2021

These authors contributed equally to this work.

[1,2,3]Sebastião Pais These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Department of Computer Science, University of Beira Interior, Covilhã,
Portugal

2]NOVA LINCS, New University of Lisboa, Lisboa, Portugal

3]GREYC, Groupe de Recherche en Informatique, Image et Instrumentation de
University of Caen Normandie, Caen, France

# Detection of Dangerous Events on Social Media: A Perspective Review

M. Luqman Jamil<EMAIL_ADDRESS><EMAIL_ADDRESS>João Cordeiro
<EMAIL_ADDRESS>*  [  [

###### Abstract

Social media is an essential gateway of information and communication for
people worldwide. The amount of time spent and reliance of people on social
media makes it a vital resource for detecting events happening in real life.
Thousands of significant events are posted by users every hour in the form of
multimedia. Some individuals and groups target the audience to promote their
agenda among these users. Their cause can threaten other groups and
individuals who do not share the same views or have specific differences. Any
group with a definitive cause cannot survive without the support which acts as
a catalyst for their agenda. A phenomenon occurs where people are fed
information that motivates them to act on their behalf and carry out their
agenda. One is benefit results in the loss of the others by putting their
lives, assets, physical and emotional health in danger. This paper introduces
a concept of dangerous events to approach this problem and their three main
types based on their characteristics: action, scenarios, and sentiment-based
dangerous events.

###### keywords:

Dangerous Events, Social Media, Event Detection, Sentiment Analysis,
Extremism, Social Network Services, Social Computing, Terrorism

##  1 Introduction

The extent of social media users consists of billions of people from all
around the world. Initially, social media was developed with the object in
mind to help connect with family and friends online. It was used to share
everyday things, events, interests, and news within closed circles of family
and friends. These were personal events, e.g. birthdays, weddings, vacations,
graduation ceremonies, and going out. After the usability of social media was
discovered, it soon caught the attention of individuals and companies that
started using social media to reach more customers. Soon after, the trend
became a global phenomenon where people started connecting worldwide based on
their common interests. The influence of social media on people’s lives and
attitudes has been widely studied and established in many different
perspectives intro1 , intro2 .

Although social media is a broad term, it mainly refers to Facebook, Twitter,
Reddit, Instagram, and YouTube. Some social media platforms allow users to
post text, photos and videos. At the same time, many other social media
applications have limited options and restrictions for sharing the type of
content. YouTube allows users to post videos, while Instagram only allows
users to share videos and photos. 4.66 billion active internet users
worldwide, and 4.2 billion users are active on social media. As of the first
quarter of 2020, Facebook has 2.6 billion monthly active users globally,
making it the most extensive social media network globally. Twitter is one of
the leading social media with 397 million users worldwide, which is becoming
increasingly prominent during events and an essential tool in politics
Statista2021. . Another study Tnewsmed  shows that Twitter is an effective and
fast way of sharing news and developing stories. This trend has continued to
grow since the last decade as the internet has become widespread.

However, the use of social media has become more complex in the last decade.
It became a broader phenomenon because of the involvement of multiple
stakeholders such as companies, groups, and other organizations. Many lobbying
and public relations firms got on board and started targeting social audiences
to change people’s perspectives and influence their decisions. Mostly these
campaigns are related to a particular individual or a company. A similar
process happens in the public sphere, where people rally against or support
their target. It played a significant role in different outcomes, affecting
countries, people, and eventually the world. One such example is “Arab Spring”
arabspring , which is an event that started in Tunisia and spread among other
regional countries. Another example of good and bad events in political
spheres of the UK and US is given in the study that uses Twitter to evaluate
the perceived impact on users goodbadevents .

The term “event” implies typically a change which is an occurrence bounded by
time and space. In the context of social media, an event can be happening on
the ground or online. Different mediums can broadcast events happenings on the
ground while people participate in the event through social media discussion.
These kinds of events can be referred to as hybrid events. An example of such
a hybrid event can be a volcano eruption where people participate in the event
using online discussions on social media while it is happening on the ground.
While some events solely happen online, such as gaming, marketing, and
learning events. Events can be communicated in text, photos and videos across
social media platforms. Many events can happen on social media platforms
simultaneously, providing beneficial and prosperous information. It provides
information about the event itself, but it also reveals sentiments and
opinions of the general public and the direction where the events are evolving
shortly. This quick interaction of users and transmission of information makes
it a dynamic process that sometimes proves hard to follow the latest
development, making it a challenging task.

Events also have time dimensions; it is equally important to determine the
time of an event after its detection. For example, an event may have occurred
in the past, happened in the present, or planned to occur in the future. Based
on that, further steps would be taken accordingly as per the requirements of
the situation. The events occurring on social media may directly impact the
personal or social life of the man/woman. The past event can tell us people’s
opinions and other factors; current events can be a great source of developing
a story, while future events can help us prepare in advance. The study
eventtime  reviews the existing research for the detection of disaster events
and classifies them in three dimensions early warning and event detection,
post-disaster, and damage assessment.

The recent example of violence in Bangladesh can explain the link between
social media with real life. On Wednesday, 15 October 2021, clashes were
sparked by videos and allegations that spread across social media that a
Qur’an, the Muslim holy book, had been placed on the knee of a statue of the
Hindu god Hanuman. The violence continued in the following days, which
resulted in the deaths of 7 people, with about 150 people injured; more than
80 special shrines set up for the Hindu festival were attacked. This case
shows social media’s severe and robust effect on our daily lives and ground
situation bngldshviolnc . This violence was termed as “worst communal violence
in years” by New York Times. Similar episodes of violence are becoming a norm
in India since the right of ring-wing politics. If there is the detection of
events occurring on social media in advance, which alerts possible coming
hazards, it can be countered in anticipation, significantly reducing the
reaction immobilization of state forces while maximizing the protection of
people at risk.

Event detection has been long addressed in the Topic Detection, and Tracking
(TDT) in academia onewdet . It mainly focuses on finding and following events
in a stream of broadcast news stories shared by social media posts. Event
Detection (ED) is further divided into two categories depending on the type of
its task; New Event Detection (NED) and Retrospective Event Detection (RED)
redev . NED focuses on detecting a newly occurred event from online text
streams, while RED aims to discover strange events from offline historical
data. Often event detection is associated with identifying the first story on
topics of interest through constant monitoring of social media and news
streams. Other related fields of research are associated with event detection,
such as; event tracking, event summarization, and event prediction. Event
tracking is related to the development of some events over time. Event
summarization outlines an event from the given data, while the event forecasts
the next event within a current event sequence. These topics are part of the
Topic Detection and Tracking (TDT) field.

Event detection is a vast research field, and there are various requirements
and challenges for each task. Various terms have been used to address
different events, making it complex to navigate the literature, sometimes
adding confusion. We propose relevant events based on their characteristics
under the umbrella term “Dangerous Events” (DE). Section 2 defines the term
“Dangerous Events” and the main grouping criteria. In order to clarify the
idea of different dangerous event types, we employ SocialNetCrawler sncrwlr
for extracting tweets that are given as examples for each type.

The rest of the paper is organized as follows: Section II presents the
definition related to dangerous events in social media. Section 3 discusses
event detection and related terms; Section 4 is dedicated to approaches and
techniques used for event detection. Section 5 presents the warning and
countering measures of dangerous events and challenges. Section 6 lists the
challenges, and open problems and the paper is finally concluded in Section 7
with some interesting future research directions.

##  2 Dangerous Events

According to Merriam-Webster merrwebdang , the word “dangerous” means
involving possible injury, pain, harm, or loss characterized by danger. In
that context, we define a dangerous event as the event that poses any danger
to an individual, group, or society. This danger can come in many shapes and
intensities. The objective is to draw a fine line between normal, harmless,
unpleasant, and extreme, abnormal and harmful events. Less sensitive,
unpleasant, and disliked events do not compel the person to feel threatened.
While, in the case of dangerous events, the person will feel fearful, unsafe,
and threatened. This provides the objective to approach the term “event” in a
broader sense to address the common element of all such events. The details of
dangers can always be discussed in detail, providing the necessity of the
situation; for example, a natural disaster proceeds urgent hate speech. In
other words, the first one requires an immediate response with no time to
lose, while the latter can allow some time to take action.

Dangerous events can be anomalies, novelty, outliers, and extreme. These terms
can be used to refer to positive or negative meanings. However, Not all
anomalies, novelties, and extremes are dangerous, but all dangerous events
fulfil one or all of those conditions (extreme, anomaly, novelty). Authors in
extremsentilax  proposed an unsupervised approach to detect extreme sentiments
on social media. It shows that Positive Extreme sentiments can be detected and
differentiated from everyday positive sentiments. Therefore, it may be
concluded that extreme negative sentiments are likely to turn into dangerous
events.

