# On the Aggression Diffusion Modeling and Minimization in Online Social
Networks

Marinos Poiitis<EMAIL_ADDRESS>0000-0002-9810-2907 Aristotle University
of ThessalonikiThessalonikiGreece ,  Athena Vakali<EMAIL_ADDRESS>1234-5678-9012 Aristotle University of ThessalonikiThessalonikiGreece and
Nicolas Kourtellis<EMAIL_ADDRESS>1234-5678-9012
Telefonica ResearchBarcelonaSpain

###### Abstract.

Aggression in online social networks has been studied mostly from the
perspective of machine learning which detects such behavior in a static
context. However, the way aggression diffuses in the network has received
little attention as it embeds modeling challenges. In fact, modeling how
aggression propagates from one user to another, is an important research topic
since it can enable effective aggression monitoring, especially in media
platforms which up to now apply simplistic user blocking techniques.

In this paper, we address aggression propagation modeling and minimization on
Twitter, since it is a popular microblogging platform at which aggression had
several onsets. We propose various methods building on two well-known
diffusion models, Independent Cascade ($IC$) and Linear Threshold ($LT$), to
study the aggression evolution in the social network. We experimentally
investigate how well each method can model aggression propagation using real
Twitter data, while varying parameters, such as seed users selection, graph
edge weighting, users’ activation timing, etc. It is found that the best
performing strategies are the ones to select seed users with a degree-based
approach, weigh user edges based on their social circles’ overlaps, and
activate users according to their aggression levels. We further employ the
best performing models to predict which ordinary real users could become
aggressive (and vice versa) in the future, and achieve up to $AUC$=0.89 in
this prediction task. Finally, we investigate aggression minimization by
launching competitive cascades to “inform” and “heal” aggressors. We show that
$IC$ and $LT$ models can be used in aggression minimization, providing less
intrusive alternatives to the blocking techniques currently employed by
popular online social network platforms.

social networks, information diffusion, aggression modeling, aggression
minimization, cascades, immunization

††ccs: Information systems Social networks††ccs: Information systems World
Wide Web††ccs: Computing methodologies Network science††ccs: Computing
methodologies Modeling and simulation

##  1\. Introduction

Online social media offer unprecedented communication opportunities, but also
come with unfortunate malicious behaviors. Cyberbullying, racism, hate speech
and discrimination are some of the online aggressive behaviors manifesting in
such platforms and often have devastating consequences for individual users
and the society as a whole. Aggression can be explicit through inappropriate
posting such as negative feelings and embarrassing photos, or implicit when it
unconsciously hurts online users (e.g., through negative gossip spreading).
Overall, online social media users are often left exposed and vulnerable to
potential aggression threats.

Inter-disciplinary studies have focused on cyber aggression from the
perspective of social psychology and computational sciences. Social learning
and bonding (Allen and Anderson, 2017; Smith et al., 2008), as well as the
theory of planned behavior (Lee and Lee, 2002) provide the basis of its
theoretical formulation. Furthermore, machine learning has been used to detect
such behavior in online platforms (e.g. (Chatzakou et al., 2017; Davidson et
al., 2017; Founta et al., 2018; Waseem and Hovy, 2016)). Even with all this
body of earlier work, online aggression has not been uniformly defined
(Corcoran et al., 2015). Hence, online aggression is formulated under varying
approaches, depending on the severity of the aggressive behavior, the type of
platform and social interactions it facilitates, the power of the aggressor
over the victim etc.

Interestingly, aggressive behavior has been found to be tightly related to the
social circle of the individual expressing it (Huang et al., 2014), while its
origins are located in the aggressive peer influence and the effect of the
initial aggressors on their network (Runions and Bak, 2015), suggesting a
potential _cascading spreading trend_ from the source of the behavior to its
connections. Additionally, the above findings have been expanded by analyzing
the underlying network structure and the pair-wise interactions, further
enhancing the intuition that _online aggression propagates akin to a diffusion
process_. Therefore, the aggression’s overall effect can be strengthened as it
propagates through the network (Henneberger et al., 2017). Surprisingly,
aggression propagation in the cyber-space has gathered little attention,
primarily due to the complexity of the problem. This is evident from the lack
of automated processes in the popular media platforms to mitigate aggression’s
negative effects, which are restricted to blocking or reporting abusive users
and removing inappropriate content (Basak et al., 2019).

The present work focuses on the online aggression propagation problem and
studies the complexities of modeling and minimizing aggression propagation. It
considers the well known diffusion models of Independent Cascading (IC) and
Linear Threshold (LT) (Kempe et al., 2003) as they provide the building blocks
for studying the diffusion process at its two fundamental types of
interactions: user-user ($IC$) and user-neighborhood ($LT$). To this end, we
formulate aggression-aware information diffusion through appropriate
parameters and enable a thorough study of the aggression dynamics. Then, we
show how $IC$ and $LT$ models can be used in aggression minimization,
providing less intrusive alternatives to the techniques currently in use by
various social media platforms.

Figure 1. The overall process of the current work. Initially, aggression
modeling is studied by setting and experimenting on its various sub-
components, including selection of seed nodes, weighting scheme of edges,
transfer of aggression and activation of nodes. The best models can then be
used to minimize the overall network’s aggression with further configurations
by applying healing strategies. Finally, the best methods are compared to the
currently used blocking mechanisms.

The overall process is outlined in Figure 1. Initially, aggression diffusion
modeling is examined, where its sub-components - seed selection strategy,
weighting scheme, activation criterion/threshold strategy and aggression
transferring - are defined and explored. The process leads to two models, an
$IC$ and an $LT$-based, with the best expressive performance. Next, aggression
minimization step exploits these two models to define and examine the method
of competitive cascades and its dynamics (i.e., healing strategy) as an
alternative to the blocking methods, which are currently used by social media
platforms, to reduce the overall aggression level of the social network.

To address the above considerations regarding online aggression diffusion and
minimization, the main contributions of this work are:

  1. (1)

C1: aggression diffusion theoretical foundation, by exploring $IC$ and $LT$ as
the basic propagation models. Upon them, the theoretical notions of user-user
interaction, initial user selection strategies and user propagation are
introduced and adjusted accordingly. The same models are also utilized in
aggression minimization methods, where various healing approaches are
formulated (Section 3).

  2. (2)

C2: aggression modeling and minimization experimentation, for both $IC$ and
$LT$, through similarity performance tests, statistical validation as well as
aggression reduction specifically for the minimization process. Real data
experimentation, extensive simulations, a modeling case study and comparison
to the blocking minimization methods currently in use by social platforms
support our findings (Section 4).

  3. (3)

C3: Exploration on results and implications, by validating the experimental
results for both modeling and minimization. Specifically, neighborhood
similarity is shown to be the most appropriate criterion according to which
user relationships are formed, while central users are the best to initiate
the diffusion process. Finally, the adjusted competitive cascades are shown to
outperform current social media banning methods achieving a reduction of
$\sim$50% for $IC$ and $\sim$15% for $LT$ (Section 4).

  4. (4)

C4: Simulation framework release for reproducibility purposes and further
experimentation or
extensions111https://anonymous.4open.science/r/95063f31-d3aa-4171-80d7-e0efb890476d/.

The rest of this paper is organized as follows. In Section 2 the related works
are reviewed. Section 3 provides the theoretical foundation of both aggression
modeling and minimization. The experimental evaluation is presented in Section
4 and Section 5 provides a discussion of this work’s results and further
improvements.

##  2\. Related Work

Information diffusion has been originally studied by probabilistic models
(Domingos and Richardson, 2001; Richardson and Domingos, 2002), which were
advanced by considering a discrete optimization problem formulation driven by
the well-known Independent Cascade ($IC$) and Linear Threshold ($LT$)
diffusion models (Kempe et al., 2003). These models capture not only the
neighborhood influence, but also the cumulative influence of a user’s social
circle where online aggression phenomena can occur (Henneberger et al., 2017).

