress made thus far, the focus of most existing work on
cyberbullying detection lies in the independent content analysis of different
comments within a social media session. We argue that such leading notions of
analysis suffer from three key limitations: they overlook the temporal
correlations among different comments; they only consider the content within a
single comment rather than the topic coherence across comments; they remain
generic and exploit limited interactions between social media users. In this
work, we observe that user comments in the same session may be inherently
related, e.g., discussing similar topics, and their interaction may evolve
over time. We also show that modeling such topic coherence and temporal
interaction are critical to capture the repetitive characteristics of bullying
behavior, thus leading to better predicting performance. To achieve the goal,
we first construct a unified temporal graph for each social media session.
Drawing on recent advances in graph neural network, we then propose a
principled graph-based approach for modeling the temporal dynamics and topic
coherence throughout user interactions. We empirically evaluate the
effectiveness of our approach with the tasks of session-level bullying
detection and comment-level case study. Our code is released to public.
111https://github.com/gesy17/TGBully

Cyberbullying Detection, User Interaction, Temporal Dynamics, Topic Coherence,
Graph Attention Network

††copyright: iw3c2w3††journalyear: 2021††conference: Proceedings of the Web
Conference 2021; April 19–23, 2021; Ljubljana, Slovenia††booktitle:
Proceedings of the Web Conference 2021 (WWW ’21), April 19–23, 2021,
Ljubljana, Slovenia††doi: 10.1145/3442381.3449828††isbn:
978-1-4503-8312-7/21/04††ccs: Computing methodologies Natural language
processing††ccs: Social and professional topics User characteristics

##  1\. Introduction

Cyberbullying is identified as a form of bullying or harassment through
electronic means, which is typically characterized by posting rumors, threats,
or pejorative labels on social media (US Legal, 2019). Due in part to the low
cost and easy access to social networking sites, Cyberbullying has become
increasingly prevalent in recent decades. According to the 2019 Youth Risk
Behavior Surveillance System (Centers for Disease Control and Prevention), an
estimated 15.7% of high school students were electronically bullied in the 12
months prior to the survey (for Disease Control and Prevention, 2019). In
response to the rapidly growing number of cyberbullying cases, numerous
efforts (e.g., (Wu et al., 2010; Zhao and Mao, 2016; Dani et al., 2017)) have
been put into computational detection and mitigation of cyberbullying
behaviors to tackle the socio-technical problem.

Figure 1. An example of cyberbullying session consisting of the original post,
user comments and posted time. The three comments in the upper part are non-
bullying comments while those in the lower part are regarded as bullying
comments.

Despite the remarkable progress made towards computational cyberbullying
detection, we argue that it is still essential to understand how users in a
social media session interact with each other in order to mitigate the long-
term impact of cyberbullying. Whereas offline bullying (or “traditional”
bullying) relates to face-to-face interactions, cyberbullying typically takes
place throughout a sequence of (intensive) user interactions in social media
sites(Cheng et al., 2020b), as illustrated in Fig. 1. Modeling user
interaction is important to cyberbullying detection for several reasons.
First, one essential aspect that differs cyberbullying from offline bullying
is its persistence & repetition (Ziems et al., 2020; Dinakar et al., 2012)
characteristic. Decades of research in psychology (Dooley et al., 2009) and
social science (Smith et al., 2008) have shown that cyberbullying is carried
out repeatedly against victims by one or more individuals. Studies of user
interactions allow us to characterize the repetition by both content and
temporal analysis. For example, recent findings (Soni and Singh, 2018; Gupta
et al., 2020) suggest that (1) Most bullying comments occurred in the first
hours after an initial post and the bullying comment counts in bullying
sessions are significantly larger than those of non-bullying sessions; and (2)
a relatively short time interval between consecutive cyberbullying comments is
observed in bullying sessions. Second, user interaction is a strong signal for
detecting cyberbullying instances. Recent analysis of cyberbullying activity
from an interdisciplinary research (Gupta et al., 2020) manifests that the
intensity of user activities within a bullying session is inherently different
from that within a non-bullying session. Third, what makes cyberbullying
detection particularly challenging compared to offline bullying is that it is
not uncommon to see that users play multiple roles such as bully, victim, and
witness in the same session (Salmivalli, 1999; Xu et al., 2012). Therefore,
modeling user interaction provides an alternative for the implicit monitor of
users’ roles over time.

However, modeling user interaction in a social media session confronts multi-
faceted challenges: (1) Sparsity. Whereas a social media session is often
involved with multiple interactions among numerous users, the interaction with
a specific user can be very low. Further, most of the comments are directed
towards the owner of the social media session, resulting in a “hub” where the
owner degree is much larger than the degrees of users who post comments
(Ravazzi et al., 2017; Huberman et al., 2009). Therefore, the user-user
interaction graph constructed from reply relation can be sparse and less
informative. (2) Repetition. Characterizing the repetition of cyberbullying
behavior is particularly challenging because we need to satisfy at least two
constraints simultaneously: similar in content and close in time (Soni and
Singh, 2018). Therefore, the desired models should be able to jointly capture
content similarities and temporal correlations. (3) User Characteristics.
Empirical findings within psychology identify that users’ personality traits
are important indicators of computer-mediated behaviors (Al-garadi et al.,
2016; Kowalski et al., 2014). However, it is rather difficult to discover such
user characteristics solely based on the content in a session due to the
extremely limited comments from an individual user.

To tackle these challenges, we propose TGBully, a Temporal-Graph-based
cyberbullying detection framework. TGBully is composed of three modules, i.e.,
a semantic context modeling module, a temporal graph interaction learning
module and a session classification module. In the first module, we contrast
the standard content analysis of a lump of comments with a hierarchical
construction of a sequence of comments, each of which comprises a sequence of
words. To incorporate characteristics of users who post these comments, we
further extract users’ historical comments, from which we infer their language
behavior. The temporal graph interaction learning module implicitly constructs
the user-user interaction graph by encoding the topic coherence and temporal
dynamics of user comments in a graph attention network (GAT). In particular,
we consider content similarities and time intervals between comments, and the
user interactions are learned through the continuous update of the edge
weights in the GAT. By doing so, TGBully models user interaction through the
propagation of interactive signals in the constructed temporal graph. The
session classification module aggregates the information representation
learned from user interactions and performs the final session-level
classification, i.e., determining whether a session is bullying or not. To
summarize, the main contributions of this work are:

  * •

We make novel observations of dynamic user interactions reflected by the
inherently related and continuously evolved comments in a social media
session. We show the importance of modeling user interactions in improving the
performance of cyberbullying detection.

  * •

We propose a principled and effective approach that models user interaction
from two key aspects: temporal dynamics and topic coherence. This is achieved
by a temporal graph constructed for each session and a GAT for learning the
interactions in the graph.

  * •

We evaluate TGBully on both session-level detection tasks and comment-level
case study. Empirical results on two benchmark datasets validate the
effectiveness of our approach and shed light on its practical implications for
policy-makers.

##  2\. Related Work

Our work is closely related to cyberbullying detection and graph attention
networks. Here, we briefly review the related literature.

###  2.1. Cyberbullying Detection

Previous work on cyberbullying detection typically builds their approaches
upon natural language processing (NLP) methods (Dinakar et al., 2011; Nand et
al., 2016; Romsaiyud et al., 2017). Their goal is to identify aggressive and
harassing mentions via text analysis. Hosseinmardi et al. (Hosseinmardi et
al., 2014) compared Linguistic Inquiry and Word Count (LIWC) values of
different categories of words, they found that words connected to negative
feelings such as “depress”, “stressful”, and “suicide” are prominent
identifiers of cyberbullying. Dani et al. (Dani et al., 2017) added sentiment
consistency features to a unigram model with TF-IDF values as content
features.