Grouping and defining dangerous events based on their characteristics is
another challenging task, and it can help address the issue of approaching
different types of dangerous events by narrowing it down to specific details.
We will define three broad categories of dangerous events with commonality
among them.

  1. 1.

Scenario-based Dangerous Events

  2. 2.

Sentiment-based Dangerous Events

  3. 3.

Action-based Dangerous Events

Figure 1 gives the depiction of dangerous events and their categories. In the
following subsections, we will outline the definition for each type of
dangerous event.

Figure 1: Dangerous Events and it’s categories.

###  2.1 Scenario-based Dangerous Events

We refer to the word “scenario” as the development of events. These events are
unplanned and unscripted, and most of the time, they occur naturally. Some
planned events can also turn into surprising scenarios. For example, a
peaceful protest can turn into a riot, like in 2020 where a peaceful protest
against corona restrictions in Germany turned into an ugly situation when the
rally was hijacked by right-wing extremists, which ended up storming
Parliament building and exhibiting right-wing symbols and slogans germanparl .

Detecting and tracking natural disasters on social media have been
investigated intensively, and studies sc-based1  have proposed different
methods to identify those disasters by various means. The aim of these studies
has been mainly to tap into the potential of social media to get the latest
updated information provided by social media users in real-time and identify
the areas where assistance is required. These categories are considered
scenario-based dangerous events in this paper, including earthquakes, force
majeure, hurricanes, floods, tornadoes, volcano eruptions, and tsunamis.
Although each calamity’s nature is different, the role of social media in such
events provides a joint base to approach them as scenario-based dangerous
events. A supposed example of scenario-based danger is obtained using the
crawler tool SocialNetCrawler, which can be accessed using the
link111http://sncrawler.di.ubi.pt/:

“@politicususa BREAKING: Scientists predict a tsunami will hit Washington, DC
on 1/18/2020 We Are Marching in DC… https://t.co/3af4ZhyV3J”

###  2.2 Sentiment-based Dangerous Events

Sentiment Analysis (SA), also known as Opinion Mining (OM), is the process of
extracting people’s opinions, feelings, attitudes, and perceptions on
different topics, products, and services. The sentiment analysis task can be
viewed as a text classification problem as the process involves several
operations that ultimately classify whether a particular text expresses
positive or negative sentiment sncamb . For example, A micro-blogging website
like Twitter is beneficial for predicting the index of emerging epidemics.
These are platforms where users can share their feelings which can be
processed to generate vital information related to many areas such as
healthcare, elections, reviews, illnesses, and others. Previous research
suggests that understanding user behaviour, especially regarding the feelings
expressed during elections, can indicate the outcome of elections saelctn .

Sentiments can be positive and negative, but for defining sentiment-based
dangerous events, the applicable sentiments are negatives and, in some
instances, negative extremes. Online radicalization can be attributed to this
threat related to extreme negative sentiments towards certain people,
countries, and governments. Such extreme negative sentiments can result in
protests, online abuse, and social unrest. Detecting these events can help
reduce its impact by allowing the concerned parties to counter beforehand. A
hypothetical example of sentiment-based dangerous example of a tweet obtained
using SocialNetCrawler is given below:

“RT @Lrihendry: When Trump is elected in 2020, I’m outta here. It’s a hate-
filled sewer. It is nearly impossible to watch the hateful at…”

###  2.3 Action-based Dangerous Events

The action involves human indulgence in an event. Various types of actions
happen on the ground that can be detected using social media. Actions can be
of many types, but we point out actions that are causing harm, loss, or threat
to any entity, which again shares the common attribute of negativity and is
highly similar to previously defined types of dangerous events. Some examples
of Action-based dangerous events can be prison breaks, terrorist attacks,
military conflicts, shootings, etc. Several studies have been published
focusing on one or more types of such actions-based events. The study antifas
focuses on anti-fascist accounts on Twitter to detect acts of violence,
vandalism, de-platforming, and harassment of political speakers by Antifa. An
assumed example of action-based example acquired using SocialNetCrawler is
given below:

“RT @KaitMarieox: This deranged leftist and LGBT activist named Keaton Hill
assaulted and threatened to kill @FJtheDeuce, a black conservati…”

##  3 Event Detection methods and techniques

Event Detection has been a popular topic in the research community. Several
methods and techniques have been proposed to detect events depending on
different requirements. These methods directly depend on the type of task and
the data available. As such, they were detecting events from image data is
undoubtedly different from text data. However, in this work, we only refer to
event detection techniques related to text data, particularly data obtained
from social media platforms.

Figure 2: Classification of ED methods.

Event detection methods and techniques revolve around a few basic approaches.
Two approaches that are being used in event detection are document-pivot and
feature-pivot. What differs in these approaches is mainly the clustering
approach, the way documents are used to feature vectors, and the similarity
metric used to identify if the two documents represent the same event or not.
Another approach is the topic modelling approach, primarily based on
probabilistic models.

It originates from the Topic Detection and Tracking task (TDT) field and can
be seen as a clustering issue. Document-pivot approach detects events by
clustering documents based on document similarity as given in Figure 3.
Documents are compared using cosine similarity with Tf-IDF (term frequency-
inverse document frequency) representations, while a Locality Sensitive
Hashing (LSH) lcltyhashing  scheme is utilized to retrieve the best match
rapidly.

Figure 3: Event Detection using Document-pivot approach 31 .

This technique was initially proposed for the analysis of timestamped document
streams. The bursty activity is considered an event that makes some of the
text features more prominent. The features can be keywords, entities and
phrases. Feature-pivot Approach clusters together with terms based on the
pattern they occur as shown in the Figure 4. A study featurenaive  uses a
Naive Bayes classifier to learn the selected features such as keywords to
identify civil unrest and protests and accordingly predict the event days.

Figure 4: Event Detection using Feature-pivot approach 31 .

Topic modelling approaches are based on probabilistic models which detect
events in social media documents in a similar way that topic models identify
latent topics in text documents. In the beginning, topic models depended on
word occurrence, where the text corpora were given as a mixture of words with
latent model topics and the set of identified topics were given as documents.
Latent Dirichlet Allocation (LDA) jelodar2019latent  is the most known
probabilistic topic modelling technique. It is a hierarchical Bayesian model
where a topic distribution is supposed to have a sparse Dirichlet prior. The
model is shown in the Figure. 5, where α is the parameter of the Dirichlet
before the per-document topic distribution θ and φ is the word distribution
for a topic. K represents the number of topics, M represents the document
number, while N gives the number of words in a document. If the word W is the
only observable variable, the learning of topics, word probabilities per
topic, and the topic mixture of each document is tackled as a problem of
Bayesian inference solved by Gibbs sampling.

Figure 5: LDA - A common topic modeling technique 31 .

Many methods are proposed for the detection of events. This event detection
(ED) methods are mainly categorized under the category of supervised and
unsupervised, as shown in Figure 2. Supervised methods include support vector
machine (SVM), Conditional random field (CRF), Decision tree (DT), Naive Bayes
(NB) and others. At the same time, the unsupervised approaches include query-
based, statistical-based, probabilistic based, clustering-based, and graph-
based.

###  3.1 Event Detection Datasets

Due to the growth of the internet and related technologies, the research in
event detection has experienced significant interest and effort. However, the
benchmark datasets for event detection witnessed slow progress. This can be
attributed to the complexity and costliness of annotating events that require
human input. There are a handful number of datasets available that covers
event detection. These datasets are mostly limited to the small size of data
and very restricted types of events. They address specific domains based on
certain features. This also raises issues while using a data-hungry deep
learning model and typically requires balanced data for each class. Some of
these datasets are briefed in the following paragraphs. Table 1 gives the
comparison of the discussed datasets and knowledge bases.

MAVEN wang2020MAVEN  which stands for MAssive eVENt detection dataset, offers
a general domain event detection dataset manually annotated by humans. It uses
English Wikipedia and FrameNet (Baker et al., 1998) documents for building the
dataset. It contains 111,611 various events and 118,732 events mentioned. The
authors claim this to be the largest available human-annotated event detection
dataset. There are 164 different types of events, representing a much wider
range of public domain events. The event types are grouped under five top-
level types: action, change, scenario, sentiment, and possession.

EventWiki ge2018eventwiki  is a knowledge base of events, which consists of
21,275 events containing 95 types of significant events collected from
Wikipedia. EventWiki gives four kinds of information: event type, event info-
box, event summary, and full-text description. Authors claim to be the first
knowledge base of significant events, whereas most knowledge bases focus on
static entities such as people, locations, and organizations.