Opinion Dynamics ($OD$) is another family of information diffusion models
which studies the effects and conditions of information propagation over the
network. Under $OD$, neighborhood influence is crucial for the diffusion
process (Fernández-Gracia et al., 2014). Another set of models are the well
known Susceptible, Infected, Recovered (SIR) models from the field of
epidemics. They are a generalization of $IC$, categorizing users into the
above three states and studying the transitions from one state to another,
under specific thresholds (Trpevski et al., 2010). Both OD and SIR are useful
in understanding the dynamics of diffusion process and determining proper
aggression propagation models. However, $OD$ methods are theoretically
inappropriate for online aggression modeling, as discussed below in Section
2.4, while a more generic investigation of the various $SIR$ models is
voluminous and is left as future work.

###  2.1. Influence Maximization

Influence Maximization proposes a real-world alternative to the above models.
Specifically, under some limitations in resources - and precisely in the
number of seed nodes that the propagation process could be initiated from
(Tong et al., 2017) \- it aims to maximize the expected influence in the
network (Borodin et al., 2010).

As a result, Influence Maximization aims at optimizing the seed selection
process by minimizing the selection cost (Chen, 2009; Long and Wong, 2011), or
even sustaining the spread of the cascade above a specified threshold (Long
and Wong, 2011). For this purpose, the greedy approach is shown to outperform
methods based on network and node properties (Kempe et al., 2003), however
many optimization methods have been proposed as the simple greedy version is
computationally infeasible. In particular Cost-Effective Lazy Forward
selection ($CELF$) leverages the mathematical set property of submodularity,
achieving an 700% improvement in execution time (Leskovec et al., 2007), while
Single Discount (SD) and Degree Discount (DD) heuristics outperform $CELF$ in
terms of execution time by more than six orders of magnitude while maintaining
a high influence spread (Chen et al., 2009). In particular, $SD$ finds the
most central nodes based on an iterative selection and removal process, while
$DD$ is fine-tuned towards $IC$ model and exploits its user activation
probability.

Besides $IC$ and $LT$ models, Simulated Annealing optimizes the spread
function of the diffusion process both in terms of time and magnitude through
a local search algorithm (Jiang et al., 2011), while heat diffusion is
utilized to simulate information diffusion (Ma et al., 2008). These natural
approaches introduce the decaying effect of time on a user’s influence and
they have largely inspired the Decaying Aggression Transfer healing mechanism
proposed in our work.

###  2.2. Competitive Cascades

In both information diffusion and influence maximization, the presence of a
parallel, sometimes negative cascade has been mentioned. Under this scenario,
each cascade selects a seed set to initiate its propagation and depending on
the assumed hypotheses, they either compete to activate a node or they spread
regardless of the competitor and measure the impact at the end of both
processes. These are known as Competitive Cascades and help us formulate the
first part of our proposed aggression minimization process.

Competitive Influence Maximization: The most usual task of competitive
cascades is to find a positive seed set that minimizes the spread of the
negative cascade (Wu and Pan, 2017; He et al., 2012). This problem is called
Influence Blocking Maximization (IBM) and has been addressed with both $IC$
and $LT$. However, these methods retrace to the discussed intractable greedy
algorithm, thus they either propose tailor-made problem variations of IBM (Wu
and Pan, 2017), or they introduce model specific structures, such as the Local
Directed Acyclic Graph, applicable only on $LT$ (He et al., 2012).
Furthermore, preference for $LT$ and authorities for $IC$ have been introduced
to address simultaneous influence of both cascades (Zhang et al., 2015; Budak
et al., 2011; Wu and Pan, 2017). These notions are useful in this work’s
competitive aggression minimization process.

###  2.3. Immunization

Social media nowadays tend to address abusive behavior restriction by blocking
abusive users or posts. This preference is grounded on either the simpler
nature of those algorithms, or due to their more straightforward application.
Hence, apart from the competitive cascades, the latest immunization or
blocking algorithms are also reviewed in this work and compared to discover
the most efficient approach in terms of aggression reduction.

State-of-the-art methods exploit the largest eigenvalue $\lambda$ of the
network adjacency matrix A which has been shown to be the only network
parameter that determines whether the diffusion would become an epidemic or
not (Prakash et al., 2012). Specifically, NetShield algorithm introduces a
ranking score called Shield score to detect the most important abusive users
(Tong et al., 2010). Furthermore, NetMelt transfers the problem to edge
deletion, as node deletion is more radical and intrusive (Tong et al., 2012).
However, such methods operate offline, i.e., they capture the network state
before the propagation process starts, in contrast to competitive cascades
that we propose.

###  2.4. Aggression Modeling and Minimization

Research on aggression has been focused on its detection rather than its
modeling. Specifically, Twitter account features have been used to detect and
categorize phenomena such as hate speech (Waseem and Hovy, 2016), while
incremental crowdsourcing techniques have been proposed to annotate abusive
tweets (Founta et al., 2018). Additionally, graph features such as peer
pressure and cumulative influence have also been highlighted (Squicciarini et
al., 2015). Furthermore, a multi-class classifier has been used for the
classification among various types of abusive behavior like racism or
homophobia (Davidson et al., 2017), while Random Forests have been exploited
to detect disturbing phenomena such as cyberbullying (Chatzakou et al., 2017).

In the context of online aggression detection, there have been some works
investigating the dynamics of the online aggressive behavior, its persistence
and spreading. More concretely, a study on Twitter data, addresses the local
ego-networks to formulate the user neighborhood and finds that the online
social influence is of significant importance to the detection of cyberbullies
(Huang et al., 2014). Expanding this work to a macroscopic view, the
underlying graph structure and pair-wise user interactions are investigated
leading to the importance of peer pressure on the spread of cyberbullying over
the social network (Squicciarini et al., 2015). Last, to theoretically
solidify these practical approaches, Runions et al. suggest that in online
social networks the reason of the aggressive behavior’s adoption is the
influence of the person who originated the content (Runions and Bak, 2015).
Despite all this volume of works on cyberbullying and more generally on online
aggression detection, the clarification of the aggression dynamics and
properties seem more like a side-effect instead of a concrete goal.

According to the authors knowledge, there is only a single approach modeling
online aggression diffusion which exploits $OD$ models (Terizi et al., 2020).
However, in social networks aggression spreads through a user post based on
this user’s social circle. Hence, cascade models are more suitable than OD for
aggression modeling, as they simulate a mass information dissemination process
rather than user-user interactions. Also, to the best of our knowledge, we are
the first to address online aggression minimization. In particular, two
different minimization methods are explored here using: (a) competitive
cascades, (b) node/edge blocking which are compared with respect to aggression
reduction efficiency.

##  3\. Methodology

In this section, we address and formalize the process pipeline and notions
presented in Figure 1. In particular, we provide the theoretical foundations
for user interactions, seed selection, aggression propagation and healing
mechanisms, and connect them to the diffusion and minimization problems, as
per Contribution 1.

###  3.1. Aggression Modeling

A social network is represented by a directed graph $G=(V,E)$, where $V$ is a
set of n nodes or users, and $E$ is a set of m edges, i.e., relationships
between users. Each node is associated with an aggression score denoted by
$A_{i}$ $\forall v_{i}\in V$. The aim is to find whether the $IC$ and $LT$ are
able to model aggression diffusion and what are the important parameters that
enable them to do so.

####  3.1.1. Seed Selection

Under aggression diffusion modeling, the seed selection process is the first
crucial component to describe. Seed nodes are the ones the diffusion process
starts. The ultimate result of the propagation model is tightly related to
these initial seed nodes. The necessity for selecting seed nodes is due to the
diffusion model being restricted by a given budget $k$, meaning that we can
only afford a specific amount of initial stimulation of $k$ seed nodes. As a
result, a sophisticated seed selection strategy could enable the diffusion
process and lead to the increase of the number of activated nodes.

Even though the model as a whole relies mostly on the continuous user
aggression scores, it is beneficial to define what an aggressive user is
distinguishing them from the normal ones, either for future work or to address
a binary version of the problem. The definition of the aggressive user given
below will be proved useful in the formulation of the various seed strategies:

###### Definition 3.1.

Given the set of aggression scores A, and a user $u$ with aggression score
$A_{u}$, $u$ is called aggressive iff $A_{u}\geq A_{Pi\%}$, where $A_{Pi\%}$
is the aggression score of the i-th percentile user of the ranked distribution
of A.

With all the necessary concepts defined and explained, the proposed seed
strategies are presented below:

  * •

All Aggressive: Set all aggressive users as seed nodes.