More recent work highlighted the importance of temporal information for
cyberbullying detection (Cheng et al., 2019; Soni and Singh, 2018; Cheng et
al., 2020a). For example, Soni and Sign (Soni and Singh, 2018) characterized
each comment on Instagram by an exponential time factor. They empirically
evaluated a set of time-related features derived from the exponential factor
and concluded several essential temporal traits of cyberbullying. Cheng et al.
(Cheng et al., 2019, 2020a) unified cyberbullying detection and time interval
prediction in a multi-task learning framework. Built upon the hierarchical
attention network, their approach aggregated contextualized comment
representation for session classification meanwhile predicted time intervals
from comment representations. In addition to text, numerous methods
incorporated social information into cyberbullying detection (Huang et al.,
2014; Chatzakou et al., 2017; Ziems et al., 2020). By correlating
cyberbullying behavior with online sociability, they primarily measured the
power imbalance between users by the social network and interaction network.
Ziems et al. (Ziems et al., 2020) utilized the neighborhood overlap, verified
status, number of followers, and recent tweets to indicate users’ visibility
among peers. Chatzakou et al. (Chatzakou et al., 2017) extracted and compared
numerous text, user, and network-based attributes. They found that compared
with aggressors, bullies are less popular and participate in fewer online
communities.

Prior research has put a large number of efforts into modeling user-generated
contents but overlooked the interactions of users themselves within a session.
Findings from various fields have evidenced the critical role user
interactions play in cyberbullying detection. We thereby complement earlier
work with the consideration of user interactions by jointly modeling the
temporal dynamics and topic coherence in an established temporal graph.

###  2.2. Graph Attention Network

Recent years have witnessed the increasing popularity of graph neural networks
(GNNs), which have become powerful tools to model real-world data in various
domains (Li et al., 2015, 2016; Hamilton et al., 2017). Among the widely
recognized GNNs, graph attention networks (GATs) (Veličković et al., 2017) use
self-attentional layers with automatically learned weights to attend over
neighborhoods’ features. Due to their prominent capability of learning salient
interaction, a surge of attempts have been made to apply GATs to various text
mining tasks. For instance, Hu et al. (Hu et al., 2019) proposed a
Heterogeneous Graph Attention networks (HGAT) for semi-supervised short text
classification. At its core, HGAT embeds heterogeneous information (topics,
short texts, and entities) via a dual-level attention mechanism. Huang et al.
(Huang and Carley, 2019) designed a novel target-dependent graph attention
network (TD-GAT) to explicitly utilize the dependency relation among words for
aspect-level sentiment classification.

Our work is more related to another line of research exploiting GAT for fact
verification (Zhong et al., 2020; Zhou et al., 2019; Liu et al., 2020). In
both cases, characterizing interactions between facts (or users in our task)
are central to the modeling process. For instance, Zhong et al. (Zhong et al.,
2020) constructed an XLNet-based word graph. Then they formed argument-level
representations using a graph convolution network to propagate over the word
graph. At last, GAT was incorporated to aggregate useful evidence for the
verification task. Similarly, Zhou et al. (Zhou et al., 2019) also employed
GAT to perform fact reasoning where each graph node was constructed by
concatenating a claim and the corresponding evidence. In these approaches,
edge weights are automatically optimized during the propagation process.
However, these approaches are not applicable to our task due to the unique
challenges of the session-based cyberbullying detection (Cheng et al., 2020b).
Therefore, to make the most of GAT, we propose to explicitly introduce the two
important bully-indicative features – temporal dynamics and topic coherence –
into the edge weight calculation.

##  3\. Problem Formulation

We use the following information extracted from a social media session for
cyberbullying detection:

  * •

Comment:  It is a sequence of comments posted by different users in a session.
There are two types of comments in general: an initial comment posted by the
owner and the subsequent comments. They comprise of the textual information of
a session. We represent words in a comment as embeddings and analyze their
contextual relation using neural networks.

  * •

Time:  It is the time stamp of each comment. Informed by previous work (Cheng
et al., 2019), we represent time by calculating the time intervals (in
minutes) between comments and the initial comment.

  * •

User:  It denotes the username of each posted comment. Since a user may
mention a previous username, we embed each username as a word and encode it
into comment representation. We also add the username embedding before the
first token of a comment to identify its poster.

  * •

History:  The historical comments of each user. We collect historical comments
from previous sessions and concatenate them into a unified paragraph. 222We
omit their relative temporal order due to the lack of such information.

Suppose that we have a corpus of sessions $\mathcal{C}$ and each session
consists of $n$ comments, denoted as
$\mathcal{S}=\\{c_{1},c_{2},...,c_{n}\\}$, where $c_{i}$ is the $i^{th}$
comment in the session. Along with the sessions is the label set $Y$
indicating whether a session is a bullying ($y=1$) or non-bullying ($y=0$)
session. A user $u\in\mathcal{U}$ posts comment
$c_{i}=\\{w_{1}^{c},w_{2}^{c},...,w_{p}^{c}\\}$ with $p$ words at time
$t_{i}$. Besides, the user $u$ also has a set of historical comments, and we
concatenate them into one paragraph with $q$ words as
$h_{u}=\\{w_{1}^{h},w_{2}^{h},...,w_{q}^{h}\\},\forall h_{u}\in\mathcal{H}$.
We then define the session-based cyberbullying detection task as:

Definition. Given the corpus of sessions $\mathcal{C}$, the associated label
set $Y$, the group of users $\mathcal{U}$, and the historical comments
$\mathcal{H}$, the goal of this work is to classify a social media session
into the bullying/non-bullying category using the multi-modal information
extracted from Comment, Time, User and History.

##  4\. TGBully: The Proposed Approach

The goal of this work is to study user interaction in a session to improve the
session-level cyberbullying detection. We herein propose TGBully, a principled
framework consisting of three modules: (1) a semantic context modeling module
that encodes each comment by considering both its textual content and user’s
language behavior reflected from her/his historical comments, (2) a temporal
graph interaction learning module that constructs a temporal graph and models
the dynamic user interaction with a bully-featured GAT. The proposed GAT
jointly captures topic coherence and temporal dynamics in the modeling
process; and (3) a session classification module that attentively aggregates
information from user interaction into a session representation, based on
which it then classifies the session into a bullying/non-bullying session. The
overall framework of TGBully is illustrated in Fig. 2.

Figure 2. Description of the overall framework. TGBully consists of three
modules: The semantic context modeling module (left) hierarchically constructs
comment sequences as $\mathbf{c}$ (blue) and encodes users’ language behavior
via the representation of historical comments $\mathbf{h}$ (yellow). The
temporal graph interaction learning module (middle) implicitly encodes the
topic coherence and temporal dynamics in the temporal interaction graph where
nodes represent user comments and edges represent time intervals between two
comments. The session classification module (right) first aggregates the
information learned from user interactions into user representation
$\mathbf{u}$ (grey). It then performs the session-level classification by
aggregating all user representations within the session into the session
representation.

###  4.1. Semantic Context Modeling

Towards the semantic understanding of texts in a session, the semantic context
modeling module utilizes a hierarchical pipeline (Yang et al., 2016; Cheng et
al., 2019) to form two types of contextualized representations. One is comment
representation, which uncovers user intent and emotion through their
utterances in the current session. The other is history representation, which
encodes user characteristics, e.g., personalities and language patterns
reflected from their historical comments.