The EventKG Abdollahi2020EventKGClickAD  is a multilingual1 resource
incorporating event-centric information extracted from several large-scale
knowledge graphs such as Wikidata, DBpedia and YAGO, as well as less
structured sources such as the Wikipedia Current Events Portal and Wikipedia
event lists in 15 languages. It contains details of more than 1,200,000 events
in nine languages. Supported languages include; English, French, German,
Italian, Russian, Portuguese, Spanish, Dutch, Polish, Norwegian, Romanian,
Croatian, Slovene, Bulgarian and Danish.

EVIN EVIN  which stands for EVents In News, describes a method that can
extract events from a news corpus and organize them in relevant classes. It
contains 453 classes of event types and 24,348 events extracted from f 300,000
heterogeneous news articles. The news articles used in this work are from a
highly diverse set of newspapers and other online news providers (e.g.,
http://aljazeera.net/, http://www.independent.co.uk,
http://www.irishtimes.com, etc.). These news articles were crawled from the
external links mentioned on Wikipedia pages while ignoring the content of
Wikipedia pages to get the articles from the original website source.

Dataset | Events | Event types | Document Source | Language | Year | Reference
---|---|---|---|---|---|---
MAVEN | 111, 611 | 164 | English Wikipedia & FrameNet | English | 2020 | wang2020MAVEN
EventWiki | 21,275 | 94 | English WIkipedia | English | 2018 | ge2018eventwiki
EventKG | 1,200,000 | undefined | DBpedia & YAGO… | Multilingual(9) | 2020 | Abdollahi2020EventKGClickAD
EVIN | 24,348 | 453 | news corpus | English | 2014 | EVIN

Table 1: Comparison of Related Event Detection Datasets

###  3.2 Supervised Methods

Supervised methods are expensive and lengthy as they require labels and
training, and this becomes difficult for more extensive data where the cost of
training the model is higher and time-consuming. Some of the supervised
methods for event detection are discussed below.

Support Vector Machines (SVM): Support vector machines are based on the
principle of minimizing structural risks statlrnth  of computer learning
theory. Minimizing structural risks is to find an assumption h for which we
can guarantee the lowest true error. The real error in h is the probability
that h will make an error in a sample test selected at random. An upper limit
can be used to connect the true error of a hypothesis h with the error of h in
the training set and the complexity of H (measured by VC-Dimension), the space
of hypotheses which contains h statlrnth . The supporting vector machines find
the hypothesis h, which (approximately) minimize this limit on the true error
by controlling effectively and efficiently the VC dimension of H.txtcatsvm

It has been confirmed in many works that SVM is one of the most efficient
algorithms for text classification. The accuracy of 87% was achieved to
classify the traffic or nontraffic events on Twitter. It was able to identify
valuable information regarding traffic events through Twitter incdetsm . SVM
combination with incremental clustering technique was applied to detect social
and real-world events from photos posted on Flicker site smedgrphmdl .

Conditional Random Fields (CRF):The CRFs is an essential type of machine
learning model developed based on the Maximum Entropy Markov Model (MEMM). It
was first proposed by Lafferty et al. (2001) as probabilistic models to
segment and label sequence data, inherit the advantages of the previous
models, increase their efficiency, overcome their defects, and solve more
practical problems crfprbmdl . A conditional Random Field (CRF) classifier was
learned to extract the artist name and location of music events from a corpus
of tweets edinsmfeeds .

Decision Tree (DT): Decision tree learning is a supervised machine learning
technique for producing a decision tree from training data. A decision tree is
also referred to as a classification tree or a reduction tree, and it is a
predictive model which draws from observations about an item to conclusions
about its target value. In the tree structure, leaves represent
classifications (also referred to as labels), non-leaf nodes are features, and
branches represent conjunctions of features that lead to the classification
artanlyzsft . A decision tree classifier called gradient boosted was used to
anticipate whether the tweets consist of an event concerning the target entity
or not.

Naïve Bayes (NB): Naïve Bayes is a simple learning algorithm that uses the
Bayes rule and a strong assumption that the attributes are conditionally
independent if the class is given. Although this independence assumption is
often violated in practice, naïve Bayes often provides competitive accuracy.
Its computational efficiency and many other distinctive features result in
naïve Bayes being extensively applied in practice.

Naïve Bayes gives a procedure for using the information in sample data to
determine the posterior probability P(y|x) of each class y, given an object x.
Once we have such estimates, they can be then used for classification or other
decision support applications. enclypmlnaivebyes

###  3.3 Unsupervised Methods

The unsupervised method does not usually require training or target labels.
However, they can depend on specific rules based on the model and
requirements. The unsupervised methods being used for event detection are
discussed below. Many unsupervised methods are developed by scientists who are
grouped into different categories that are described in the following
subsections.

Query Based Methods: Query-based methods are based on queries and simple rules
to identify planned rules from multiple websites. e.g., YouTube, Flicker,
Twitter. An event’s temporal and spatial information was extracted and then
used to inquire about other social media websites to obtain relevant
information.qureybsd  The query-based method requires predefined keywords if
there are many keywords to avoid unimportant events.

Statistical Based Methods: Different researchers under this category
introduced many methods. For example, the average frequency of unigrams was
calculated to find the significant unigrams (keywords) and combine those
unigrams to illustrate the trending events. unsuptopkextrct  The attempt was
made to detect the hot events by identifying burst features (i.e., unigram)
during different time windows. Each unigram bursty feature signal was then
converted into a frequency domain. They were using Discrete Fourier
Transformation (DFT). However, DFT was not able to detect the period when
there is a burst which is very important in ED processtwitternewsrepo .

Wavelet Transformation(WT): Another technique called Wavelet Transformation
(WT) was introduced to assign signals to each unigram feature. WT technique is
different from DFT in term of isolating time and frequency and provide better
resultsedintwitter . A new framework was proposed that integrated different
unsupervised techniques. For example, LDA, NER, bipartite graph clustering
algorithm based on relation and centrality scores to discover hidden events
and extract their essential information such as time, location, and people
that have been involved edsocialweb .

Named Entity Relation(NER): Named Entity Relation (NER) identifies increasing
weights for the proper noun features. A proposed technique applied tweet
segmentation to get the sentences containing more phrasing words instead of
unigrams. Later, they computed the TFIDF of these sentences and user frequency
and increased weights for the proper noun features identified by Named Entity
Relation (NER). Li et al. (2012a)first applied tweets classified them using
K-Nearest Neighbor (KNN) to identify the events from tweets published by
Singapore userstwitterbsded .

Weiler et al. (2014) eventidentity  used shifts of terms computed by Inverse
Document Frequency (IDF) over a simple sliding window model to detect events
and trace their evolution. Petrović et al. ( 2010) fsdtwitter  modified and
used Locality Sensitive Hashing (LSH) to perform First Story Detection (FSD)
task on Twitter.

Probabilistic Based Methods: Latent Dirichlet Allocation (LDA) and
Probabilistic Latent Semantic Indexing (PLSI) are topic modelling methods that
are being used for event detection. In LDA, each document has many topics, and
it is supposed to have a group of topics for each document. The model is shown
in Figure 6.

LDA worked well with news articles and academic abstracts, but it fell short
for small texts. However, the LDA model has been improved by adding tweet
pooling schemes and automatic labelling. Pooling schemes include basic scheme,
author scheme, burst term scheme, temporal scheme, and hashtag scheme tweets
published under the same hashtag. The experiment results proved that the
hashtag scheme produced the best clusters results ldatpc . However, LDA
defines the number of topics and terms per topic in advance, inefficiently
implementing it over social media.

Figure 6: Topic Modeling in LDA srvyedtxtstrm .

Clustering-Based Method: Clustering-based methods mainly rely on selecting the
most informative features, which contribute to event detection, unlike
supervised methods, which need labelled data for prediction. It contributes to
detecting events more accurately.

Figure 7: Clustering-Based method srvyedtxtstrm .

Many clustering-based methods exist for text data, and K-means is a famous
clustering algorithm. A novel dual-level clustering was proposed to detect
events based on news representation with time2vec dualclstr . Clustering-based
methods have been employed in various ways and other techniques such as NER,
TFIDF and others in different tasks, but the ideal clustering technique is
still yet to come.

Graph-Based Methods: Graph-based methods consist of nodes/vertices
representing entities and edges representing the relationship between the
nodes. Valuable information can be extracted from these graphs by grouping a
set of nodes based on the set of edges. Each generated group is called a
cluster/graph structure, and it is also known as community, cluster or module.
The links between different nodes are called intra-edges. Meanwhile, links
that connect different communities are called inter-edges.

Figure 8: Graph-based clustering method srvyedtxtstrm .