  * •

Top Aggressive: From the set of aggressive users, select the k users with the
highest aggression score as seed nodes, where k is given by the user. If
$k>|aggressive$ $users|$, use All Aggressive strategy.

  * •

Single Discount (SD): Use algorithm presented in (Chen et al., 2009).
Iteratively, place the user with the highest degree in seed nodes set, remove
said node and proceed.

  * •

Degree Discount (DD): Also presented in (Chen et al., 2009). It works like
$SD$ but it is fine grained towards $IC$ model, taking advantage of the prior
activation probability.

  * •

Random: Choose users as seeds randomly.

####  3.1.2. Weighting Scheme

Next step is to define the various weighting schemes we apply on the network
edges. Each scheme captures real world properties that can impact the
aggression diffusion in a different way, as they can affect the probability of
propagating aggression. In the following, we assume that the user is embedded
in directional edges:

  * •

Jaccard overlap (Jaccard): In social media, the friends and followers of a
user can heavily influence the user’s own beliefs. If two connected users
appear to have similar or even identical social circles, this could point to
very similar beliefs or behaviors. Therefore, given two connected nodes $u$
and $v$, and their corresponding sets of neighbors $N_{u}$ and $N_{v}$, the
Jaccard overlap of their edge is defined as weight $w_{uv}=\frac{N_{u}\cap
N_{v}}{N_{u}\cup N_{v}}\in[0,1]$.

  * •

Power score (Power): Given a node $u$, the $Power$ $score$ $P_{u}$ is defined
as the ratio of the in-degree (incoming edges) over out-degree (outgoing
edges), $P_{u}=\frac{inDegree_{u}}{outDegree_{u}}\in[0,1]$. The higher the
Power score of a node, the more dominant the influence the node receives from
its in-neighbors in comparison to the degree that it influences its out-
neighbors. However, for this measure to be applied as an edge weight, it has
to be considered in a pairwise fashion. Therefore, given an edge from node $u$
to $v$, we define $P_{uv}=\frac{P_{v}}{P_{u}}\in[0,1]$ to capture $v$’s in-
neighbor and $u$’s out-neighbor influence.

  * •

Weighted overlap (Weighted): Given two nodes $u$ and $v$, their Jaccard
overlap $w_{uv}$ and their Power score $P_{uv}$, the Weighted overlap is
defined as $Pw_{uv}=P_{uv}*w_{uv}\in[0,1]$. This metric combines the previous
two weights.

Next, and using the above notations and metrics, we describe the $IC$ and $LT$
diffusion models, given a constant seed budget $k$.

####  3.1.3. Independent Cascade (IC)

At time step $t=0$ the process starts with an initial seed set $S$ of active
nodes with $|S|\leq k$ and proceeds in discrete time steps according to the
following stochastic rule. At each time step $t$, a set of active nodes
$Active^{t}$ are present in the network. Each active node $u\in Active^{t}$,
that was activated in time step $t$, has a single chance to activate each one
of its inactive neighbors, $v$, with probability $p$. This probability is
defined with respect to the very same notions that were introduced in
weighting schemes above. That is, node $v$ is activated by node $u$ with:
$p\leq w_{uv}$, $p\leq P_{uv}$ or $p\leq Pw_{uv}$ when the activation
criterion is Jaccard, Power or Weighted, respectively.

Under the aggressive $IC$ model, activation means that $u$ transfers its
aggression score to user $v$, that is $A_{v}=A_{u}$. However, there is a case
where multiple nodes $u\in N_{v}$ try to activate the same node $v$
simultaneously, and succeed. Then, the decision of whose aggression score to
transfer follows one of the below strategies:

  * •

Random: The activation order is arbitrary and the aggression score of a
randomly selected node $u\in N_{v}$ from the successful ones (i.e., $N_{v}\cap
Active^{t}$), is transferred to the node being activated $v$.

  * •

Top: The most aggressive node among the successful ones transfers its
aggression score to $v$.

  * •

Cumulative: The cumulative aggression score of all nodes is transferred
according to their contribution, a metric capturing peer pressure, also
verified in (Squicciarini et al., 2015). For example, if
$S=\\{u_{1},u_{2},u_{3}\\}$ the set of nodes that succeeded, then
$A_{v}=\sum_{i\in S}w_{i}*A_{i}$ with $w_{i}=\frac{A_{i}}{\sum_{j\in S}A_{j}}$
and $A_{i}$ their respective aggression scores.

If node $u$ succeeds, then node $v$ gets activated in step $t+1$; but
regardless of $u$’s success, it cannot make any further attempts to activate
$v$ in a following step. The $IC$ process terminates when no more activations
are possible.

Figure 2. A single step of the aggression-based IC diffusion process. Node
labels represent aggression scores, while edge labels are Jaccard weights
capturing the two participating nodes’ neighborhood similarity. In this
scenario, the cumulative activation criterion is used, meaning that when two
nodes activate a third node, they both contribute to its aggression score
change, while the seed nodes are selected randomly. Bold lines denote the
activators in this specific scenario.

To clarify the aggression-related $IC$ diffusion process, Figure 2 presents a
single step of a given scenario. Specifically, in time step $t=0$, only the
seed (black) nodes A, B and C are active, which have been picked randomly,
i.e. according to Random seed strategy. The edge weights have been defined
based on Jaccard weighting scheme, capturing the similarity between the
neighborhoods of the two nodes participating in the corresponding edge, and
only a sample of them is presented for clarity purposes. To give an example,
the weight of the edge between nodes C and D is the number of common neighbors
divided by the number of common and uncommon ones, i.e $\frac{1}{5}$. These
very same weights act as the activation probability as well, meaning that an
already active node could possibly activate the specific inactive neighbor
with a probability equal to their edge’s weight. On time step $t=1$, suppose
that two new (gray) nodes - D and E - get activated. D is activated by two
activators simultaneously (B and C), while E is activated by a single one
(i.e., B), a process denoted by the bold edges. Here, it should be clarified
that since the process is probabilistic, different nodes could have been
activated by different combinations of activators in a different scenario.
Continuing, in the case of E, the aggression score of B who activates E, is
transferred as is to E, changing it from $0.5$ to $0.6$. In the case of D
though, the Cumulative activation criterion is used to break the tie as it is
not clear which node among B and C should transfer its aggression. Therefore D
adopts both of its activators’ aggression scores, according to the definition
of Cumulative criterion above, meaning that D’s aggression score becomes
$\frac{0.8}{1.4}*0.8+\frac{0.6}{1.4}*0.6=0.714$.

####  3.1.4. Linear Threshold (LT)

In the $LT$ model, each node $v$ is assigned a threshold $\theta_{v}$. The
process starts with an initially activated seed set $S$ with $|S|\leq k$,
similarly to the $IC$ process. In step $t$, node $v$ is influenced by each
neighbor $u\in N_{v}$, according to a weight which should respect the node
threshold selection. Specifically, the alternative activation criteria at time
step $t$ are:

  * •

Aggression: When aggression scores are used as node thresholds, that is :
$\theta_{v}=A_{v}\forall v\in V$, then $\theta_{v}\leq\sum_{u\in N_{v}\cap
Active^{t}}w_{e_{vu}}$, where $w_{e_{vu}}$ is the weight of edge $e_{vu}$.

  * •

Power: When power scores are used as node thresholds, that is
$\theta_{v}=P_{v}\forall v\in V$, then $\theta_{v}\leq\sum_{u\in N_{v}\cap
Active^{t}}w_{e_{vu}}$.

Finally, under the aggressive $LT$ model, in contrast to the $IC$ case,
activation has a single interpretation which is that the active neighbors of
node $v$ transfer their average aggression score to $v$:
$A_{v}=\frac{\sum_{u\in\bar{N}_{v}}A_{u}}{|\bar{N}_{v}|}\in[0,1]$ where
$\bar{N}_{v}$ is the set of active neighbors of $v$. In this context, the
process unfolds until there are no nodes left to become active according to
the thresholds.

Figure 3. A single step of the aggression-based LT diffusion process. Node
labels represent aggression scores while edge labels are Jaccard weights,
capturing the two participating nodes’ neighborhood similarity. In this
scenario, Aggression threshold strategy is used, meaning that nodes transfer
their aggression explicitly based on their tendency towards it instead of some
other attribute of theirs, while the seed nodes are selected randomly.