####  4.1.1. Comment Encoding

The comment encoding mainly consists of four layers. The first layer is the
word embedding, which transforms a $p$-word comment
$c=\\{w_{1}^{c},w_{2}^{c},...,w_{p}^{c}\\}$ to a sequence of high dimensional
vectors $\\{\mathbf{w}_{1}^{c},\mathbf{w}_{2}^{c},...,\mathbf{w}_{p}^{c}\\}$
via an embedding matrix. The second is a word-level gated recurrent units
(GRU) (Cho et al., 2014) layer. It aims to encode long-term contextual
dependency among words. It has been shown that GRU achieves better performance
than typical LSTM (Hochreiter and Schmidhuber, 1997) on small datasets (Gruber
and Jockisch, 2020), therefore, we use a stack of bidirectional GRUs to model
word sequences from both forward and backward directions. The contextualized
representation $\mathbf{r}_{i}^{w}$ of the $i^{th}$ word $\mathbf{w}_{i}^{c}$
is calculated as:

(1)
$\mathbf{r}_{i}^{w}=[\overrightarrow{GRU}(\mathbf{w}_{i}^{c}),\overleftarrow{GRU}(\mathbf{w}_{i}^{c})].$

However, the words in a comment are not equally significant to cyberbullying.
To automatically select informative words from a comment, we herein employ
word-level attention as the third layer. For the $j^{th}$ comment, the
aggregated comment representation $\mathbf{c}_{j}$ is learned as:

(2)
$\alpha_{i}^{w}=\frac{\exp(\tanh(\mathbf{v}_{w}^{T}\times\mathbf{r}_{i}^{w}+b_{w}))}{\Sigma_{k=1}^{p}\exp(\tanh(\mathbf{v}_{w}^{T}\times\mathbf{r}_{k}^{w}+b_{w}))},$

(3)
$\mathbf{c}_{j}=\displaystyle\sum_{i=1}^{p}\alpha_{i}^{w}\mathbf{r}_{i}^{w},$

where $\mathbf{v}_{w}$ and $b_{w}$ are the parameters of the attention
network. $\alpha_{i}^{w}$ is the attention weight of the $i^{th}$ word. The
final layer of this sub-module consists of comment-level bidirectional GRUs
that model comments in a temporal order. Then the bi-GRU output representation
of the $j^{th}$ comment is formulated as:

(4)
$\mathbf{r}_{j}^{c}=[\overrightarrow{GRU}(\mathbf{c}_{j}),\overleftarrow{GRU}(\mathbf{c}_{j})].$

####  4.1.2. History Encoding

As the number of comments posted by a user is rather limited, the goal of
history encoding is to better capture users’ language behavior and personality
traits mirrored by their historical comments. Survey studies find that
bullying behavior has strong connections with both hostility and neuroticism
(Arıcak, 2009; Corcoran et al., 2012). Neurotic users are more emotional and
sometimes even more irrational, thereby, they are more likely to engage in
cyberbullying activities (Al-garadi et al., 2016). Further, it has been
advocated that the use of historical information in social media is effective
for personality prediction (Golbeck et al., 2011). We first associate the
learned user characteristics with corresponding comments in the current
session. We then use the same semantic context modeling approach described
above for user history modeling. In particular, given $n$ users’ history in a
session as inputs, where each history comment comprises $q$ words, the output
contextualized history sequence is
$\\{\mathbf{r}_{1}^{h},\mathbf{r}_{2}^{h},...,\mathbf{r}_{n}^{h}\\}$.

###  4.2. Interaction Learning in Temporal Graph

The Temporal graph interaction learning module seeks to learn user interaction
and capture the repetitive characteristics of cyberbullying behavior. To
accomplish this goal, the module first constructs a temporal interaction graph
for each social media session. Since GAT can selectively aggregate principle
neighbor information via its attentional layers, we use it to implicitly
encodes the topic coherence and temporal dynamics between different users in
the graph.

We begin by illustrating how the temporal graph interaction learning module
can effectively address the two major challenges for modeling user
interactions in a session. To recall, as most comments are directed towards
the owner of the session, the first challenge is that social graph based on
reply and friendship relation can be sparse; the second challenge relates to
the unique characteristic of cyberbullying – repetition: content similarity
and time proximity. We argue that these challenges can be effectively tackled
by jointly modeling the topic coherence and the temporal dynamics in a
session. The two aspects improve our understandings of the discussed topics
and temporal correlation across time.

We first construct the temporal graph where the nodes are user comments and
the edges are time intervals between comments. Each comment in our graph is
inherently connected, consequently, they will not be bounded by sparse and
uneven reply relationships between their users. We then propose to encode the
topic coherence and temporal dynamics in the temporal graph jointly in a
unified framework. The affinity level of two neighbor user comments, namely,
their content similarity and time proximity, is automatically determined and
propagated through the edges of the temporal graph. At its core, the temporal
graph interaction learning module leverages GATs to aggregate neighbor
information (i.e., comments) according to their affinity levels.

####  4.2.1. Temporal Graph

The user interaction is implicitly learned by capturing their comment
interaction in the temporal graph. Specifically, we first transform the
session into a fully connected graph, where we establish connectivity between
two arbitrary comments. In the temporal graph, comments are viewed as graph
nodes and time intervals between them are viewed as edges. Informed by recent
advances in GNNs (Huang and Carley, 2019; Hu et al., 2019), we incorporate a
graph attention neural network (Veličković et al., 2018) to automatically
aggregate information from neighbor nodes to the central node. Different
significance levels of neighbor nodes are weighted by their edge weights. The
GAT network encodes the temporal-level and topic-level relatedness between the
central node and neighbors by introducing them into the calculation of edge
weights. Then it propagates interactive signals to the central nodes by
aggregating neighbors nodes according to their edge weights. Additionally, we
consider self-propagation to reserve the original content information of each
node. Since our temporal graph is already fully-connected, instead of stacking
multiple graph layers, we only perform one-hop graph aggregation333Empirical
experimental results also prioritize the one-hop graph aggregation.. For the
$j^{th}$ comment in the temporal graph, we denote its node vector as
$\mathbf{g}_{j}$, which is calculated as:

(5)
$\mathbf{g}_{j}=\sum_{\mathbf{r}_{k}^{c}\in\\{N_{j}\cup\mathbf{r}_{j}^{c}\\}}\pi(\mathbf{r}_{k}^{c},\mathbf{r}_{j}^{c},t_{k,j})\mathbf{W}_{c}\mathbf{r}_{k}^{c},$

where $N_{j}$ is the set of $j^{th}$ comment’s neighbor nodes,
$\mathbf{W}_{c}$ is the transformation matrix of the graph network.
$t_{k,j}=t_{j}-t_{k}$ is the time interval between two comments.
$\pi(\mathbf{r}_{k}^{c},\mathbf{r}_{j}^{c},t_{k,j})$ denotes the weight of
edge connecting node $\mathbf{r}_{k}^{c}$ and $\mathbf{r}_{j}^{c}$. It is a
decaying factor that controls the relative contribution each neighbor makes to
the central node. We detail the implementation of
$\pi(\mathbf{r}_{k}^{c},\mathbf{r}_{j}^{c},t_{k,j})$ later.

####  4.2.2. Topic coherence

Semantic relatedness between comments in a session can reveal users’ reactions
and attitudes towards the commenting behavior of other users. For instance, a
group of users may repeat the offensive terms used in a previous post to
humiliate the victim. Such language repetition reflects and reinforces their
aggressive motivations and potential power imbalance between bullies and
victims. Crucially, capturing such topic-level repetition is useful for
cyberbullying detection.

Our proposed approach is based on an important observation: In the embedding
space, two comments usually locate in close proximity if they share similar
content or they are coherent in the topical discussion. To measure such
proximity, we compute the embedding similarity $\pi_{time}$ of comments by
taking dot products of their transformed node vectors in the graph. Formally,

(6)
$\pi_{topic}(\mathbf{r}_{k}^{c},\mathbf{r}_{j}^{c})=(\mathbf{r}_{k}^{c}\mathbf{W}_{o})^{T}\mathbf{r}_{j}^{c},$

where $\mathbf{r}_{k}^{c}$ and $\mathbf{r}_{j}^{c}$ are two nodes representing
two arbitrary user comments. $\mathbf{W}_{o}$ is the graph transformation
matrix.