###  3.4 Semi-Supervised Methods

Semi-supervised learning combines both supervised and unsupervised learning
methods. Typically, a small number of labelled and largely unlabeled data is
used for training purposes. Sometimes they are also referred to as the hybrid
method. If there is a vast number of unlabeled data combined with insufficient
labelled data, it can affect the classification accuracy. It is also referred
to as imbalanced training data.

Similarly, if there is no labelled data for a particular class, the
classification can become inefficient and accurate. Some of the semi-
supervised methods include self-training, generative models and graph-based
methods. A semi-supervised algorithm based on tolerance roughest and ensemble
learning is recommended for such kinds of problems ensamblern . The missing
class is extracted by approximation from the dataset and used as the labelled
sample. The ensemble classifier iteratively builds the margin between positive
and negative classes to estimate negative data further since negative data is
mixed with the positive data. Therefore, classification is done without
training samples by applying a hybrid approach, and it saves the cost of
getting labelled data manually, especially for larger datasets.

##  4 Discussion

This section discusses different works related to event detection that are
categorized under the types proposed earlier in this work. The types of events
are scenario-based, sentiment-based, and action-based dangerous events. Each
work is described in this section and its event type and technique.
Furthermore, this section also discusses the research related to event
prediction. Table 3 illustrates different type of events detection from social
media.

###  4.1 Detection of Different/Dangerous Events on Social Media

Nourbakhsh et al. nourbakhsh2017breaking  address natural and artificial
disasters on social media. They identified events from local news sources that
may become global breaking news within the next 24 hours. They used Reuters
News Tracer, a real-time news detection and verification engine. It uses a
fixed sphere decoding (FSD) algorithm to detect breaking stories in real-time
from Twitter. Each event is shown as a cluster of tweets engaging with that
story. By considering different data features, they applied SGD and SVM
classifier that detects breaking disasters from postings of local authorities
and local news outlets.

Sakaki et al. equake2010  leverage Twitter for detecting earthquake occurrence
promptly. They propose a method to scrutinize the real-time interaction of
earthquakes events and, similar to detect a target event. Semantic analyses
were deployed on tweets to classify them into positive and negative classes.
The target for classification is two keywords; earthquake or shaking, which
are also addressed as query words. Total of 597 positive samples of tweets
that report earthquake occurrence is used as training data. They also
implemented filtering methods to identify the location, and an application
called the earthquake reporting system in Japan.

Liu et al. crisisbert  aims for crisis events. They propose a state-of-the-art
attention-based deep neural networks model called CrisisBERT to embed and
classify crisis events. It consists of two phases which are crisis detection
and crisis recognition. In addition, another model for embedding tweets is
also introduced. The experiments are conducted on C6 and C36 datasets.
According to the authors, these models surpass state-of-the-art performance
for detection and recognition problems by up to 8.2% and 25.0%, respectively.

Archie et al. hurrearth  proposed an unsupervised approach for the detection
of sub-events in major natural disasters. Firstly, noun-verb pairs and phrases
are extracted from tweets as an important sub-event prospect. In the next
stage, the semantic embedding of extracted noun-verb pairs and phrases is
calculated and then ranked against a crisis-specific ontology called
management of Crisis (MOAC) ontology. After filtering these obtained candidate
sub-events, clusters are formed, and top-ranked clusters describe the highly
important sub-events. The experiments are conducted on Hurricane Harvey and
the 2015 Nepal Earthquake datasets. According to the authors, the approach
outperforms the current state-of-the-art sub-event identification from social
media data.

Forests fire have become a global phenomenon due to rising droughts and
increasing temperatures which is often attributed to global warming and
climate change. The work firehaze  tests the usefulness of social media to
support disaster management. However, the primary data for dealing with such
incidents come from NASA satellite imagery. The authors use GPS-stamped tweets
posted during 2014 from Sumatra Island, Indonesia, which experiences many haze
events. Twitter has proven to be a valuable resource during such events,
confirmed by performing analysis on the dataset. Furthermore, the authors also
announced the development of a tool for disaster management.

Huang et al. emergemcyweibo  focuses on emergency events. They consider the
various type of events under the term “emergency events”. It includes
infectious disease, explosions, typhoons, hurricanes, earthquakes, floods],
tsunamis, wildfires, and nuclear disasters. The model must automatically
identify the attribute information 3W (What, When, and Where) of emergency
events to respond in time. Their proposed solution contains three phases, the
classification phase, the extraction phase, and the clustering phase, and it
is based on the Similarity-Based Emergency Event Detection (SBEED) framework.
The experiment is done using the Weibo dataset. Different classification
models such as KNN, Decision Trees, Naïve Bayes, Linear SVC (RBF), and Text-
CNN are used in the classification phase. Secondly, time and location are
extracted from the classification obtained. Lastly, an unsupervised dynamical
text clustering algorithm is deployed to cluster events depending on the text-
similarity of type, time and location information. The authors claim
superiority of the proposed framework having good performance and high
timeliness that can be described what emergency, and when and where it
happened.

Pais et al. present an unsupervised approach to detect extreme sentiments on
social networks. Online wings of radical groups use social media to study
human sentiments engaging with uncensored content to recruit them. They use
people who show sympathy for their cause to further promote their radical and
extreme ideology. The authors developed a prototype system composed of two
components, i.e., Extreme Sentiment Generator (ESG) and Extreme Sentiment
Classifier (ESC). ESG is a statistical method used to generate a standard
lexical resource called ExtremesentiLex that contains only extreme positive
and negative terms. This lexicon is then embedded to ESC and tested on five
different datasets. ESC finds posts with extremely negative and positive
sentiments in these datasets. The result verifies that the posts that were
previously classified as negatives or positives are, in fact, extremely
negatives or positives in most cases.

COVID-19 pandemic has forced people to change their lifestyles. Lockdown
further pushed people to use social media to express their opinions and
feelings. It provides a good source for studying users’ topics, emotions, and
attitudes discussed during the pandemic. The authors of work covsent
collected two massive COVID-19 datasets from Twitter and Instagram. They
explore data with different aspects, including sentiment analysis, topics
detection, emotions, and geo-temporal. Topic modelling on these datasets with
distinct sentiment types (negative, neutral, positive) shows spikes on
specific periods. Sentiment analysis detects spikes on specific periods and
identifies what topics led to those spikes attributed to economy, politics,
health, social, and tourism. Results showed that COVID-19 affected significant
countries and experienced a shift in public opinion. Much of their attention
was towards China. This study can be very beneficial to read people’s
behaviour as an aftermath; Chinese people living in those countries also faced
discrimination and even violence because of the Covid-19 linked with China.

Plaza-del-Arco et al. hateoffense  investigate the link of hate speech and
offensive language(HOF) with relevant concepts. Hate speech targets a person
or group as a negative opinion, and it is related to sentiment analysis and
emotion analysis as it causes anger and fear inside the person experiencing
it. The approach consists of three phases and is based on multi-task learning
(MTL). The setup is based on BERT, a transformer-based encoder pre-trained on
a large English corpus. Four sequence classification heads are added to the
encoder, and the model is fine-tuned for multi-class classification tasks. The
sentiment classification task categorizes tweets into positive and negative
categories, while emotion classification classifies tweets into different
emotion categories (anger, disgust, fear, joy, sadness, surprise, enthusiasm,
fun, hate, neutral, love, boredom, relief, none). The offence target is
categorized as an individual, group, and unmentioned to others. Final
classification detects HOF and classifies tweets into HOF and non-HOF.

Kong et al. farrightextreme  explore a method that explains how extreme views
creep into online posts. Qualitative analysis is applied to make ontology
using Wikibase. It proceeded from the vocabulary of annotations such as the
opinions expressed in topics and labelled data collected from three online
social networking platforms (Facebook, Twitter, and Youtube). In the next
stage, a dataset was created using keyword search. The labelled dataset is
then expanded to using a looped machine learning algorithm. Two detailed case
studies are outlined with observations of problematic online speech evolving
the Australian far-right Facebook group. Using our quantitative approach, we
analyzed how problematic opinions emerge. The approach exhibits how
problematic opinions appear over time and how they coincide.

Demszky et al.plrztnpolitk  highlights four linguistic dimensions of political
polarization in social media, which includes; topic choice, framing, affect
and apparent force. These features are quantified with existing lexical
methods. Clustering of tweet embeddings is proposed to identify important
topics for analysis in such events. The method is deployed on 4.4M tweets
related to 21 mass shootings. Evidence proves the discussions on these events
are highly polarized politically, and it is driven by the framing of biased
differences rather than topic choice. The measures in this study provide
connecting evidence that creates a big picture of the complex ideological
division penetrating public life. The method also surpasses LDA-based
approaches for creating common topics.