Figure 3 presents a single step of a scenario for the aggression-based $LT$
diffusion model. Similar to the case of the $IC$ scenario, the seed (black)
nodes A, B and C initiating the diffusion process have been randomly selected.
Also, Jaccard has been applied to weigh the graph, by capturing the
neighborhood similarity of two connected nodes, a notion that is very useful
considering that $LT$ exploits a node’s neighborhood influence by definition.
Additionally, Aggression threshold strategy is enabled, forcing inactive nodes
to adopt the aggression of their activators based on their own aggressiveness
instead of some other metric or feature they may have at their disposal. $LT$
model differentiates from the $IC$ on two crucial mechanisms. First, no tie
braking rule is necessary, as all the active neighbors contribute equally in
the aggression transferring process. Second, the process is not stochastic,
meaning that given the specific scenario only a single outcome is possible, as
there are no probabilities involved, just deterministic thresholds. This is
why there are no bold edges here, in contrast to the ones appearing in the
$IC$ scenario. For completeness of the example, we examine time step $t=1$ of
the figure, when the two seed nodes B and C activate their (grey) neighbors D
and F. These are to be activated because their thresholds are low enough. The
activation is executed by transferring both of their aggression scores,
expressed as the average of the two values. Specifically, D’s threshold (which
is its aggression score $A_{D}=0.15$) is lower than its B and C neighbors’
influence, calculated as the average of the participating edge weights
$\frac{1/5+1/5}{2}=0.2$. Therefore, this leads to D’s aggression score to
change from $0.15$ to $\frac{0.6+0.8}{2}=0.7$. In a similar manner, F is
updated from $0.2$ to $0.7$.

###  3.2. Aggression Minimization

Given the two models ($IC$ and $LT$), we address the problem of aggression
minimization by means of two different approaches, called competitive and
blocking aggression minimization, respectively.

####  3.2.1. Competitive Aggression Minimization (CAM)

Under this minimization method, there are two competing diffusion processes, a
negative - also called aggressive - and a positive or educational cascade. The
goal of the latter is to minimize the spread of the negative one. For the
negative cascade and hence the aggression diffusion, we use the best model
discovered in aggression modeling (as explained earlier), while for the
positive one, we use the corresponding cascading model and test the
alternative configurations.

###### Problem 1 (CAM).

Given a directed network $G=(V,E)$, with $V$ and $E$ denoting the node and
edge sets, respectively, an integer budget $k$ and two competing diffusion
processes, the aggressive and educational, with the latter having a set of
parameters $\theta(\cdot)$, the $CAM$ problem is to find a positive seed set
$S$ with $|S|\leq k$ and $\theta^{\ast}(\cdot)$, s.t.
$\theta^{\ast}(\cdot)=\arg\min_{\theta}A(G)$ w.r.t. $S$, where
$A(G)=\sum_{v\in V}A_{v}$ is the overall aggression score in the network.

We now define the rules that describe the educational cascade. Under $IC$, the
activation probabilities follow the same notions of the negative cascade with
respect to weighting schemes: Jaccard overlap, Power score or Weighted
overlap. However, under $LT$, the activation criterion as well as node
thresholds should depend on Power only. Aggression would be an inappropriate
threshold as the educational cascade intents to mitigate the effect of
aggression on the network, instead of maximizing the overall aggression score.

Using a predefined negative seed set, and according to the best strategy of
aggression modeling scenario, we allow the two processes unfold
simultaneously. If at time step $t$, both cascades reach the same node $v$,
then $v$ gets positively activated, simulating the fact that most probably an
educated or aware user would stop manifesting aggressive behavior.

Regarding the meaning of activation from the perspective of the positive
cascade (the healing effect), the seed nodes of the positive cascade are
considered as the most sensitized as the educational piece of information
initiates its propagation from them. Hence, the healing effect on them is
drastic and depends on their respective aggression scores. More formally, the
new aggression score of the positive seed nodes follows the below rule:

$A^{\prime}_{s}=\begin{cases}0&\text{if }p\geq A_{s}\\\
A_{s}&otherwise\end{cases}$

Apart from seed nodes, we discern four possible cases:

  * •

Vaccination: activating user $v$ results to $A^{\prime}_{v}\rightarrow 0$,
i.e., the user becomes normal immediately. This hypothesis is strict and
serves as an upper bound to the minimization’s performance as it captures the
ideal case of a user becoming completely sensitized by the educational cascade
without ever reverting to the aggressive behavior.

  * •

Aggression transfer: activating user $v$ results to:

    * –

IC: $A^{\prime}_{v}\rightarrow A_{u}$, with user $u$ activating $v$

    * –

LT: $A^{\prime}_{v}\rightarrow\frac{\sum_{u\in N_{v}}A_{u}}{|N_{u}|}$

This is the case where aggression is transferred according to the underlying
mechanics of the diffusion model, as described in Section 3.1. Therefore,
reducing the overall aggression of the network depends solely on the influence
of the positive cascade’s seed nodes, considering they are the only ones
starting with a reduced aggression as described above.

  * •

Decaying aggression transfer: activating user $v$ is affected by a decaying
factor $\lambda$ capturing the distance from the source of the information.
Hence, $\lambda=\frac{1}{hops}$ and:

    * –

IC: $A^{\prime}_{v}\rightarrow\lambda*A_{u}$, with user $u$ activating $v$

    * –

LT: $A^{\prime}_{v}\rightarrow\frac{\sum_{u\in N_{v}}\lambda*A_{u}}{|N_{u}|}$

This mechanism simulates the fading effect of the positive cascade, meaning
that users immediately exposed to it are more prone to change in comparison to
those getting affected indirectly e.g. by a friend’s re-posting activity.

  * •

Hybrid: using a combination of cases $1$ and $3$, this case aims to capture
the different potential reactions of the various users when they become
exposed to the positive cascade. Formally:

$A^{\prime}_{v}=\begin{cases}\text{Vaccination}&\text{, if }p\geq A_{v}\\\
\text{Decaying aggression transfer}&,otherwise\end{cases}$

Finally, with respect to the seed strategy of the positive cascade, two
alternatives are considered. On one hand, the seed strategy of the negative
cascade can be used, but this scenario favors the negative cascade as the
decision would be based on its own specifics. On the other hand, one of the
other strategies proposed in aggression modeling section can be exploited,
regardless of the seed strategy of the negative cascade.

####  3.2.2. Blocking Aggression Minimization (BAM)

In contrast to CAM, in Blocking Aggression Minimization (BAM) there is only a
single cascade, the aggressive one. The aim here is to target specific nodes
or edges to immunize. That means removing them to optimally suppress
aggression diffusion over the network. In the case of social networks, node
removal is equal to banning a user which is a drastic measure. This is why we
consider the case of edge removal also. Formally the problem is defined below:

###### Problem 2 (BAM-N).

(node version)

Given a directed network $G=(V,E)$, with $V$ and $E$ denoting the node and
edge sets, respectively, and an integer budget $k$, the problem is to find a
subset of nodes $S\subseteq V$ with $|S|=k$, s.t. $S=\arg\min_{s\in
P(V)}A(G)$, where $A(G)=\sum_{v\in V}A_{v}$ is the overall aggression score in
the network, and $P(V)$ is the Power Set of $V$.

###### Problem 3 (BAM-E).

(edge version)

Given a directed network $G=(V,E)$, with $V$ and $E$ denoting the node and
edge sets, respectively, and an integer budget $k$, the problem is to find a
subset of edges $S\subseteq E$ with $|S|=k$, s.t. $S=\arg\min_{s\in
P(E)}A(G)$, where $A(G)=\sum_{v\in V}A_{v}$ is the overall aggression score in
the network, and $P(E)$ is the Power Set of $E$.

Based on the major finding of (Prakash et al., 2012) and (Wang et al., 2003),
for a large family of diffusion processes, the only network parameter that
determines whether this diffusion would become an epidemic or not is the
largest eigenvalue $\lambda$ of the adjacency matrix A. For this reason, in
our experiments next, we use NetShield (Tong et al., 2010) to solve the BAM-N
problem, and NetMelt (Tong et al., 2012) for BAM-E problem. Motivated by the
results of these two studies, coupled with the competitive minimization
explained earlier, we also implement an aggression-related variation. In
particular, instead of using the initial edge weights, we exploit the product
of the aggression scores of source and destination nodes. That, is for pair
$(u,v)$ in the adjacency matrix, the cell value now becomes $A_{u}*A_{v}$.
However, it should be noted that $BAM$ is a problem that exploits offline
methods, in contrast to $CAM$ which adjusts to the dynamics of the aggression
diffusion. Concluding, the above problems are tested and validated in the
experimental section below.