####  4.2.3. Temporal dynamics

In addition to the topic coherence, temporal proximity is also an important
feature for repetition. Instead of being immediate one-off responses, previous
research discovered that cyberbullying sessions receive a more steady stream
of comments that are consecutive in time (Soni and Singh, 2018). It is also
common to observe bursts of comments in which several bully users gang up on a
victim in a bullying session (Al Mazari, 2013). Both the steady stream of
replies and a sudden burst of comments can be recognized by capturing
intervals between each chronological pair of comments. Thus, we further
integrate the time-lapse factor into the constructed temporal graph.
Specifically, we calculate the time interval of comments and project it into
the graph embedding space as below:

(7) $\pi_{time}(t_{k,j})=W_{t}t_{k,j}=W_{t}(t_{j}-t_{k}),$

where $W_{t}$ is the time transformation coefficient and $t_{k,j}$ denotes the
time interval.

####  4.2.4. Joint modeling

The ultimate goal of this work is to improve cyberbullying detection with user
interaction modeling. Here, we detail how to learn the topic coherence and
temporal dynamics of user interaction in a unified framework such that the
cyberbullying detection performance can be optimized. One empirical
observation is that the affinity level of two comments should be enhanced when
they: (1) discuss consistent and coherent topics; and (2) appear closely in
the time scope. Therefore, we integrate the topical factor $\pi_{topic}$ and
the temporal factor $\pi_{time}$ into the learning of the edge weight
$\pi(\mathbf{r}_{k}^{c},\mathbf{r}_{j}^{c},t_{k,j})$:

(8)
$\pi(\mathbf{r}_{k}^{c},\mathbf{r}_{j}^{c},t_{k,j})=\tanh(\pi_{topic}(\mathbf{r}_{k}^{c},\mathbf{r}_{j}^{c})+\pi_{time}(t_{k,j})).$

Note that for simplicity, we only utilize dot products of two nodes and add
the results to the transformed time intervals. Further explorations of the
edge weight calculation, such as alternative forms of attention mechanism and
different ways of combining the two components, are left as future work. By
modeling user interactions via the bully-featured GAT – GAT that encodes topic
coherence and temporal dynamic jointly, we mimic the information flow in the
social network to capture the evolution of the discussed topics and the
repetitive characteristics of cyberbullying.

###  4.3. Session Classification

The session classification module aims to classify a social media session into
a bullying or non-bullying session. In particular, it first aggregates user’s
history representation $\mathbf{r^{h}}$ and graph representation $\mathbf{g}$
into a unified user representation $\mathbf{u}$. Then it incorporates a user-
level attention network to select informative users to form the final session
representation $\mathbf{s}$, which is used for the final classification task.

While examining users’ historical comments, an important notion is that some
users show similar language behavior over time. For example, a user who
constantly bullies others comments with offensive remarks whenever s/he is
involved in an online discussion. By contrast, a user who occasionally bullies
others may make offensive comments only once. Under such circumstance, her/his
historical comments may not be informative enough to recognize the user’s
bullying tendency in the current session. Considering the relative importance
of historical information for each user, we here use a gating mechanism to
automatically weigh the relative importance of comments in the current session
and user history. For the $j^{th}$ comment in session $s$, the merged user
representation $\mathbf{u}_{j}$ is formulated as below:

(9)
$\mathbf{u}_{j}=\beta_{j}\odot\mathbf{r}_{j}^{h}+(1-\beta_{j})\odot\mathbf{g}_{j},$
(10)
$\beta_{j}=\sigma(\mathbf{W}_{h}\mathbf{r}_{j}^{h}+\mathbf{W}_{g}\mathbf{g}_{j}+b_{c}),$

where $\mathbf{W}_{h}$, $\mathbf{W}_{g}$ and $b_{c}$ are trainable parameters,
$\sigma$ is the sigmoid activation function, and $\beta_{j}$ is the
controlling weight. Since different user comments are not equally relevant to
cyberbullying detection, we aggregate them using a user-level attention
mechanism. The final session representation $\mathbf{s}$ is learned as:

(11)
$\alpha_{j}^{s}=\frac{\exp(\tanh(\mathbf{v}_{s}^{T}\times\mathbf{u}_{j}+b_{s}))}{\Sigma_{k=1}^{n}\exp(\tanh(\mathbf{v}_{s}^{T}\times\mathbf{u}_{k}+b_{s}))},$

(12) $\mathbf{s}=\displaystyle\sum_{j=1}^{n}\alpha_{j}^{s}\mathbf{u}_{j},$

where $\mathbf{v}_{s}$ and $b_{s}$ are parameters of the attention network,
and $\alpha_{j}^{s}$ is the attention weight of the $j^{th}$ user. For
session-level cyberbullying detection, we feed $\mathbf{s}$ into a single-
layer dense network and predict its label $y$, where $y=1$ indicates a
bullying case.

##  5\. Experiments

In this section, we begin by an introduction of the benchmark datasets for
session-based cyberbullying detection. We then detail the experimental setup
including the baselines, evaluation metrics , and implementations. The focus
of this section lies in answering the following questions:

  * •

RQ1: How does TGBully fare against existing cyberbullying detection models
w.r.t. predictive performance?

  * •

RQ2: How does each module of TGBully contribute to the overall model
performance?

  * •

RQ3: As real-world cyberbullying datasets are often imbalanced, how does the
data imputation techniques (e.g., oversampling) influence the session-level
cyberbullying detection?

  * •

RQ4: Apart from the session-level prediction, can TGBully effectively
recognize cyberbullying instances at the comment-level?

###  5.1. Data

Table 1. Statistics of the Instagram and Vine dataset.

Datasets | # Sessions | # Bully | # Non-bully | # Comments
---|---|---|---|---
Instagram | 2,218 | 678 | 1,540 | 155,260
Vine | 970 | 304 | 666 | 78,250

We conduct experiments on two benchmark datasets collected from Instagram
(Hosseinmardi et al., 2015) and Vine (Rafiq et al., 2015). Both datasets were
crawled using a snowball sampling method and manually annotated via the crowd-
sourcing platform CrowdFlower. Definitions of cyberbullying were provided for
five annotators at the beginning of the labeling process. A session is
considered a cyberbullying session if no less than three people labeled it as
bully. Sessions were removed from the dataset if they contain less than 15
comments. Statistics of the two datasets are illustrated in Table 1.

Instagram: Instagram444https://www.instagram.com/ is a media-based social
network where users post, like, and comment on images. It has been reported as
one of the top five networks with the highest percentage of users reporting
experience of cyberbullying (the Label Anti Bullying Charity, 2013). For each
media-session, this dataset provides the labeled image content, the textual
content of comments and the arrival time of comments. In addition, the user of
each comment is identified along with social network properties (user IDs of
their likes, followers, and followings) and historical information (comments
and likes in earlier sessions). In total, this dataset contains 2,218 media
sessions with an average number of 72 comments in each session. For each
comment, the average number of words is 12.

Vine: Vine555https://vine.co/ is a mobile application that allows users to
upload, like, and comment on six-second looping videos. For each video
session, along with the video content, this dataset also records content,
time, and associated user IDs of each comment. However, different from the
Instagram dataset, this dataset doesn’t provide users’ historical comments.
Instead, it provides user profile information such as location, profile
description, and social network properties (user IDs of their followers and
followings). In implementation, we substitute user historical comments in the
history encoding module with their profile description. On average, there are
81 comments per session and 8 words per comment.