While most typical use of social media is focused on disease outbreaks,
protests, and elections, Khandpur et al. cyberattack  explored social media to
uncover ongoing cyber-attacks. The unsupervised approach detects cyber-attacks
such as; breaches of private data, distributed denial of service (DDOS)
attacks, and hijacking of accounts while using only a limited set of event
trigger as a fixed input.

Coordinated campaigns aim to manipulate and influence users on social media
platforms. Pacheco et al. coordcomp  work aim to unravel such campaigns using
an unsupervised approach. The method builds a coordination network relying on
random behavioural traces shared between accounts. A total of five case
studies are presented in work, including U.S. elections, Hong Kong protests,
the Syrian civil war, and cryptocurrency manipulation. Networks of coordinated
Twitter accounts are discovered in all these cases by inspecting their
identities, images, hashtag similarities, retweets, or temporal patterns. The
authors propose using the presented approach for uncovering various types of
coordinated information warfare scenarios.

Coordinated campaigns can also influence people towards offline violence. Xian
Ng et al. coordinariots  investigates the case of capital riots. They
introduce a general methodology to discover coordinated by analyzing messages
of user parleys on Parler. The method creates a user-to-user coordination
network graph prompted by a user-to-text graph and a similarity graph. The
text-to-text graph is built on a textual similarity of posts shared on Parler.
The study of three prominent user groups in the 6 January 2020 Capitol riots
detected networks of coordinated user clusters that posted similar textual
content in support of different disinformation narratives connected to the
U.S. 2020 elections.

Wanzheng Zhu and Suma Bhat drgEuphemisticPD  studies the specific case of the
use of euphemisms by fringe groups and organizations that is expression
substituted for one considered to be too harsh. The work claims to address the
issue of Euphemistic Phrase detection without human effort for the first time.
Firstly the phrase mining is done on raw text corpus to extract standard
phrases; then, word embedding similarity is implemented to select candidates
of euphemistic phrases. In the final phases, those candidates are ranked using
a masked language model called SpanBERT.

Yang Yang et al. hmntrfk  explore the use of Network Structure Information
(NSI) for detecting human trafficking on social media. They present a novel
mathematical optimization framework that combines the network structure into
content modelling to tackle the issue. The experimental results are proven
effective for detecting information related to human trafficking.

Tweets | Proposed dangerous event type
---|---
“RT @KaitMarieox: This deranged leftist and LGBT activist named Keaton Hill assaulted and threatened to kill @FJtheDeuce, a black conservati…” | Action-based dangerous event
“RT @Lrihendry: When Trump is elected in 2020, I’m outta here. It’s a hate-filled sewer. It is nearly impossible to watch the hateful at…” | Sentiment-based dangerous event
“Scientists predict a tsunami will hit Washington, DC on 1/18/2020 We Are Marching in DC… https://t.co/3af4ZhyV3J” | Scenario-based dangerous event
Table 2: Presumed types of dangerous events for tweets.

Authors present Table 2 to clarify the intent of this work by providing an
example of the collected tweets and their presumed techniques. Based on the
existing methods for event detection, it gives a clear objective for using
these methods for detecting dangerous events.

###  4.2 Event Prediction

Event prediction is a complex issue that revolves around many dimensions.
Various events are challenging to predict before they become apparent. For
example, it is impossible to predict in case of natural disasters, and they
can only be detected after the occurrence. Some events can be predicted while
they are still in the evolving phase. Authors of nourbakhsh2017breaking
identify events from local news sources before they may become breaking news
globally. The use case of Covid-19 can be regarded as an example where it
started locally and became a global issue later.

A dataset is obtained from a recent Kaggle competition to explore the
usability of a method for predicting disaster in tweets. The work in
twitterbertpred  tests the efficiency of BERT embedding, which is an advanced
contextual embedding method that constructs different vectors for the same
word in various contexts. The result shows that the deep learning model
surpasses other typical existing machine learning methods for disaster
prediction from tweets.

Zhou et al. covidfatlerate  proposed a novel framework called as Social Media
enhAnced pandemic suRveillance Technique (SMART) to predict Covid-19 confirmed
cases and fatalities. The approach consists of two parts where firstly,
heterogeneous knowledge graphs are constructed based on the extracted events.
Secondly, a module of time series prediction is constructed for short-and
long-term forecasts of the confirmed cases and fatality rate at the state
level in the United States and finally discovering risk factors for
intervening COVID-19. The approach exhibits an improvement of 7.3% and 7.4%
compared to other state-of-the-art methods.

Most of the other existing research targets particular scenarios of event
prediction with limited scope. Keeping in mind the complexity of this problem,
we only present a few related works, and the generalization is obscure.

| Event Type | Technique | Reference | Dataset | Year
---|---|---|---|---|---
Scenario-based |  |  |  |  |
| Natural Disasters | SVM/SGD | nourbakhsh2017breaking  | Twitter | 2017
| Earthquake | Classification(SVM) | equake2010  | Twitter | 2010
| Crisis | CrisisBERT | crisisbert  | Twitter (C6,C36) | 2021
| Earthquake &Hurricane | Unsupervised | hurrearth  | Twitter | 2019
| Fire and Haze Disaster | Classification (hotspots) | firehaze  | NASA &Twitter | 2017
| Emergency | Text-CNN, Linear SVC & Clustering | emergemcyweibo  | Weibo | 2021
| … | … | … | … | …
Sentiment-based |  |  |  |  |
| Extreme Sentiments | Unsupervised learning | extremsentilax  | misc. | 2020
| Covid19 Sentiments | word2vec | covsent  | Twitter & Instagram | 2021
| Hate speech & offensive Language | BERT | hateoffense  | HASOC(Twitter) | 2021
| Far-right Extremism | Classification | farrightextreme  | Facebook, Twitter &Youtube | 2021
| Political Polarization | Clustering | plrztnpolitk  | Twitter | 2019
| … | … | … | … | …
Action-based |  |  |  |  |
| Cyber attack | Unsupervised | cyberattack  | Twitter | 2017
| Coordinated campaigns | Unsupervised | coordcomp  | misc. | 2021
| Riots | Clustering | coordinariots  | Parler | 2021
| Drugs Trafficking | SPANBert | drgEuphemisticPD  | Text Corpus(subreddit) | 2021
| Human Trafficking | Classification (NSI) | hmntrfk  | Wiebo | 2018
| … | … | … | … | …
Table 3: Dangerous Events categorized under relevant types

##  5 Conclusion

In this work, we laid the basis of the term “Dangerous Events” and explored
different existing techniques and methods for detecting events on social
media. Dangerous events contain a broad meaning, but we keep it essential to
define the term better. We believe much more can be included in dangerous
events, as we explored in the discussion section. Categorizing dangerous
events into sub-categories can help specify the event and its features. The
subcategories consist of scenario-based, sentiment-based and action-based
dangerous events. The usefulness of social media these days provides a
significant advantage in detecting such events initially. While in some cases,
significant events also originate from social media and manifest in real life,
such as; mass protests, communal violence and radicalization.

The events on social media are mainly polarized. People use it to express
their likes or dislike, which can also be classified as positive or negative.
Not all extreme events are dangerous, but all dangerous events are extreme, as
people can show their happiness using extreme emotions, which is an anomaly
under normal circumstances. There is a common element in all dangerous events:
we all want to avoid it because of the harm it brings. We believe there is an
excellent scope for related work in future. As a proposal, we suggest a
dataset containing all types of dangerous events. Secondly, different
techniques can be applied to this dataset to further deepen the usefulness and
evolve a technique that can be generalized for all kinds of such events.
Considering the limitations of event detection, techniques covering only
specific events, a joint base can help discover the universally applicable
method.

Acknowledgments

This work was supported by National Founding from the FCT- Fundacão para a
Ciência e a Tecnologia, through the MOVES Project-PTDC/EEI-AUT/28918/2017 and
by operation Centro-01-0145-FEDER-000019-C4-Centro de Competências em Cloud
Computing, co-financed by the European Regional Development Fund (ERDF)
through the Programa Operacional Regional do Centro (Centro 2020), in the
scope of the Sistema de Apoio à Investigação Científica e Tecnológica.

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# Lipschitz continuity of the solutions to the Dirichlet problems for the
invariant laplacians

Congwen Liu<EMAIL_ADDRESS>CAS Wu Wen-Tsun Key Laboratory of Mathematics,
School of Mathematical Sciences, University of Science and Technology of China
Hefei, Anhui 230026, People’s Republic of China.  and  Heng Xu
<EMAIL_ADDRESS>School of Mathematical Sciences, University of
Science and Technology of China
Hefei, Anhui 230026, People’s Republic of China.