##  4\. Experimental Results

In this section, and as per our Contribution 2, we present the results of the
experimental process, first with respect to aggression modeling using $IC$ and
$LT$ methods (Sec. 4.2 and 4.3), and then aggression minimization using
competing cascades or blocking (Sec. 4.4).

Table 1. Validation vector transition features $t_{0}$ | $t_{i}$
---|---
n | {n $||$ a$\\}_{i}$
a | {n $||$ a$\\}_{i}$
N-N | {N-N $||$ N-A $||$ A-N $||$ A-A$\\}_{i}$
N-A | {N-N $||$ N-A $||$ A-N $||$ A-A$\\}_{i}$
A-N | {N-N $||$ N-A $||$ A-N $||$ A-A$\\}_{i}$
A-A | {N-N $||$ N-A $||$ A-N $||$ A-A$\\}_{i}$

###  4.1. Experimental Setup

Dataset: In this work, we utilize an unlabeled Twitter dataset comprised of
$81,306$ users and $1,768,149$ directed edges between them (Leskovec and
Mcauley, 2012). To reduce noise in the converging process, we use the strongly
connected component of this network, which consists of $68,413$ nodes and
$1,685,163$ directed edges. The edge weights are decided according to the
proposed weighting schemes described in Section 3. In order to acquire labels
to bootstrap our propagation algorithms, following similar methodology as in
(Terizi et al., 2020), we apply the prediction algorithm proposed in
(Chatzakou et al., 2017), which provides a probability for a user to be
aggressive or not, based on their characteristics in the network and their
activity. We tune this algorithm to use only the available network-related
features and compute the initial user aggression scores of all users, which is
equal to the probability of a user being aggressive. With this classifier,
$5,594$ users (or $\sim$8% of the network) were given a non-zero aggression
score.

Metrics used to measure aggression change: Similar to (Terizi et al., 2020),
we measure the state of aggression of users, and how it changes through
simulated time using a vector of $26$ metrics. The 6 core metrics enlisted
below capture the network state with respect to users and edges and their
label at time $t_{i}$:

  * •

n: portion of normal users in the network

  * •

a: portion of aggressive users in the network

  * •

N-N: portion of edges that both users $i$ and $j$ are normal

  * •

N-A: portion of edges that user $i$ is normal & $j$ aggressive

  * •

A-N: portion of edges that user $i$ is aggressive & $j$ normal

  * •

A-A: portion of edges that $i$ and $j$ are aggressive users

By combining these core metrics, Table 1 presents 20 additional metrics
capturing the transitions through simulated time between $t_{i}$ and initial
state $t_{0}$.

Measuring metrics in ground truth: We were provided with the ground truth
scores for the above metrics from (Terizi et al., 2020), for both state of
individual users and their edges, as well as change of state. This ground
truth vector was computed on a smaller labeled Twitter network of 401 users,
published by (Chatzakou et al., 2017) and re-crawled in 2019 (i.e., two
snapshots of state in 2016 and 2019, and provided by (Terizi et al., 2020)).

Comparing simulation and ground truth data: The above set of metrics is
computed for all simulated models and all their time steps and compared with
the ground truth vector using Cosine Similarity, i.e.,
Cosine(ground_truth_vector, simulation_vector). In fact, we tried other
similarity metrics to capture the agreement with ground truth, such as Pearson
and Spearman correlations, but Cosine Similarity produced more stable results.

The careful reader will notice that the state of users is binary (normal and
aggressive). Hence, to compute the cosine of the two vectors at each simulated
step, the users’ state must be dichotomized. For this, we experimented with
different thresholds ($T_{A}$) on the users’ aggression scores
($T_{A}=\\{0.1,0.2,...,0.9\\}$) and concluded that results with $T_{A}=0.4$
exhibit best overall similarity with ground truth data. Thus, all results
presented next are based on $T_{A}=0.4$.

Statistical Tests: Regarding $IC$ modeling, we run every experiment 10 times
due to the inherent randomized nature of the activation process. Additionally,
when Random is used as the seed strategy, each experiment is executed 10 more
times, leading to a total of 100 executions. To present these results, we use
Cumulative Distribution Functions (CDF). To validate whether there is
significant difference in parameter values, we employ One-Way ANOVA, followed
by a pairwise post-hoc Tukey’s HSD test, to spot the exact value of the
significant parameter. We identify statistically significant differences at
p-value $<0.001$ for both tests. For the $LT$ modeling, we run each experiment
only once, except for the case of Random seed strategy, for which we execute
each setup 10 times and acquire average performance. These results are
presented in a table where appropriate.

Simulation Framework: We designed and implemented a modular and extensible
simulation framework to execute the modeling and minimization experiments. It
is written in Python, it is open sourced, and allows fine-grained control of
simulation parameters, to enable reproducible experiments, and future
extensions (Contribution 4).

###  4.2. Aggression Modeling

(a) $WS$: Weighted; $AC$: Random

(b) $WS$: Power; $AC$: Cumulative

(c) $WS$: Jaccard; $AC$: Top

Figure 4. Cosine similarities for $IC$ based on various seed strategies,
Weighting Schemes ($WS$) and Activation Criteria ($AC$). Omitted combinations
presented similar behavior.

In the next paragraphs, we present and analyze the results for selecting 1)
Seed set strategy, 2) Weighting scheme, and 3) Activation criterion (for $IC$)
and Threshold strategy (for $LT$).

####  4.2.1. Seed Selection Strategy

For the rest of the experimental process, we distinguish the results of $IC$
and $LT$-based models due to the different presentation processes.
Experimenting on different seed sizes (budget $k$) shows no actual accuracy
implications. Instead, it only affects the duration of the diffusion process.
Specifically, increasing (decreasing) the seed size just forces the process to
terminate sooner (later). For this reason, seed size is set to $5594$, to
match the number of aggressive users according to the definition presented in
subsection 3.1.1. Regarding seed strategy, Top Aggressive is unnecessary as
the selected seed size allows the use of all aggressive users.

Figure 4 presents the results regarding $IC$-based models. For brevity, the
exhaustive list of experiments is not presented, as similar patterns were
observed. From these results, we note a small, yet statistically significant
dominance of the network-feature strategies, Single Discount ($SD$) and Degree
Discount ($DD$). Additionally, for Power\- and Weighted-based graphs, $SD$ and
$DD$ do not have significant differences, while in Jaccard, $SD$ is prevalent.
Hence, for the $IC$ models, $SD$ constitutes the appropriate seed strategy due
to its reduced computation cost compared to $DD$. The same observation is also
applicable to the $LT$-based configurations (Table 2). It is noted, however,
that $DD$, although applicable, is not compatible with the theoretical
background of the $LT$ model.

####  4.2.2. Weighting Scheme

(a) $AC$: Random

(b) $AC$: Top

(c) $AC$: Cumulative

(d) $WS$: Jaccard.

Figure 5. Cosine similarities for $IC$ based on different Activation Criteria
($AC$) or Weighting Schemes ($WS$). Seed strategy: $SD$.

Given the selected seed strategy $SD$, we now analyze the results on different
weighting schemes. Figure 5 presents the results of the $IC$-based models
while Table 2 does so for the $LT$ ones. It is shown that for both $IC$ and
$LT$, Jaccard is the best weighting scheme, regardless of the activation
criterion, also confirmed statistically with ANOVA and Tukey tests. This means
that a user’s social circle and therefore its neighborhood similarity is more
appropriate to model online aggression instead of its power and credibility
per se. Additionally, Weighted and Power as weighting schemes do not present
any significant differences constituting them inappropriate weighting schemes.
Thus, we conclude with Jaccard as the appropriate weighting scheme.