Table 2. Evaluation results of different methods on the Instagram dataset.

| 50% | 100%
---|---|---
| Recall | F1 | AUC | Recall | F1 | AUC
LR | 52.74$\pm$4.48 | 64.05$\pm$2.93 | 73.89$\pm$1.94 | 59.06$\pm$3.79 | 69.12$\pm$2.86 | 76.97$\pm$1.75
SVM | 55.93$\pm$4.80 | 66.31$\pm$3.36 | 75.28$\pm$2.20 | 59.32$\pm$3.85 | 69.03$\pm$2.94 | 76.94$\pm$1.77
XGBoost | 69.05$\pm$5.71 | 72.25$\pm$1.59 | 79.63$\pm$1.16 | 71.55$\pm$3.64 | 75.25$\pm$2.09 | 81.68$\pm$1.57
CNN | 67.41$\pm$4.53 | 66.67$\pm$0.95 | 83.39$\pm$1.15 | 70.51$\pm$5.84 | 67.12$\pm$1.53 | 84.00$\pm$1.37
LSTM | 64.18$\pm$2.24 | 66.63$\pm$1.37 | 82.22$\pm$2.45 | 70.01$\pm$8.37 | 67.42$\pm$1.60 | 83.87$\pm$1.59
SelfAtt | 66.73$\pm$6.24 | 72.17$\pm$5.40 | 90.30$\pm$1.92 | 74.05$\pm$4.88 | 74.45$\pm$3.00 | 89.32$\pm$1.37
HAN | 71.69$\pm$4.24 | 75.01$\pm$1.74 | 90.07$\pm$1.40 | 78.21$\pm$7.02 | 76.99$\pm$1.99 | 91.16$\pm$1.02
SICD | 62.53$\pm$4.53 | 69.71$\pm$2.38 | 79.92$\pm$1.27 | 76.87$\pm$3.79 | 70.55$\pm$1.29 | 86.24$\pm$1.16
Soni & Sign | 65.67$\pm$5.32 | 70.68$\pm$1.87 | 84.19$\pm$0.82 | 72.06$\pm$4.01 | 72.59$\pm$1.43 | 88.06$\pm$1.24
HANCD | 78.39$\pm$3.69 | 76.40$\pm$2.17 | 91.63$\pm$2.13 | 76.91$\pm$4.56 | 77.55$\pm$2.14 | 91.32$\pm$1.29
TGBully | 77.64$\pm$4.70 | 78.12$\pm$2.04 | 91.81$\pm$0.98 | 82.57$\pm$3.10 | 80.97$\pm$2.03 | 92.91$\pm$1.30

Table 3. Evaluation results of different methods on the Vine dataset.

| 50% | 100%
---|---|---
| Recall | F1 | AUC | Recall | F1 | AUC
LR | 47.42$\pm$5.32 | 55.92$\pm$3.21 | 63.72$\pm$2.25 | 50.00$\pm$4.06 | 59.65$\pm$3.31 | 65.85$\pm$2.30
SVM | 47.69$\pm$4.98 | 57.22$\pm$3.17 | 64.31$\pm$2.42 | 48.48$\pm$3.28 | 60.38$\pm$2.76 | 66.81$\pm$1.78
XGBoost | 63.11$\pm$4.95 | 61.32$\pm$2.89 | 69.82$\pm$1.88 | 64.54$\pm$3.77 | 64.00$\pm$3.12 | 70.75$\pm$1.42
CNN | 60.50$\pm$6.90 | 53.97$\pm$2.43 | 67.07$\pm$0.70 | 61.61$\pm$2.27 | 56.56$\pm$1.40 | 72.65$\pm$2.01
LSTM | 60.12$\pm$8.23 | 57.76$\pm$1.78 | 73.64$\pm$4.17 | 62.76$\pm$9.77 | 58.91$\pm$2.61 | 73.04$\pm$5.05
SelfAtt | 53.00$\pm$7.29 | 59.46$\pm$4.33 | 81.77$\pm$1.60 | 59.83$\pm$6.56 | 64.49$\pm$3.18 | 81.39$\pm$1.74
HAN | 71.11$\pm$6.28 | 66.54$\pm$2.11 | 82.13$\pm$1.77 | 70.04$\pm$2.93 | 68.10$\pm$2.26 | 82.32$\pm$1.18
SICD | 61.72$\pm$5.24 | 58.43$\pm$5.75 | 76.74$\pm$1.64 | 66.82$\pm$3.92 | 62.76$\pm$2.38 | 78.56$\pm$2.52
Soni & Sign | 64.50$\pm$5.98 | 60.47$\pm$2.43 | 80.27$\pm$1.35 | 64.48$\pm$5.24 | 63.49$\pm$1.97 | 80.33$\pm$3.27
HANCD | 68.39$\pm$3.94 | 67.10$\pm$3.08 | 83.62$\pm$2.08 | 71.05$\pm$4.66 | 68.01$\pm$2.86 | 82.76$\pm$3.13
TGBully | 75.17$\pm$6.52 | 68.64$\pm$3.78 | 83.54$\pm$3.35 | 70.18$\pm$6.65 | 69.35$\pm$2.04 | 83.05$\pm$2.67

###  5.2. Experimental Setup

For each dataset, we randomly sample 80% sessions for training, 10% for
validation, and 10% for testing. We assume each session is independent and do
not consider cross-session correlation. Session length is set as 140 and
sentence length is 30. We use the pretrained 400-dim Twitter embedding (Godin
et al., 2015) as the textual input. The sentence-level Bi-GRU has 2$\times$32
hidden states and the session-level Bi-GRU has 2$\times$64\. The dropout rate
is 0.2. We repeat each experiment 5 times and report their average
performances and standard deviations.

Evaluation Metrics. We use three widely recognized evaluation metrics in
cyberbullying detection tasks, i.e., recall, micro F1, and AUC. We pay special
attention to recall scores as the frequency of bullying sessions in the real-
world is typically much lower compared to that of non-bullying sessions.
Further, we report the micro F1 score for the interest of cyberbullying
instances.

Baselines. We compare TGBully with three categories of baselines: feature-
based models (LR, SVM and XGBoost), neural text classification models (CNN,
LSTM, SelfAtt and HAN), and cyberbullying detection models (SICD, Soni & Sign
and HANCD). Specifically,

  * •

LR. Logistic Regression is a commonly used binary classifier. It estimates and
optimizes parameters of a logistic model.

  * •

SVM (Cortes and Vapnik, 1995). Support Vector Machine (SVM) is a non-
probabilistic binary classifier. It maximizes the functional margin between
positive and negative classes.

  * •

XGBoost (Chen and Guestrin, 2016). It is a highly scalable and efficient tree
boosting system. The decision tree-based algorithm uses several regularization
and parallelization techniques to improve overall performances.

  * •

CNN (Kim, 2014). The original implementation uses convolutional neural network
(CNN) to classify single sentences. We apply it at comment level and average
comment representations to form session representations.

  * •

LSTM (Hochreiter and Schmidhuber, 1997). It designs long short-term memory
units for context modeling. Similarly, we use it to model comments and
classify sessions with averaged comment representations.

  * •

SelfAtt (Vaswani et al., 2017). It incorporates multi-head self-attention
mechanism to model context dependency in sentences. In addition to word-level
attention, we apply a similar comment-level attention to model relation
between different comments.

  * •

HAN (Yang et al., 2016). A hierarchical bi-directional GRU framework is used
to model sessions at both word level and comment level.

  * •

SICD (Dani et al., 2017). A sentiment enhanced feature-based cyberbullying
model. User-post relationship and sentiment similarity between posts are used
to guide the prediction.

  * •

Soni & Sign (Soni and Singh, 2018). A cyberbullying model that exploits
temporal information and activity levels in sessions. Textual, social and
temporal features are simultaneously utilized for classification.

  * •

HANCD (Cheng et al., 2019). A multi-task cyberbullying model based on HAN. It
adds a time interval prediction subtask to exploit temporal dynamics within
sessions.

For LR, SVM, and XGBoost, we use word-level tf-idf of user comments as textual
features and number of likes and follows as social features. We use the same
word embedding for TGBully and all neural text classification models.
Hyperparameters are tuned to optimize performance. We set the parameters as
reported in the original papers for all cyberbullying detection models. One of
the challenges in cyberbullying detection is the data scarcity problem.
Therefore, to investigate how much these models rely on the labeled data, we
also construct another training set by randomly sampling half of the original
training data. We report results for both cases with half and full training
data.