###### Abstract.

This short note is motivated by an attempt to understand the distinction
between the Laplace operator and the hyperbolic Laplacian on the unit ball of
$\mathbb{R}^{n}$, regarding the Lipschitz continuity of the solutions to the
corresponding Dirichlet problems.

We investigate the Dirichlet problem

$\begin{cases}\Delta_{\vartheta}u=0,&\text{ in }\,\mathbb{B}^{n},\\\
u=\phi,&\text{ on }\,\mathbb{S}^{n-1},\end{cases}$

where

$\Delta_{\vartheta}:=(1-|x|^{2})\bigg{\\{}\frac{1-|x|^{2}}{4}\Delta+\vartheta\sum_{j=1}^{n}x_{j}\frac{\partial}{\partial
x_{j}}+\vartheta\left(\frac{n}{2}-1-\vartheta\right)I\bigg{\\}}.$

We show that the Lipschitz continuity of boundary data always implies the
Lipschitz continuity of the solutions if $\vartheta>0$, but does not when
$\vartheta\leq 0$.

###### Key words and phrases:

Lipschitz continuity; Dirichlet boundary value problem; invariant laplacians

###### 2010 Mathematics Subject Classification:

Primary 35J25; Secondary 31B05.

The first author was supported by the National Natural Science Foundation of
China grant 11971453.

##  1\. Introduction

Let $\mathbb{B}^{n}$ be the open unit ball in $\mathbb{R}^{n}$ ($n\geq 2$) and
$\mathbb{S}^{n-1}$ the unit sphere.

It is known, even for $n=2$, that the Lipschitz continuity of the boundary
data $\phi:\mathbb{S}^{n-1}\to\mathbb{R}^{n}$ does not imply the Lipschitz
continuity of the solution of the Dirichlet boundary value problem

(1) $\begin{cases}\Delta u=0,&\text{in }\,\mathbb{B}^{n},\\\ u=\phi,&\text{on
}\,\mathbb{S}^{n-1}.\end{cases}$

See [2, Example 1] for a counterexample. In [2], the authors also showed that
the Lipschitz continuity of $\phi$ implies the Lipschitz continuity of the
solution provided that the harmonic extension $P[\phi]$ of $\phi$ is a
$K$-quasiregular mapping. Kalaj [9] obtained a related result, but under
additional assumption of $C^{1,\alpha}$ regularity of $\phi$.

Recently, Chen et al. [4] investigate solutions of the hyperbolic Poisson
equation $\Delta_{h}u=\psi$, where

$\Delta_{h}u(x):=(1-|x|^{2})^{2}\Delta
u(x)+2(n-2)(1-|x|^{2})\sum_{j=1}^{n}x_{j}\frac{\partial}{\partial x_{j}}u(x)$

is the Laplace-Beltrami operator on the unit ball $\mathbb{B}^{n}$. Among the
other things, the authors showed that, in contrast with the above, the
Lipschitz continuity of $\phi$ implies the Lipschitz continuity of the
solution of the Dirichlet boundary value problem

(2) $\begin{cases}\Delta_{h}u=0,&\text{in }\,\mathbb{B}^{n},\\\
u=\phi,&\text{on }\,\mathbb{S}^{n-1},\end{cases}$

with no other assumption than that $n\geq 3$. See [4, Theorem 1.2].

This short note is motivated by an attempt to understand the distinction
between the two boundary value problems. To this end, we consider a family of
differential operators

$\Delta_{\vartheta}u(x):=(1-|x|^{2})\bigg{\\{}\frac{1-|x|^{2}}{4}\Delta
u(x)+\vartheta\sum_{j=1}^{n}x_{j}\frac{\partial u}{\partial
x_{j}}(x)+\vartheta\left(\frac{n}{2}-1-\vartheta\right)u(x)\bigg{\\}}.$

We call $\Delta_{\vartheta}$ the (Möbius) invariant Laplacians, since

$\Delta_{\vartheta}\left\\{\left(\mathrm{det}\,\psi^{\prime}(x)\right)^{\frac{n-2-2\theta}{2n}}f(\psi(x))\right\\}=\left(\mathrm{det}\psi^{\prime}(x)\right)^{\frac{n-2-2\theta}{2n}}\left(\Delta_{\vartheta}f\right)(\psi(x))$

for every $f\in C^{2}(\mathbb{B}^{n})$ and for every Möbius transformation
$\psi$ (see [7, Proposition 3.2]). These differential operators are closely
related to polyharmonic functions. In particular, the differential operator
$L_{\vartheta}:=\frac{4}{1-|x|^{2}}\Delta_{\vartheta}$ plays a crucial role in
the modified Almansi representation for polyharmonic functions in [8] as well
as in the cellular decomposition theorem for polyharmonic functions in [3, 8].
We refer the reader to [6, 7, 11] for more information about these
differential operators.

We investigate the Dirichlet problem

(3) $\begin{cases}\Delta_{\vartheta}u=0,&\text{ in }\,\mathbb{B}^{n},\\\
u=\phi,&\text{ on }\,\mathbb{S}^{n-1},\end{cases}$

which includes as special cases the Dirichlet problems (1) and (2), in view of

$\Delta_{0}=\frac{(1-|x|^{2})^{2}}{4}\Delta\quad\text{and}\quad\Delta_{\frac{n}{2}-1}=\frac{1}{4}\,\Delta_{h}.$

Unlike in [2, 4], for convenience, we work with functions instead of mappings.

Our main result is as follows.

######  Theorem 1.1.

Suppose $n\geq 2$ and $\vartheta>-1/2$.

  1. (i)

If $\vartheta\leq 0$, there exists a Lipschitz continuous function
$\phi:\mathbb{S}^{n-1}\to\mathbb{R}$ such that the solution of (3) is not
Lipschitz continuous.

  2. (ii)

If $\vartheta>0$, the Lipschitz continuity of boundary data always implies the
Lipschitz continuity of the solutions.

######  Remark 1.

When $\vartheta=n/2-1$, $\Delta_{\vartheta}=\frac{1}{4}\,\Delta_{h}$ and the
assumption $\vartheta>0$ reads $n>2$. Theorem 1.1 explains why the assumption
$n\geq 3$ in [4, Theorem 1.2] is necessary.

##  2\. Preliminaries

In this section, we gather some technical lemmas needed for Sections 3 and 4,
and fix our notation.

According to [6], the Poisson kernel associated to $\Delta_{\vartheta}$ is
given by

$P_{\vartheta}(x,\zeta):=c_{n,\vartheta}\frac{(1-|x|^{2})^{1+2\vartheta}}{|x-\zeta|^{n+2\vartheta}},\qquad(x,\zeta)\in\mathbb{B}^{n}\times\mathbb{S}^{n-1},$

with
$c_{n,\vartheta}:=\frac{\Gamma\left(\frac{n}{2}+\vartheta\right)\Gamma(1+\vartheta)}{\Gamma\left(\frac{n}{2}\right)\Gamma(1+2\vartheta)}$
and the Poisson integral of a function $\phi\in C(\mathbb{S}^{n-1})$ is given
by

(4)
$\mathcal{P}_{\vartheta}[\phi](x):=\int_{\mathbb{S}^{n-1}}P_{\vartheta}(x,\zeta)\phi(\zeta)d\sigma(\zeta),$

where $\sigma$ is the surface measure on $\mathbb{S}^{n-1}$ normalized so that
$\sigma(\mathbb{S}^{n-1})=1$.

We record the following from [6, Theorem 2.4].

######  Lemma 2.1.

The Dirichlet problem (3) has a solution for all $\phi\in C(\mathbb{S}^{n-1})$
if and only if $\vartheta>-1/2$. Moreover, when $\vartheta>-1/2$, the solution
is given by $u=\mathcal{P}_{\vartheta}[\phi]$.

Straightforward calculation yields the following.

######  Lemma 2.2.

For $k=1,\ldots,n$, we have

(5) $\displaystyle\frac{\partial P_{\vartheta}}{\partial
x_{k}}(x,\zeta)~{}=~{}$
$\displaystyle-2(1+2\vartheta)c_{n,\vartheta}\frac{(1-|x|^{2})^{2\vartheta}x_{k}|x-\zeta|^{2}}{|x-\zeta|^{n+2\vartheta+2}}$
$\displaystyle\quad-~{}(n+2\vartheta)c_{n,\vartheta}\frac{(1-|x|^{2})^{1+2\vartheta}(x_{k}-\zeta_{k})}{|x-\zeta|^{n+2\vartheta+2}}.$

A number of hypergeometric functions will appear throughout. We use the
classical notation

$\>_{2}\\!F_{1}\left(a,\,b;\,c;\,\lambda\right):=\sum_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}}{(c)_{k}}\frac{\lambda^{k}}{k!}$

with $c\neq 0,-1,-2,\ldots$, where $(a)_{k}$ stands for the Pochhammer symbol,
which is defined as

$(a)_{k}:=\begin{cases}1,&\text{ if }k=0,\\\ a(a+1)\ldots(a+k-1),&\text{ if
}k\geq 1.\end{cases}$

We refer to [1, Chapter 2] for the properties of these functions. Here, we
only record two formulas for later reference.