####  4.2.3. Activation Criterion and Threshold Strategy

Up to now, we investigated the various possible configurations from a
macroscopic point of view, to conclude to $SD$ as the best seed selection
strategy and Jaccard as the best weighting scheme. Next, we look into
activation criteria and threshold strategies to select best possible setups.
Specifically, Figure 5(d) shows that Cumulative activation strategy is the
most dominant, while there is no clear distinction between Top and Random
strategies, validated by ANOVA and Tukey’s HSD test too. This prevalence is
explained by the crucial effect that a user’s neighborhood has on them, as it
is also pointed out by the General Aggression Model (Allen and Anderson,
2017).

Proceeding to $LT$ models, Table 2 shows that Aggression-based thresholds
perform slightly better and, hence, they should be used to model aggression
propagation with respect to Jaccard and $SD$. This observation is intuitive,
as it suggests that the higher the aggressiveness of a user, the easier to
propagate it - enabling the overall aggression diffusion process over $LT$.

Table 2. Cosine similarity for $LT$ models, with different Weighting Schemes, Seed Strategies and Threshold Strategies. * means that the reported cosine similarity is insensitive to the specific parameter’s values, given the rest of the configuration parameters. The best value is highlighted in bold. Weighting Scheme | Seed Strategy | Threshold Strategy | Cosine Similarity
---|---|---|---
Jaccard | Random | * | 0.690
Jaccard | All Aggressive | * | 0.688
Jaccard | $SD$ |  Aggression — Power |  0.691 — 0.690
Power | * | * | 0.689
Weighted | * | * | 0.689

####  4.2.4. Snapshot Evolution

In the previous experimental cases, the diffusion process was let to unfold
until completion (i.e., after all possible time steps), and then the
similarity with ground truth was calculated. However, to investigate if there
is an intermediate time step (snapshot) in the propagation that can better
match the ground truth change of state, we created snapshots of the diffusion
process at the end of every time step and tracked the corresponding results.
Additionally, in order to assess the validity and superiority of the
aggression-related mechanics of the best models, a comparison to IC and LT
baseline models is also conducted, presented in Figure 6. In particular, the
IC baseline is characterized by a random activation probability, random seed
node selection and random aggression transmission, which is needed to capture
the actual performance. Analogously, the LT baseline is based on random node
weights to use as thresholds and random seed node selection as well. This
randomness leads to the report of their mean performance out of 10 experiment
repetitions.

In both $IC$ and $LT$ aggression-based models, the performance decreases
rapidly within the first few steps and gradually stabilizes in the last ones.
This trend shows that aggression probably does not propagate that deeply into
the network (remember that steps here mean graph hops). Interestingly, the
stable behavior for the last steps validates the integrity of the convergence
value studied earlier. Experimenting with two more similarity metrics (Pearson
and Spearman correlations), it was found that Pearson behaves like Cosine, but
Spearman presents an early increase with a steady drop afterwards. However,
Cosine was preferred for our analysis, since the normality requirement for
Pearson is not satisfied and Cosine is more sensitive to subtle changes than
Spearman.

Moreover, the comparison to the baselines shows that - for both IC and LT
models - adopting aggression-related mechanics in the weighting scheme, seed
selection process and activation/threshold strategy offers additional value
and enhances the performance of the modeling. This means that the original
diffusion models, although applicable, do not have enough expressive power to
capture and describe the complex nature of online aggression spreading. It
also suggests that the proposed mechanics are promising and towards the right
direction to better understand aggression diffusion over social networks.

Figure 6. Snapshot evolution for the best $IC$ (left) and $LT$ (right)
configurations and comparison to the random baseline models

Takeaways: Summarizing, aggression diffusion over social networks can be
described by a cascading model, such as $IC$ or $LT$, fulfilling the first
part of Contributions 2 and 3. Considering that the original $IC$ and $LT$ are
not able to capture online aggression dynamics, new mechanisms have to be
created. For both models, the best seed strategy is $SD$, meaning that the
models able to capture the online aggression diffusion more accurately should
initiate their process from the most central nodes in the network, e.g. users
with many friends and followers around whom large user communities are formed.
Moreover, for both cases, the best performing weighting scheme is Jaccard
which expresses that relations are formed based on users’ neighborhood
similarity. The explanation of this outcome is that to predict aggression
diffusion one should examine the social interactions instead of individual
user attributes. Specific to $IC$-based models, activation criterion should be
set to Cumulative, enabling the whole neighborhood of a node to affect its
aggression state, while for $LT$-based models, an Aggression-based threshold
strategy is preferred, as it enables the overall dissemination process.
Lastly, the diffusion process should unfold until convergence to achieve a
stable state.

###  4.3. Case Study: modeling aggression diffusion

To evaluate the best $IC$ and $LT$ models (as concluded earlier), we apply
them to the small labeled Twitter network of the prediction algorithm
(Chatzakou et al., 2017). There are labels for two network snapshots (for 2016
and 2019), while users are labeled as normal or aggressive in both. Using the
user labels of the first snapshot, we run the $IC$ and $LT$ models and measure
the $AUC$ of the predicted labels, vs. the ones of the 2nd snapshot. The $AUC$
achieved by $IC$ (with $SD$, Jaccard, and Cumulative) is $0.82$, while for
$LT$ (with $SD$, Jaccard and Aggression) is $0.89$. Thus, both configurations
perform well in modeling aggression diffusion, and $LT$ seems to provide
slightly better performance, with fewer false positives (aggressors).

###  4.4. Aggression Minimization

The second part of the experimental process pertains to aggression
minimization ($CAM$ and $BAM$ problems) and aids in Contributions 2 and 3. To
address the $CAM$ problem, a competitive cascade process is exploited aiming
at decreasing the final aggression score of the network, while for $BAM$, the
proposed node (edge) blocking mechanisms select the nodes (edges) to remove
before the launch of the aggressive cascade.

####  4.4.1. Competitive Aggression Minimization

With respect to $CAM$, the positive cascade should be similar to the negative
one to compete in equal terms, i.e., for both $IC$ and $LT$ models, and both
positive and negative cascades the chosen weighting scheme is Jaccard. For the
$IC$, the activation strategy is Cumulative, and for $LT$ the threshold
strategy is Aggression. The experimentation comprises of the different seed
strategies for the positive cascade, as well as the various healing
mechanisms.

Here, we focus on the best configurations of the modeling phase mentioned
earlier. Figure 7 presents aggression evolution during the competitive cascade
process, for various seed and healing strategies on the $IC$ models. Y-axis
presents the ratio of loss or gain with respect to the case where no healing
is applied. First, regarding the seed strategy of the positive cascade, Random
presents the best results for every healing mechanism reaching 57.8%
aggression reduction, while $DD$ follows with 57.3% and $SD$ and All
Aggressive come last. Thus, Random is the best option when computational cost
is important, while $DD$ could be chosen if stability is necessary.

(a) All Aggressive seed strategy

(b) Random seed strategy

(c) $DD$ seed strategy

(d) $SD$ seed strategy

Figure 7. Healing strategy comparison on $IC$ models, under $CAM$ problem

With respect to the healing strategies, regardless of the positive seed
strategy, Transfer presents the worst performance with a best-case reduction
of 8%, due to the simple aggression score transferring, no matter the result
on the corresponding node. The rest of the healing mechanisms perform
similarly, irrespectively of the seed strategy. In particular, Vaccination is
the most dominant mechanism, achieving 57.3% aggression reduction, while
Decaying transfer and Hybrid follow with $\sim$57%. These results are
intuitive, since Vaccination makes the strong assumption that nodes get
completely healed, whereas Decaying transfer is a more relaxed and realistic
version of Vaccination, and Hybrid lies in between them.

Another critical factor of a healing strategy’s efficiency is the number of
nodes that get activated by the negative cascade, since the same aggression
reduction on a larger number of activated nodes expresses a more efficient
mechanism. In particular, all healing mechanisms, except Transfer, activate
about the same number of nodes (plots omitted for brevity). Combining the
results on aggression reduction with number of activated nodes, we conclude
that Vaccination is the best healing mechanism, but due to its mostly
theoretical nature, Decaying transfer could be assumed to minimize aggression
in $IC$-based models.