###  5.3. Performance Evaluation

To answer the question in RQ1, we compare the performances of all baseline
methods with TGBully’s on the Instagram and Vine datasets. The results are
reported in Table 2 and 3.

Comparing TGBully with other cyberbullying baselines that also consider
temporal analysis, we observe that: (1) TGBully consistently achieves the best
performances in most cases with different datasets and training ratios. This
is in part because our approach effectively models user interactions in a
graph attention neural network with a focus on temporal dynamics and topic
coherence of a session. The joint modeling of the two bully-indicative aspects
further leads to better recognition of the repetitive characteristics of
cyberbullying behavior. The improvement also comes from the modeling of user
personality traits and language behavior from users’ historical comments while
learning user interaction. (2) Large variations are observed in the results of
all methods, especially when there is less training data. This reveals the
necessity of using larger datasets in order to further improve the performance
of cyberbullying detection. This result is also evidenced by recent findings
in (Cheng et al., 2020b), which states that TGBully improves the performance
by a smaller margin over baselines with the Vine dataset as Vine has fewer
samples and we do not have access to the historical comments of its users.

By comparing different methods’ performances, we also conclude that: (1)
Neural models achieve substantially better performances than feature-based
methods, especially w.r.t. Recall and AUC. Rather than representing session
information with raw features, neural networks incorporate pretrained word
embeddings that have rich semantic meanings and are fine-tuned during the
training process. Another important reason is that neural models can better
capture context dependency between words, leading to high-quality session
representations, even with highly imbalanced datasets. (2) Another observation
is that explicitly modeling the hierarchical structure can further improve the
performance. Hierarchical neural models (SelfAtt, HAN, HANCD, and TGBully)
outperform standard neural models (CNN and LSTM) partly because they better
capture the interaction between different comments and consider their relative
importance in a session. (3) Introducing additional temporal and sentiment
information enhances both feature-based models and text classification models.
For instance, SICD and Soni & Sign introduce sentiment and temporal modeling
respectively, and both methods outperform LR and SVM methods by a large
margin. Similarly, HANCD and TGBully improve over HAN model by modeling the
temporal dynamics in a session.

Figure 3. TGBully and its variants on the Instagram dataset.

(a) A bullying example.

(b) A non-bullying example.

Figure 4. Visualization of various types of weights within a session. Shades
of purple and blue denote user-level and word-level attention weights
respectively. Green mesh grids denote graph weights between users (i.e., their
comments), where grid at the $i^{th}$ row and $j^{th}$ column represents
weight $w_{ij}$. Darker color represents higher weights.

###  5.4. Ablation Study

This set of experiments seeks to examine the impact of each module of TGBully
on the overall performance (RQ2). Here, we use the Instagram dataset as a
working example because similar findings can be observed with Vine. Results
are reported in Fig. 3. We use the leave-one-out scheme to evaluate the impact
of each of the three aspects – topic coherence, temporal dynamics, and
historical comments – on the performance of TGBully. We also include HAN for
reference because TGBully will be converted to HAN when the modeling of the
three aspects is removed simultaneously.

From Fig. 3, we observe that all three aspects are important to session-level
cyberbullying detection. In particular, the performance of TGBully exacerbates
the most when topic coherence is removed from the learning process. Therefore,
it is crucial to model semantic interaction between comments. By explicitly
connecting the edge weights to the modeling of topic coherence, TGBully
enforces the correlations between semantically related comments meanwhile
reduces the noise induced by unrelated comments. Modeling temporal dynamics
also brings substantial performance improvement. Results show that
transforming time intervals between comments into edge weights in the temporal
graph can help capture the evolving characteristics of sessions. Moreover, the
consideration of historical comments also enhances the performance of TGBully,
indicating the effectiveness of modeling users’ personalities and language
behavior for cyberbullying detection. By combining all three aspects together,
TGBully can achieve the best performance overall.

###  5.5. Influence of Data Imbalance

Cyberbullying datasets are typically imbalanced. A common solution to
extremely skewed data is data imputation, e.g., data oversampling. To further
examine how the common data imputation techniques impact the model performance
(RQ3), we artificially create a balanced dataset by oversampling data from the
minority class. Specifically, We use SMOTE (Chawla et al., 2002) to synthesize
new bullying examples in the feature space and illustrate the performance
comparisons of various methods in Table 4. We observe that oversampling brings
converse effects to different methods. It improves performances of LR, SVM,
XGBoost, and LSTM, and decreases that of HAN, HANCD, and TGBully. Two reasons
may account for this. First, methods such as LR, SVM, XGBoost, and LSTM only
exploit semantic information of comments. Since they capture information from
a single modality, their classifiers may benefit from a more balanced data
distribution in the feature space. Second, other approaches (HAN, HANCD, and
TGBully) exploit multi-model information, e.g., comment hierarchy, temporal
dynamics , and user history. Nevertheless, synthetic data assumes independence
between different modalities and ignores their correlation. Thus, they may add
extra noise to models and harm their performances.

Table 4. Performance comparison of different methods with oversampled dataset.

| Oversampled | Original
---|---|---
| Recall | F1 | AUC | Recall | F1 | AUC
LR | 69.50 | 71.27 | 78.81 | 59.06 | 69.12 | 76.97
SVM | 60.99 | 69.08 | 77.06 | 59.32 | 69.03 | 76.94
XGBoost | 73.05 | 75.74 | 81.90 | 71.55 | 75.25 | 81.68
LSTM | 70.44 | 68.75 | 85.25 | 70.01 | 67.42 | 83.87
HAN | 77.93 | 76.75 | 91.47 | 78.21 | 76.99 | 91.16
HANCD | 77.36 | 77.30 | 90.66 | 76.91 | 77.55 | 91.32
TGBully | 79.02 | 78.67 | 91.81 | 82.57 | 80.97 | 92.91

###  5.6. Case Study

To answer RQ4, we investigate the capability of our model to detect bullying
instances at the comment-level. Here, we use the comment-level attention
weights that indicate the relative importance of each user comment for
classification. The underlying assumption is that comments with higher
attention weights are more likely to be bullying instances. To validate this,
we ask three graduate students (two from computer science and one from
psychology) to manually annotate the user comments in a total number of 100
sessions. The final label of each comment is determined by a majority vote.
Among the 100 sessions with comment-level labels, we randomly select a
cyberbullying session and a non-bullying session for illustration. We aim to
visually testify the effectiveness of our method and answer RQ4. The results
can be seen in Fig. 4. In addition to the user-level attention weights, we
also present graph weights and word-level attention weights in TGBully.

From Fig. 4, we make the following observations: (1) TGBully can effectively
recognize bullying users (comments) based on the learned attention weights. As
illustrated in Fig. 4(a), as the $2^{nd}$, $4^{th}$ and $5^{th}$ comments
contain humiliating words, they are predicted with higher weights indicating a
high probability of being bullying comments. We found that the results are
lined with ground-truth labels. In contrast, comments in Fig. 4(b) are non-
bullying and their attention weights are more evenly distributed. (2) TGBully
encodes temporal dynamics and topic coherence between comments via the GAT.
Comparing Fig. 4(a) and Fig. 4(b), we find that graph weights of bullying
sessions are relatively dense and concentrate on specific nodes. For instance,
the $2^{nd}$, $4^{th}$ and $5^{th}$ comments convey similar semantic meaning.
Accordingly, their interactions are highlighted by the graph weights.
Meanwhile, apart from being coherent in topic, the $4^{th}$ and $5^{th}$
comments are also consecutive in time. As a result, their correlations are
more obvious than that between the $2^{nd}$ and $4^{th}$ comments. By
contrast, weights of the non-bullying session largely concentrate on
diagonals, indicating that they tend to focus on self interactions of comments
and reflect limited cross-comment interactions. Despite the overlapped words
and temporal proximity between comments, our method doesn’t capture
significant cross-comment interaction. This may happen because the GAT model
only aims to capture specific bully-related interaction. (3) TGBully
effectively selects offensive words. For instance, words such as “terrible”,
“ni**a”, and “bit*h” in Fig. 4(b) have larger attention weights. We also
notice that TGBully recognizes some salient words in a non-bullying session,
which indicates it is discriminative to bullying words.