(6) $\>{}_{2}\\!F_{1}\left(a,\,b;\,c;\,\lambda\right)~{}=~{}$
$\displaystyle(1-\lambda)^{c-a-b}\>_{2}\\!F_{1}\left(c-a,\,c-b;\,c;\,\lambda\right).$
(7)
$\displaystyle\frac{d}{d\lambda}\>_{2}\\!F_{1}\left(a,\,b;\,c;\,\lambda\right)~{}=~{}$
$\displaystyle\frac{ab}{c}\>_{2}\\!F_{1}\left(a+1,\,b+1;\,c+1;\,\lambda\right).$

######  Lemma 2.3.

We have

(8)
$\int_{\mathbb{S}^{n-1}}P_{\vartheta}(x,\zeta)~{}d\sigma(\zeta)~{}=~{}c_{n,\vartheta}\,\>{}_{2}\\!F_{1}\left(-\vartheta,\,\frac{n}{2}-1-\vartheta;\,\frac{n}{2};\,|x|^{2}\right)$

and for $k=1,\ldots,n$,

(9) $\int_{\mathbb{S}^{n-1}}\frac{\partial P_{\vartheta}}{\partial
x_{k}}(x,\zeta)~{}d\sigma(\zeta)~{}=~{}C(n,\vartheta)\,\>_{2}\\!F_{1}\left(1-\vartheta,\,\frac{n}{2}-\vartheta;\,\frac{n}{2}+1;\,|x|^{2}\right)x_{k},$

with

$C(n,\vartheta):=\frac{-2\vartheta(n-2-2\vartheta)}{n}\,c_{n,\vartheta}.$

###### Proof.

The first identity follows from the formula ([6, Lemma 2.1])

(10)
$\int_{\mathbb{S}^{n-1}}\frac{d\sigma(\zeta)}{|x-\zeta|^{2\lambda}}~{}=~\>{}_{2}\\!F_{1}\left(\lambda,\,\lambda-\frac{n}{2}+1;\,\frac{n}{2};\,|x|^{2}\right),\quad
x\in\mathbb{B}^{n},$

and (6). The identity (9) follows by differentiating both sides of (8) and
applying (7). ∎

######  Lemma 2.4.

When $\vartheta>0$, the functions

$x~{}\longmapsto~{}\int_{\mathbb{S}^{n-1}}\frac{\partial
P_{\vartheta}}{\partial x_{k}}(x,\zeta)~{}d\sigma(\zeta),\quad k=1,\ldots,n,$

are all bounded on $\mathbb{B}^{n}$.

###### Proof.

It follows from (9) that

$\displaystyle\left|\int_{\mathbb{S}^{n-1}}\frac{\partial
P_{\vartheta}}{\partial x_{k}}(x,\zeta)~{}d\sigma(\zeta)\right|~{}\lesssim~{}$
$\>{}_{2}\\!F_{1}\left(1-\vartheta,\,\frac{n}{2}-\vartheta;\,\frac{n}{2}+1;\,|x|^{2}\right)$
$\displaystyle~{}=~{}$
$\displaystyle\sum_{j=0}^{\infty}\frac{(1-\vartheta)_{j}(\frac{n}{2}-\vartheta)_{j}}{j!\,(\frac{n}{2}+1)_{j}}|x|^{2j}.$

Here and below, we use the notation $f_{1}\lesssim f_{2}$ if there exists a
uniform constant $C>0$, such $f_{1}\leq Cf_{2}$. Also, $f_{1}\approx f_{2}$
means that $f_{1}\lesssim f_{2}$ and $f_{2}\lesssim f_{1}$. Note that the
coefficients in the last series are of order $j^{-1-2\vartheta}$, as
$j\to\infty$. This proves the assertion. ∎

The following lemma may be well known, we include a short proof for the
convenience of the reader.

######  Lemma 2.5.

Suppose that $p>q\geq 0$. We have

(11) $\int_{\mathbb{S}^{n-1}}\frac{|\zeta-e_{1}|^{q}}{|\zeta-
re_{1}|^{n-1+p}}\,d\sigma(\zeta)~{}\lesssim~{}\frac{1}{(1-r)^{p-q}},\qquad
r\in[0,1).$

###### Proof.

The special case $q=0$ of the lemma is well-known, see for instance [5, Lemma
2.9].

For $q>0$, noting that

$|\zeta-e_{1}|^{q}\leq\left(|\zeta-re_{1}|+|re_{1}-e_{1}|\right)^{q}\leq
2^{q}\big{[}|\zeta-re_{1}|^{q}+(1-r)^{q}\big{]},$

we have

$\displaystyle\int_{\mathbb{S}^{n-1}}\frac{|\zeta-e_{1}|^{q}}{|\zeta-
re_{1}|^{n-1+p}}\,d\sigma(\zeta)~{}\lesssim~{}$
$\displaystyle\int_{\mathbb{S}^{n-1}}\frac{d\sigma(\zeta)}{|\zeta-
re_{1}|^{n-1+p-q}}$
$\displaystyle\quad~{}+~{}(1-r)^{q}\int_{\mathbb{S}^{n-1}}\frac{d\sigma(\zeta)}{|\zeta-
re_{1}|^{n-1+p}},$

and (11) follows from the special case $q=0$. ∎

##  3\. Proof of Theorem 1.1, Part (i)

In the case $\vartheta=0$, as is mentioned in the introduction, [2, Example 1]
is such an example.

Now we assume that $\vartheta<0$. Take $\phi\equiv 1$. By (9) and (7),

$\displaystyle\sum_{k=1}^{n}x_{k}\frac{\partial}{\partial
x_{k}}\mathcal{P}_{\vartheta}[\phi](x)~{}=~{}$ $\displaystyle
C(n,\vartheta)\,\>_{2}\\!F_{1}\left(1-\vartheta,\,\frac{n}{2}-\vartheta;\,\frac{n}{2}+1;\,|x|^{2}\right)\,|x|^{2}$
$\displaystyle=~{}$ $\displaystyle
C(n,\vartheta)\,\>_{2}\\!F_{1}\left(\frac{n}{2}+\vartheta,\,1+\vartheta;\,\frac{n}{2}+1;\,|x|^{2}\right)\,|x|^{2}\,(1-|x|^{2})^{2\vartheta}.$

Since $\vartheta<0$, by the same argument as in the proof of Lemma 2.4, we see
that the hypergeometric function in the last line is bounded both from above
and below, hence

(12) $\sum_{k=1}^{n}x_{k}\frac{\partial}{\partial
x_{k}}\mathcal{P}_{\vartheta}[\phi](x)~{}\approx~{}(1-|x|^{2})^{2\vartheta}$

near the boundary $\mathbb{S}^{n-1}$. Note that

$\big{|}\nabla\left(\mathcal{P}_{\vartheta}[\phi]\right)(x)\big{|}~{}\geq~{}\left|\sum_{k=1}^{n}x_{k}\frac{\partial}{\partial
x_{k}}\mathcal{P}_{\vartheta}[\phi](x)\right|.$

So (12) implies that $\nabla\left(\mathcal{P}_{\vartheta}[\phi]\right)$ is
unbounded in $\mathbb{B}^{n}$. Consequently, $\mathcal{P}_{\vartheta}[\phi]$
is not Lipschitz continuous in $\mathbb{B}^{n}$.

######  Remark 2.

When $n=3$, there is a more explicit example:

$\displaystyle\mathcal{P}_{\vartheta}[1](x)~{}=~{}$
$\displaystyle\frac{\Gamma\left(\frac{3}{2}+\vartheta\right)\Gamma(1+\vartheta)}{\Gamma\left(\frac{3}{2}\right)\Gamma(1+2\vartheta)}\>_{2}\\!F_{1}\left(-\vartheta,\,\frac{1}{2}-\vartheta;\,\frac{3}{2};\,|x|^{2}\right)$
$\displaystyle=~{}$ $\displaystyle
2^{-1-2\vartheta}\frac{(1+|x|)^{1+2\vartheta}-(1-|x|)^{1+2\vartheta}}{|x|},$

which is clearly not Lipschitz continuous when $\vartheta<0$. Here, in the
last equality we have used the formula (see [10, p. 39])

$(1+z)^{1-2a}-(1-z)^{1-2a}=2z(1-2a)\>_{2}\\!F_{1}\left(a,\,a+\frac{1}{2};\,\frac{3}{2};\,z^{2}\right).$

##  4\. Proof of Theorem 1.1, Part (ii)

Let $\phi$ be a Lipschitz continuous function on $\mathbb{S}^{n-1}$, i.e.,
$|\phi(\zeta)-\phi(\eta)|\leq L|\zeta-\eta|$ for some constant $L$ and all
$\zeta,\eta\in\mathbb{S}^{n-1}$.