Proceeding to the $LT$ models and mentioning that $DD$ is not a viable option
as it has been explained already, Figure 8 presents that $SD$ is dominant,
achieving up to 92% reduction, with Random following closely with 90%. Moving
to the healing strategies, Vaccination achieves the best reduction of
approximately 92%, but, in contrast to the $IC$ case, it is the only one that
preserves the reduction during the whole process, as it completely deactivates
the affected nodes, and, thus, they have zero effect on their neighbors during
the subsequent steps. On the contrary, the rest of methods can not preserve
the reduced aggression levels suggesting a potential inefficiency of
competitive cascades on $LT$ models. Hence, it is observed that in $LT$
models, the most possible and realistic reduction starts from 5% and can raise
up to 20% using as healing mechanism the Hybrid.

Regarding the activated nodes during each propagation process, Vaccination
activates the least number of nodes, while Decaying transfer stimulates 2x
more nodes compared to Hybrid in most cases. Also, considering the theoretical
nature of Vaccination, Decaying Transfer should be prioritized as a healing
mechanism.

####  4.4.2. Blocking Aggression Minimization

Moving to the second minimization problem ($BAM$), the results of four methods
combining: 1) node or 2) edge blocking (removal), while using a) Jaccard as
edge weight in the adjacency matrix A, or b) the modified version with the
aggression scores (please see details in Sec. 4.4) are briefly discussed and
compared to a no-immunization process as baseline. For node and edge blocking,
NetShield (Tong et al., 2010) and NetMelt (Tong et al., 2012) algorithms were
used, respectively, to select nodes or edges to remove.

(a) All Aggressive seed strategy

(b) Random seed strategy

(c) $SD$ seed strategy

Figure 8. Healing strategy comparison on $LT$ models, under $CAM$ problem

For $IC$, the aggression-based variations do not present significant
improvement over the baseline. Aggression reduction is possible - although
smaller than competitive cascades - by exploiting node removal, achieving a
reduction of 8%. We note that the number of removed nodes is $5594$, equal to
the seed size of the competitive cascade process. However, when edge blocking
is examined, the overall aggression score is, in fact, doubled, regardless of
the type of the matrix A, due to the combination of significant edge removal
and the degree-based seed strategy of the underlying diffusion process.
Moreover, to let NetMelt reach the same number of nodes as NetShield,
$\sim$$470k$ edges have to be removed, which is unrealistic.

For $LT$, edge removal (with both types of A) presents a switching behavior of
1% increase and decrease in overall aggression, in relation to the baseline of
no immunization, leading to an insignificant outcome on aggression
minimization. Node removal, though, performs better, since the reduction
reaches 6% for the Jaccard-based A, and 8% for the aggression-based case. In
general, the aggression alternative of the adjacency matrix seems to benefit
the process, but more in-depth research is needed to understand its tradeoffs.
Lastly, regarding the number of activated nodes (results omitted), in
comparison to the competitive cascades, the immunization methods here tend to
activate 4x times more nodes overall.

Takeaways: Aggression minimization through competitive cascades outperforms
blocking techniques. Specifically, using $IC$ models, a reduction of 57% of
the initial total aggression score of the network is achieved, in contrast to
the 8% of the blocking methods. Using $LT$ models, even though competitive
cascades are more efficient in terms of aggression reduction, the gradual
aggression score increase during the diffusion process suggests that node
blocking should be preferred. However, it should be noted here that even if
blocking mechanisms were undisputedly superior to competitive cascades in
terms of aggression reduction, they are also too invasive. This means that
they disrupt the network heavily as they completely remove nodes and edges, as
it is also observed in Twitter. Therefore, even with the proper selection of
e.g. users to block, the effects of this action are observable by many users
and disrupt their experience. These observations cover the second and last
part of Contributions 2 and 3.

##  5\. Discussion & Conclusion

In this work, the dual problem of aggression modeling and aggression
minimization in online social networks, and specifically Twitter, was
examined. To the best of our knowledge, aggression modeling has been addressed
only in (Terizi et al., 2020), where it was approached through opinion
dynamics instead of cascading models. Both approaches present comparable
performance in modeling real Twitter data on aggression diffusion, while
having some distinguishing features regarding the process and convergence
conditions. We believe our work advances the community’s understanding on
aggression diffusion one step further, while determining the most prevalent of
the two methods is left for a follow-up work. Apart from aggression modeling,
however, we are the first to address aggression minimization. We exploit
models from the domains of competitive cascades and immunization and discover
strategies to implement them as minimization methods.

Results: First, we showed that $IC$ and $LT$ are suitable for modeling
aggression diffusion, as they can reach a cosine similarity of $\sim$70% with
ground truth data. The results showed that by focusing on the most central
users - instead of the most aggressive ones - to initiate the process, online
aggression diffusion can be better modeled by Twitter or other external
entities. Additionally, capturing neighborhood similarity is vital for
describing aggression dissemination, hence Jaccard overlap should be used as
the weighting scheme for graph edges. Moreover, Cumulative activation
criterion of $IC$ engulfs the team influence on information propagation, while
for $LT$, Aggression threshold strategy suggests that high aggressiveness
propagates easier, enabling aggression diffusion.

With respect to aggression minimization, competitive cascades achieved a total
of 57% reduction in $IC$, and 5-20% in $LT$ models. The similar behavior of
the most prevalent seed strategies allows a trade-off between stability ( $DD$
and $SD$ strategies) and computational cost (Random). Moreover, Decaying
Transfer was the most appropriate healing mechanism, despite Vaccination’s
higher performance. This is because Vaccination naively presumes that
positively affected users, e.g., through a cyberbullying sensitizing campaign,
will not switch back to abusive behavior. Node blocking methods reached
$\sim$8% reduction in both $IC$ and $LT$. Edge blocking is inappropriate, as
it interferes with the degree of nodes, making it incompatible with the
degree-based $SD$ seed strategy of the aggression cascade. Furthermore, in
$LT$ scenario, competitive cascades could not preserve their efficiency and
hence the more stable blocking methods should be preferred.

Implications of our work: Next, we unfold and discuss what the real-world
implications of our work are. Considering the case of a Twitter community
moderator, the above observations provide some tools to control and provision
the community’s function. That is, to ensure the interception of aggression
spread, such moderators should closely monitor the online behavior of central
and influential users. This is in case they turn aggressive, as they could
initiate a burst of aggression increase in their community and Twitter as a
whole. However, if aggression is already established in a specific community,
the moderators should regulate the activity of users sharing many common
friends and followers, with the already aggressive ones. Lastly, to measure
and avoid underestimating the impact of aggressive activity, they should adopt
a holistic approach by considering the cumulative effect of one’s neighbors on
their aggressive behavior.

Regarding the observations on competitive cascades and blocking techniques,
and to reduce aggression in online social networks, we discuss what would the
actual real world recommendations be. First, it is clear by now that the
already used methods of banning users - and by extension removing multiple
edges - are too invasive and disrupt the network and user experience. This
fact may even be the reason behind Twitter’s reluctance to ban users, even
when they are evidently aggressive and disruptive. Instead, our results show
that a simple awareness campaign (e.g., sharing posts for eliminating racism
or hate speech) can go a long way, as it could both be less intrusive and lead
to larger aggression reduction. Even better, community moderators could focus
such campaigns on Twitter users with a large audience (popular or highly
influential users) to reduce the time needed to reach users who are in the
periphery of the communities, as in this way Decaying Transfer (the best
healing strategy we found), would operate on its maximum effect. Last but not
least, these campaigns could help users improve their behavior in other social
media as well, an outcome that is not ensured by blocking mechanisms.

The methods presented in this work consist the starting point of understanding
and addressing online aggression diffusion and minimization. In this context
the most promising future research directions are:

  1. (1)

Explore further the possible weighting schemes, seed strategies,
activation/threshold criteria and aggression transferring mechanisms, to
better capture aggression dynamics on real networks.

  2. (2)

Investigate various thresholds that enable an epidemic spread of aggression
under different virus-spreading models.

  3. (3)

Examine ways to incorporate offline blocking methods into the diffusion
process and dynamically minimize aggression.

Many challenges lurk in addressing the above directions. However, all of these
future steps are feasible and promote our knowledge on addressing online
aggression once and for all.