##  6\. Conclusion and Future Work

In this work, we make a novel observation that in a social media session, user
interaction is embodied by the relations between their posted comments, and
such interaction dynamically evolves with time. We further show that the topic
coherence and temporal dynamics in the modeling of user interactions jointly
contribute to capturing the repetitive characteristic of cyberbullying
behavior. The core contribution of this work is the proposed TGBully, a
temporal graph-based cyberbullying detection framework with a focus on user
interaction. TGBully consists of three modules. The first semantic context
modeling module hierarchically constructs user comments from sequences of
words and infers users’ language behavior and personality from their
historical comments. The second temporal graph interaction learning module
first constructs the user interaction graph. It then encodes the topic
coherence and temporal proximity through a bully-featured GAT. The third
session classification module aggregates the information learned from user
interaction in a session and performs the final session-level classification.
We empirically evaluate the effectiveness of our approach with the tasks of
session-level bullying detection and comment-level case study.

Our work opens several key avenues for future work. One such direction is to
combine the explicit reply/social network with the introduced implicit
temporal graph to together model the user interaction in a social media
session. Additionally, we can incorporate other important aspects of
cyberbullying into the temporal interaction graph. For instance, we can
consider different roles of users (e.g., bullies and victims) and study how
the role of a user evolves over time in the graph modeling process. Further
investigations are needed to apply TGBully to detecting comment-level
cyberbullying instances at scale.

##  7\. Ethics Statement

The common understanding of the importance of data privacy and security is
shared by the authors. To secure user privacy, we replaced usernames and ids
(include their mentioning in other users’ comments) with numerical indices
before the annotating and prepossessing process. The insulting or offensive
terms and the figures used in this paper are for illustration only, they do
not represent the ethical attitude of the authors.

## ACKNOWLEDGEMENTS

This material is based upon work supported by the Office of Naval Research
(ONR) Grant N00014-21-1-4002, Association for Research Libraries (ARL) Grant
W911NF2020124, and National Science Foundation (NSF) Grant #2036127.