We shall show that

(13)
$\sup_{x\in\mathbb{B}^{n}}|\nabla\left(\mathcal{P}_{\vartheta}[\phi]\right)(x)|<\infty,$

which would clearly imply the Lipschitz continuity of
$\mathcal{P}_{\vartheta}[\phi]$.

To do this, we note that

(14)
$\left|\nabla\left(\mathcal{P}_{\vartheta}[\phi]\right)\right|=\left|\left(\nabla\left(\mathcal{P}_{\vartheta}[\phi\circ
T]\right)\right)\circ T^{-1}\right|$

holds for every $\phi\in C(\mathbb{S}^{n-1})$ and every orthogonal
transformation $T$. Also, given any orthogonal transformation $T$, the
function $\phi\circ T$ is also Lipschitz continuous on $\mathbb{S}^{n-1}$,
with the same Lipschitz constant. For any $x\in\mathbb{B}^{n}$, we can choose
an orthogonal transformation $T$ such that $T^{-1}x=re_{1}$ with $r=|x|$, so
that

$\left|\nabla\left(\mathcal{P}_{\vartheta}[\phi]\right)(x)\right|=\left|\left(\nabla\left(\mathcal{P}_{\vartheta}[\phi\circ
T]\right)\right)(re_{1})\right|.$

We are reduced to prove the following.

######  Claim 1.

$\frac{\partial\mathcal{P}_{\vartheta}[\phi]}{\partial x_{k}}(re_{1})$,
$k=1,\ldots,n$, are bounded functions of $r$ on $[0,1)$.

We consider two different cases.

Case 1: $k\in\\{2,\ldots,n\\}$.

By taking $x=re_{1}$ in (9), we have

$\displaystyle\int_{\mathbb{S}^{n-1}}\frac{\partial P_{\vartheta}}{\partial
x_{k}}(re_{1},\zeta)~{}d\sigma(\zeta)~{}=~{}0,$

and hence

(15) $\frac{\partial\mathcal{P}_{\vartheta}[\phi]}{\partial
x_{k}}(re_{1})=\int_{\mathbb{S}^{n-1}}\frac{\partial P_{\vartheta}}{\partial
x_{k}}(re_{1},\zeta)[\phi(\zeta)-\phi(e_{1})]~{}d\sigma(\zeta).$

Now, taking $x=re_{1}$ in (5), we get

$\frac{\partial P_{\vartheta}}{\partial
x_{k}}(re_{1},\zeta)=(n+2\vartheta)c_{n,\vartheta}\,\frac{(1-r^{2})^{1+2\vartheta}\zeta_{k}}{|re_{1}-\zeta|^{n+2\vartheta+2}}.$

Substituting this into (15), we obtain

$\displaystyle\left|\frac{\partial\mathcal{P}_{\vartheta}[\phi]}{\partial
x_{k}}(re_{1})\right|~{}\lesssim~{}$
$\displaystyle(1-r^{2})^{1+2\vartheta}\int_{\mathbb{S}^{n-1}}\frac{|\zeta_{k}|\,|\phi(\zeta)-\phi(e_{1})|}{|re_{1}-\zeta|^{n+2\vartheta+2}}d\sigma(\zeta)$
$\displaystyle~{}\lesssim~{}$
$\displaystyle(1-r^{2})^{1+2\vartheta}\int_{\mathbb{S}^{n-1}}\frac{|\zeta-
e_{1}|^{2}d\sigma(\zeta)}{|re_{1}-\zeta|^{n+2\vartheta+2}},$

where we used the condition $|\phi(\zeta)-\phi(e_{1})|\leq L|\zeta-e_{1}|$ and
the easy observation that $|\zeta_{k}|\leq|\zeta-e_{1}|$ in the last
inequality. This, together with Lemma 2.5, shows that
$\frac{\partial\mathcal{P}_{\vartheta}[\phi]}{\partial x_{k}}(re_{1})$ is a
bounded function of $r$ on $[0,1)$.

Case 2: $k=1$.

It follows from (5) that

$\displaystyle\left|\frac{\partial P_{\vartheta}}{\partial
x_{1}}(re_{1},\zeta)\right|~{}\leq~{}$ $\displaystyle
2(1+2\vartheta)c_{n,\vartheta}\frac{(1-r^{2})^{2\vartheta}}{|re_{1}-\zeta|^{n+2\vartheta}}~{}+~{}(n+2\vartheta)c_{n,\vartheta}\frac{(1-r^{2})^{1+2\vartheta}|r-\zeta_{1}|}{|re_{1}-\zeta|^{n+2\vartheta+2}}$
$\displaystyle~{}\leq~{}$ $\displaystyle
2(1+2\vartheta)c_{n,\vartheta}\frac{(1-r^{2})^{2\vartheta}}{|re_{1}-\zeta|^{n+2\vartheta}}~{}+~{}(n+2\vartheta)c_{n,\vartheta}\frac{(1-r^{2})^{1+2\vartheta}|\zeta-
e_{1}|}{|re_{1}-\zeta|^{n+2\vartheta+2}}$
$\displaystyle\qquad\quad~{}+~{}(n+2\vartheta)c_{n,\vartheta}\frac{(1-r^{2})^{1+2\vartheta}(1-r)}{|re_{1}-\zeta|^{n+2\vartheta+2}}.$

Combining with the condition $|\phi(\zeta)-\phi(e_{1})|\leq L|\zeta-e_{1}|$,
this yields

$\displaystyle\bigg{|}\frac{\partial\mathcal{P}_{\vartheta}[\phi]}{\partial
x_{1}}$ $\displaystyle(re_{1})\bigg{|}$ $\displaystyle~{}\leq~{}$
$\displaystyle\left|\int_{\mathbb{S}^{n-1}}\frac{\partial
P_{\vartheta}}{\partial
x_{1}}(re_{1},\zeta)\phi(e_{1})d\sigma(\zeta)\right|~{}+~{}\left|\int_{\mathbb{S}^{n-1}}\frac{\partial
P_{\vartheta}}{\partial
x_{1}}(re_{1},\zeta)[\phi(\zeta)-\phi(e_{1})]d\sigma(\zeta)\right|$
$\displaystyle\lesssim~{}$
$\displaystyle\underbrace{|\phi(e_{1})|\,\left|\int_{\mathbb{S}^{n-1}}\frac{\partial
P_{\vartheta}}{\partial
x_{1}}(re_{1},\zeta)d\sigma(\zeta)\right|}_{I_{1}}~{}+~{}\underbrace{(1-r^{2})^{2\vartheta}\int_{\mathbb{S}^{n-1}}\frac{d\sigma(\zeta)}{|re_{1}-\zeta|^{n+2\vartheta}}}_{I_{2}}$
$\displaystyle\qquad\quad~{}+~{}\underbrace{(1-r^{2})^{1+2\vartheta}\int_{\mathbb{S}^{n-1}}\frac{|\zeta-
e_{1}|}{|re_{1}-\zeta|^{n+2\vartheta+2}}d\sigma(\zeta)}_{I_{3}}$
$\displaystyle\qquad\qquad\quad~{}+~{}\underbrace{(1-r^{2})^{2+2\vartheta}\int_{\mathbb{S}^{n-1}}\frac{d\sigma(\zeta)}{|re_{1}-\zeta|^{n+2\vartheta+2}}}_{I_{4}}.$

Since $\vartheta>0$, by Lemma 2.4, $I_{1}$ is bounded. Also, in view of Lemma
2.5 , $I_{2},I_{3},I_{4}$ are all bounded. Consequently,
$\frac{\partial\mathcal{P}_{\vartheta}[\phi]}{\partial x_{1}}(re_{1})$ is a
bounded function of $r$ on $[0,1)$.

This concludes the proof of the claim and hence the proof of the theorem.

### Acknowledgement

The first author was supported by the National Natural Science Foundation of
China grant 11971453.

### Data availability

We do not analyse or generate any datasets, because our work proceeds within a
theoretical and mathematical approach. One can obtain the relevant materials
from the references below.

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