## References

  * (1)
  * Allen and Anderson (2017) Johnie J Allen and Craig A Anderson. 2017.  General aggression model.  _The International Encyclopedia of Media Effects_ (2017), 1–15.
  * Basak et al. (2019) Rajesh Basak, Shamik Sural, Niloy Ganguly, and Soumya K Ghosh. 2019.  Online public shaming on twitter: Detection, analysis, and mitigation.  _IEEE Transactions on Computational Social Systems_ 6, 2 (2019), 208–220.
  * Borodin et al. (2010) Allan Borodin, Yuval Filmus, and Joel Oren. 2010.  Threshold models for competitive influence in social networks. In _International workshop on internet and network economics_. Springer, 539–550.
  * Budak et al. (2011) Ceren Budak, Divyakant Agrawal, and Amr El Abbadi. 2011\.  Limiting the spread of misinformation in social networks. In _Proceedings of the 20th international conference on World wide web_. ACM, 665–674.
  * 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_. ACM, 13–22.
  * Chen (2009) Ning Chen. 2009\.  On the approximability of influence in social networks.  _SIAM Journal on Discrete Mathematics_ 23, 3 (2009), 1400–1415.
  * Chen et al. (2009) Wei Chen, Yajun Wang, and Siyu Yang. 2009.  Efficient influence maximization in social networks. In _Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining_. ACM, 199–208.
  * Corcoran et al. (2015) Lucie Corcoran, Conor Guckin, and Garry Prentice. 2015\.  Cyberbullying or cyber aggression?: A review of existing definitions of cyber-based peer-to-peer aggression.  _Societies_ 5, 2 (2015), 245–255.
  * Davidson et al. (2017) Thomas Davidson, Dana Warmsley, Michael Macy, and Ingmar Weber. 2017.  Automated hate speech detection and the problem of offensive language. In _Eleventh international aaai conference on web and social media_.
  * Domingos and Richardson (2001) Pedro Domingos and Matt Richardson. 2001.  Mining the network value of customers. In _Proceedings of the seventh ACM SIGKDD international conference on Knowledge discovery and data mining_. ACM, 57–66.
  * Fernández-Gracia et al. (2014) Juan Fernández-Gracia, Krzysztof Suchecki, José J Ramasco, Maxi San Miguel, and Víctor M Eguíluz. 2014.  Is the voter model a model for voters?  _Physical review letters_ 112, 15 (2014), 158701\.
  * Founta et al. (2018) Antigoni Maria Founta, Constantinos Djouvas, Despoina Chatzakou, Ilias Leontiadis, Jeremy Blackburn, Gianluca Stringhini, Athena Vakali, Michael Sirivianos, and Nicolas Kourtellis. 2018.  Large scale crowdsourcing and characterization of twitter abusive behavior. In _Twelfth International AAAI Conference on Web and Social Media_.
  * He et al. (2012) Xinran He, Guojie Song, Wei Chen, and Qingye Jiang. 2012\.  Influence blocking maximization in social networks under the competitive linear threshold model. In _Proceedings of the 2012 siam international conference on data mining_. SIAM, 463–474.
  * Henneberger et al. (2017) Angela K Henneberger, Donna L Coffman, and Scott D Gest. 2017\.  The effect of having aggressive friends on aggressive behavior in childhood: using propensity scores to strengthen causal inference.  _Social Development_ 26, 2 (2017), 295–309.
  * 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.
  * Jiang et al. (2011) Qingye Jiang, Guojie Song, Cong Gao, Yu Wang, Wenjun Si, and Kunqing Xie. 2011\.  Simulated annealing based influence maximization in social networks. In _Twenty-fifth AAAI conference on artificial intelligence_.
  * Kempe et al. (2003) David Kempe, Jon Kleinberg, and Éva Tardos. 2003\.  Maximizing the spread of influence through a social network. In _Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining_. ACM, 137–146.
  * Lee and Lee (2002) Jintae Lee and Younghwa Lee. 2002.  A holistic model of computer abuse within organizations.  _Information management & computer security_ 10, 2 (2002), 57–63.
  * Leskovec et al. (2007) Jure Leskovec, Andreas Krause, Carlos Guestrin, Christos Faloutsos, Christos Faloutsos, Jeanne VanBriesen, and Natalie Glance. 2007\.  Cost-effective outbreak detection in networks. In _Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining_. ACM, 420–429.
  * Leskovec and Mcauley (2012) Jure Leskovec and Julian J Mcauley. 2012.  Learning to discover social circles in ego networks. In _Advances in neural information processing systems_. 539–547.
  * Long and Wong (2011) Cheng Long and Raymond Chi-Wing Wong. 2011.  Minimizing seed set for viral marketing. In _2011 IEEE 11th International Conference on Data Mining_. IEEE, 427–436.
  * Ma et al. (2008) Hao Ma, Haixuan Yang, Michael R Lyu, and Irwin King. 2008\.  Mining social networks using heat diffusion processes for marketing candidates selection. In _Proceedings of the 17th ACM conference on Information and knowledge management_. ACM, 233–242.
  * Prakash et al. (2012) B Aditya Prakash, Deepayan Chakrabarti, Nicholas C Valler, Michalis Faloutsos, and Christos Faloutsos. 2012\.  Threshold conditions for arbitrary cascade models on arbitrary networks.  _Knowledge and information systems_ 33, 3 (2012), 549–575.
  * Richardson and Domingos (2002) Matthew Richardson and Pedro Domingos. 2002.  Mining knowledge-sharing sites for viral marketing. In _Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining_. ACM, 61–70.
  * Runions and Bak (2015) Kevin C Runions and Michal Bak. 2015.  Online moral disengagement, cyberbullying, and cyber-aggression.  _Cyberpsychology, Behavior, and Social Networking_ 18, 7 (2015), 400–405.
  * 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.
  * Squicciarini et al. (2015) Anna Squicciarini, Sarah Rajtmajer, Y Liu, and Christopher Griffin. 2015.  Identification and characterization of cyberbullying dynamics in an online social network. In _Proceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining 2015_. 280–285.
  * Terizi et al. (2020) Chrysoula Terizi, Despoina Chatzakou, Evaggelia Pitoura, Panayiotis Tsaparas, and Nicolas Kourtellis. 2020\.  Angry Birds Flock Together: Aggression Propagation on Social Media.  _arXiv preprint arXiv:2002.10131_ (2020).
  * Tong et al. (2017) Guangmo Tong, Weili Wu, Shaojie Tang, and Ding-Zhu Du. 2017\.  Adaptive influence maximization in dynamic social networks.  _IEEE/ACM Transactions on Networking (TON)_ 25, 1 (2017), 112–125.
  * Tong et al. (2012) Hanghang Tong, B Aditya Prakash, Tina Eliassi-Rad, Michalis Faloutsos, and Christos Faloutsos. 2012\.  Gelling, and melting, large graphs by edge manipulation. In _Proceedings of the 21st ACM international conference on Information and knowledge management_. ACM, 245–254.
  * Tong et al. (2010) Hanghang Tong, B Aditya Prakash, Charalampos Tsourakakis, Tina Eliassi-Rad, Christos Faloutsos, and Duen Horng Chau. 2010.  On the vulnerability of large graphs. In _2010 IEEE International Conference on Data Mining_. IEEE, 1091–1096.
  * Trpevski et al. (2010) Daniel Trpevski, Wallace KS Tang, and Ljupco Kocarev. 2010\.  Model for rumor spreading over networks.  _Physical Review E_ 81, 5 (2010), 056102.
  * Wang et al. (2003) Yang Wang, Deepayan Chakrabarti, Chenxi Wang, and Christos Faloutsos. 2003.  Epidemic spreading in real networks: An eigenvalue viewpoint. In _22nd International Symposium on Reliable Distributed Systems, 2003. Proceedings._ IEEE, 25–34.
  * Waseem and Hovy (2016) Zeerak Waseem and Dirk Hovy. 2016.  Hateful symbols or hateful people? predictive features for hate speech detection on twitter. In _Proceedings of the NAACL student research workshop_. 88–93.
  * Wu and Pan (2017) Peng Wu and Li Pan. 2017\.  Scalable influence blocking maximization in social networks under competitive independent cascade models.  _Computer Networks_ 123 (2017), 38–50.
  * Zhang et al. (2015) Huiyuan Zhang, Huiling Zhang, Xiang Li, and My T Thai. 2015\.  Limiting the spread of misinformation while effectively raising awareness in social networks. In _International Conference on Computational Social Networks_. Springer, 35–47.