## References

  * (1)
  * Al-garadi et al. (2016) Mohammed Ali Al-garadi, Kasturi Dewi Varathan, and Sri Devi Ravana. 2016.  Cybercrime detection in online communications: The experimental case of cyberbullying detection in the Twitter network.  _Computers in Human Behavior_ 63 (2016), 433–443.
  * Al Mazari (2013) Ali Al Mazari. 2013\.  Cyber-bullying taxonomies: Definition, forms, consequences and mitigation strategies. In _2013 5th International Conference on Computer Science and Information Technology_. IEEE, 126–133.
  * Arıcak (2009) Osman Tolga Arıcak. 2009\.  Psychiatric symptomatology as a predictor of cyberbullying among university students.  _Eurasian Journal of Educational Research (EJER)_ 34 (2009).
  * Chatzakou et al. (2017) Despoina Chatzakou, Nicolas Kourtellis, Jeremy Blackburn, Emiliano De Cristofaro, Gianluca Stringhini, and Athena Vakali. 2017\.  Mean birds: Detecting aggression and bullying on twitter. In _Proceedings of the 2017 ACM on web science conference_. 13–22.
  * Chawla et al. (2002) Nitesh V Chawla, Kevin W Bowyer, Lawrence O Hall, and W Philip Kegelmeyer. 2002.  SMOTE: synthetic minority over-sampling technique.  _JAIR_ 16 (2002), 321–357.
  * Chen and Guestrin (2016) Tianqi Chen and Carlos Guestrin. 2016.  Xgboost: A scalable tree boosting system. In _SIGKDD 2016_. 785–794.
  * Cheng et al. (2019) Lu Cheng, Ruocheng Guo, Yasin Silva, Deborah Hall, and Huan Liu. 2019.  Hierarchical attention networks for cyberbullying detection on the instagram social network. In _SIAM 2019_. SIAM, 235–243.
  * Cheng et al. (2020a) Lu Cheng, Kai Shu, Siqi Wu, Yasin N Silva, Deborah L Hall, and Huan Liu. 2020a.  Unsupervised Cyberbullying Detection via Time-Informed Gaussian Mixture Model. In _CIKM 2020_.
  * Cheng et al. (2020b) Lu Cheng, Yasin Silva, Deborah Hall, and Huan Liu. 2020b.  Session-based Cyberbullying Detection: Problems and Challenges.  _IEEE Internet Computing, Special Issue on Cyber-Social Health: Promoting Good and Countering Harm on Social Media_ (2020).
  * Cho et al. (2014) Kyunghyun Cho, Bart van Merriënboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. 2014\.  Learning Phrase Representations using RNN Encoder–Decoder for Statistical Machine Translation. In _EMNLP 2014_. 1724–1734.
  * Corcoran et al. (2012) Lucie Corcoran, Irene Connolly, and Mona O’Moore. 2012\.  Cyberbullying in Irish schools: An investigation of personality and self-concept.  _The Irish Journal of Psychology_ 33, 4 (2012), 153–165.
  * Cortes and Vapnik (1995) Corinna Cortes and Vladimir Vapnik. 1995.  Support-vector networks.  _Machine learning_ 20, 3 (1995), 273–297.
  * Dani et al. (2017) Harsh Dani, Jundong Li, and Huan Liu. 2017.  Sentiment informed cyberbullying detection in social media. In _ECML-PKDD 2017_. Springer, 52–67.
  * Dinakar et al. (2012) Karthik Dinakar, Birago Jones, Catherine Havasi, Henry Lieberman, and Rosalind Picard. 2012\.  Common sense reasoning for detection, prevention, and mitigation of cyberbullying.  _TIIS_ 2, 3 (2012), 1–30.
  * Dinakar et al. (2011) Karthik Dinakar, Roi Reichart, and Henry Lieberman. 2011\.  Modeling the detection of textual cyberbullying. In _Proceedings of the Social Mobile Web_. Citeseer.
  * Dooley et al. (2009) Julian J Dooley, Jacek Pyżalski, and Donna Cross. 2009\.  Cyberbullying versus face-to-face bullying: A theoretical and conceptual review.  _Zeitschrift für Psychologie/Journal of Psychology_ 217, 4 (2009), 182–188.
  * for Disease Control and Prevention (2019) Centers for Disease Control and Prevention. 2019.  Youth Risk Behavior Survey Data.  https://www.cdc.gov/healthyyouth/data/yrbs/
  * Godin et al. (2015) Fréderic Godin, Baptist Vandersmissen, Wesley De Neve, and Rik Van de Walle. 2015.  Multimedia Lab @ ACL WNUT NER Shared Task: Named Entity Recognition for Twitter Microposts using Distributed Word Representations. In _Proceedings of the Workshop on Noisy User-generated Text_. 146–153.  https://doi.org/10.18653/v1/W15-4322
  * Golbeck et al. (2011) Jennifer Golbeck, Cristina Robles, Michon Edmondson, and Karen Turner. 2011.  Predicting personality from twitter. In _2011 IEEE third international conference on privacy, security, risk and trust and 2011 IEEE third international conference on social computing_. IEEE, 149–156.
  * Gruber and Jockisch (2020) Nicole Gruber and Alfred Jockisch. 2020.  Are GRU Cells More Specific and LSTM Cells More Sensitive in Motive Classification of Text.  _J. Front. Artif. Intell_ 3 (2020), 1–6.
  * Gupta et al. (2020) Aabhaas Gupta, Wenxi Yang, Divya Sivakumar, Yasin N Silva, Deborah L Hall, and Maria Camila Nardini Barioni. 2020.  Temporal Properties of Cyberbullying on Instagram. In _The WebConf 2020 Companion_.
  * Hamilton et al. (2017) Will Hamilton, Zhitao Ying, and Jure Leskovec. 2017\.  Inductive representation learning on large graphs. In _NIPS 2017_. 1024–1034.
  * Hochreiter and Schmidhuber (1997) Sepp Hochreiter and Jürgen Schmidhuber. 1997.  Long short-term memory.  _Neural computation_ 9, 8 (1997), 1735–1780.
  * Hosseinmardi et al. (2014) Homa Hosseinmardi, Richard Han, Qin Lv, Shivakant Mishra, and Amir Ghasemianlangroodi. 2014.  Analyzing negative user behavior in a semi-anonymous social network.  _CoRR abs_ 1404 (2014).
  * Hosseinmardi et al. (2015) Homa Hosseinmardi, Sabrina Arredondo Mattson, Rahat Ibn Rafiq, Richard Han, Qin Lv, and Shivakant Mishra. 2015.  Analyzing labeled cyberbullying incidents on the instagram social network. In _International conference on social informatics_. Springer, 49–66.
  * Hu et al. (2019) Linmei Hu, Tianchi Yang, Chuan Shi, Houye Ji, and Xiaoli Li. 2019.  Heterogeneous graph attention networks for semi-supervised short text classification. In _EMNLP-IJCNLP 2019_. 4823–4832.
  * Huang and Carley (2019) Binxuan Huang and Kathleen M Carley. 2019.  Syntax-Aware Aspect Level Sentiment Classification with Graph Attention Networks. In _EMNLP-IJCNLP 2019_. 5472–5480.
  * Huang et al. (2014) Qianjia Huang, Vivek Kumar Singh, and Pradeep Kumar Atrey. 2014\.  Cyber bullying detection using social and textual analysis. In _Proceedings of the 3rd International Workshop on Socially-Aware Multimedia_. 3–6.
  * Huberman et al. (2009) Bernardo Huberman, Daniel M Romero, and Fang Wu. 2009\.  Social networks that matter: Twitter under the microscope.  _First Monday_ (2009).
  * Kim (2014) Yoon Kim. 2014.  Convolutional Neural Networks for Sentence Classification. In _EMNLP 2014_. 1746–1751.
  * Kowalski et al. (2014) Robin M Kowalski, Gary W Giumetti, Amber N Schroeder, and Micah R Lattanner. 2014.  Bullying in the digital age: A critical review and meta-analysis of cyberbullying research among youth.  _Psychological bulletin_ 140, 4 (2014), 1073\.
  * Li et al. (2015) Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. 2015\.  Gated graph sequence neural networks. In _ICLR 2016_.
  * Li et al. (2016) Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. 2016\.  Semi-supervised classification with graph convolutional networks. In _ICLR 2017_.
  * Liu et al. (2020) Zhenghao Liu, Chenyan Xiong, Maosong Sun, and Zhiyuan Liu. 2020\.  Fine-grained fact verification with kernel graph attention network. In _ACL 2020_. 7342–7351.
  * Nand et al. (2016) Parma Nand, Rivindu Perera, and Abhijeet Kasture. 2016\.  “How Bullying is this Message?”: A Psychometric Thermometer for Bullying. In _COLING 2016_. 695–706.
  * Rafiq et al. (2015) Rahat Ibn Rafiq, Homa Hosseinmardi, Richard Han, Qin Lv, Shivakant Mishra, and Sabrina Arredondo Mattson. 2015.  Careful what you share in six seconds: Detecting cyberbullying instances in Vine. In _ASONAM 2015_. IEEE, 617–622.
  * Ravazzi et al. (2017) Chiara Ravazzi, Roberto Tempo, and Fabrizio Dabbene. 2017\.  Learning influence structure in sparse social networks.  _IEEE Transactions on Control of Network Systems_ 5, 4 (2017), 1976–1986.
  * Romsaiyud et al. (2017) Walisa Romsaiyud, Kodchakorn na Nakornphanom, Pimpaka Prasertsilp, Piyaporn Nurarak, and Pirom Konglerd. 2017.  Automated cyberbullying detection using clustering appearance patterns. In _2017 9th International Conference on Knowledge and Smart Technology_. IEEE, 242–247.
  * Salmivalli (1999) Christina Salmivalli. 1999\.  Participant role approach to school bullying: Implications for interventions.  _Journal of adolescence_ 22, 4 (1999), 453–459.
  * Smith et al. (2008) Peter K Smith, Jess Mahdavi, Manuel Carvalho, Sonja Fisher, Shanette Russell, and Neil Tippett. 2008\.  Cyberbullying: Its nature and impact in secondary school pupils.  _Journal of child psychology and psychiatry_ 49, 4 (2008), 376–385.
  * Soni and Singh (2018) Devin Soni and Vivek Singh. 2018.  Time reveals all wounds: Modeling temporal dynamics of cyberbullying sessions. In _ICWSM 2018_. AAAI press, 684–687.
  * the Label Anti Bullying Charity (2013) Ditch the Label Anti Bullying Charity. 2013\.  Ditch the Label Anti Bullying Charity: The annual cyberbullying survey 2013.  https://www.ditchthelabel.org/wp-content/uploads/2016/07/cyberbullying2013.pdf.  Accessed: 2020-09-18.
  * US Legal (2019) Inc. US Legal. 2019\.  Cyber Bullying Law and Legal Definition.  https://definitions.uslegal.com/c/cyber-bullying/
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need. In _NIPS 2017_. 5998–6008.
  * Veličković et al. (2017) Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. 2017.  Graph attention networks. In _NIPS 2018_.
  * Veličković et al. (2018) Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. 2018.  Graph Attention Networks. In _ICLR 2018_.
  * Wu et al. (2010) Hao Wu, Jiajun Bu, Chun Chen, Can Wang, Guang Qiu, Lijun Zhang, and Jianfeng Shen. 2010.  Modeling dynamic multi-topic discussions in online forums. In _AAAI 2010_. Citeseer.
  * Xu et al. (2012) Jun-Ming Xu, Kwang-Sung Jun, Xiaojin Zhu, and Amy Bellmore. 2012\.  Learning from Bullying Traces in Social Media. In _NAACL 2012_. Association for Computational Linguistics, 656–666.
  * Yang et al. (2016) Zichao Yang, Diyi Yang, Chris Dyer, Xiaodong He, Alex Smola, and Eduard Hovy. 2016\.  Hierarchical attention networks for document classification. In _NAACL 2016_. 1480–1489.
  * Zhao and Mao (2016) Rui Zhao and Kezhi Mao. 2016.  Cyberbullying detection based on semantic-enhanced marginalized denoising auto-encoder.  _IEEE Transactions on Affective Computing_ 8, 3 (2016), 328–339.
  * Zhong et al. (2020) Wanjun Zhong, Jingjing Xu, Duyu Tang, Zenan Xu, Nan Duan, Ming Zhou, Jiahai Wang, and Jian Yin. 2020\.  Reasoning over semantic-level graph for fact checking. In _ACL 2020_. 6170–6180.
  * Zhou et al. (2019) Jie Zhou, Xu Han, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun. 2019.  GEAR: Graph-based Evidence Aggregating and Reasoning for Fact Verification. In _ACL 2019_. 892–901.
  * Ziems et al. (2020) Caleb Ziems, Ymir Vigfusson, and Fred Morstatter. 2020\.  Aggressive, Repetitive, Intentional, Visible, and Imbalanced: Refining Representations for Cyberbullying Classification. In _ICWSM 20_ , Vol. 14. 808–819.