# “Double vaccinated, 5G boosted!”: Learning Attitudes towards COVID-19
Vaccination from Social Media

Ninghan Chen  University of LuxembourgL-4364Esch-sur-AlzetteLuxembourg
<EMAIL_ADDRESS>,  Xihui Chen  University of LuxembourgL-4364Esch-sur-
AlzetteLuxembourg<EMAIL_ADDRESS>,  Zhiqiang Zhong  University of
LuxembourgL-4364Esch-sur-AlzetteLuxembourg<EMAIL_ADDRESS>and  Jun Pang
University of LuxembourgL-4364Esch-sur-AlzetteLuxembourg<EMAIL_ADDRESS>
(2022)

###### Abstract.

To address the vaccine hesitancy which impairs the efforts of the COVID-19
vaccination campaign, it is imperative to understand public vaccination
attitudes and timely grasp their changes. In spite of reliability and
trustworthiness, conventional attitude collection based on surveys is time-
consuming and expensive, and cannot follow the fast evolution of vaccination
attitudes. We leverage the textual posts on social media to extract and track
users’ vaccination stances in near real time by proposing a deep learning
framework. To address the impact of linguistic features such as sarcasm and
irony commonly used in vaccine-related discourses, we integrate into the
framework the recent posts of a user’s social network neighbours to help
detect the user’s genuine attitude. Based on our annotated dataset from
Twitter, the models instantiated from our framework can increase the
performance of attitude extraction by up to 23% compared to state-of-the-art
text-only models. Using this framework, we successfully validate the
feasibility of using social media to track the evolution of vaccination
attitudes in real life. We further show one practical use of our framework by
validating the possibility to forecast a user’s vaccine hesitancy changes with
information perceived from social media.

social media, deep learning, graph neural networks, COVID-19, vaccination
attitude

††copyright: acmcopyright††journalyear: 2022††doi: XXXXXXX.XXXXXXX††copyright:
none††ccs: Computing methodologies Information extraction

##  1\. Introduction

Vaccination is now unanimously accepted as an effective approach to combat the
ongoing global COVID-19 pandemic, caused by the contagious coronavirus SARS-
CoV-2 (Minaee et al., 2021). Despite the decreased efficacy against the
infection of the recent variants, a high-level uptake of the currently
available vaccines is still believed as key to restrain the numbers of severe
diseases, deaths, and particularly hospitalisation, which is crucial for
medical systems to remain operating as normal (Andrews et al., 2022).
Regrettably, similar to the vaccines of other infectious diseases, not
everyone is willing to be vaccinated (Dubé et al., 2013). The impact of
_vaccine hesitancy_ has been widely recognised and extensively studied in a
number of countries for various groups of people, e.g., healthcare workers and
immigrants. Many related factors and their influences are evaluated and
compared, e.g., education, income and gender (Cascini et al., 2021). These
scientific findings have provided policymakers with valuable references to
design strategies to reduce vaccine hesitancy and fix the stagnant uptake
ratios.

The success of these studies relies on the collection and accurate
understanding of the public’s attitudes towards vaccination. Social surveys
with well-defined questions, due to their reliability and trustworthiness,
have been adopted as the dominant source of public opinions in the literature.
However, as conducting surveys is expensive and time-consuming, they tend to
fall behind the fast development of the COVID-19 pandemic (Budd et al., 2020),
and thus fail in capturing the evolution of vaccine hesitancy. Continuous
tracking of public vaccination attitudes can help healthcare bodies to
identify the significant fluctuations to make a timely intervention or fast
capture the public’s response to implemented policies. Moreover, it allows for
analysing dynamic factors such as occasional social protests and
misinformation propagation, in addition to the static ones like demographic
profiles which rarely change in short periods.

In recent years, social media has attracted the attention of data analysts as
an auxiliary data source to complement public health surveillance (PHS) in
spite of its inherent bias, e.g., regarding population sampling (Budd et al.,
2020; Aiello et al., 2020). Due to social distancing and fear of the unknown,
people spend more time than ever on social media. As social media posts have
proved to encode posters’ subjective opinions (Johnson et al., 2020; Gunaratne
et al., 2019), in this paper, we aim to leverage the enormous daily posts
during the COVID-19 pandemic to extract users’ vaccination attitudes and track
their temporal changes.

We take advantage of the recent advances of deep learning in natural language
processing (NLP), and propose a framework that can accurately classify a
textual post according to the vaccination stance expressed by its originator.
Our framework effectively addresses the challenge impairing existing NLP
models’ performance, i.e., the linguistic features such as sarcasm and irony,
which are quite common in vaccination-related discourse. Consider the
following example: “ _I wouldn’t do it for their vaccine, I’m waiting for the
6G_ ”. The user expresses his/her support for vaccination by making fun of the
conspiracy theory that chips are implanted with vaccine injection. After
experimenting with the state-of-the-art text feature-based classification
methods, we only get an accuracy of $0.65$, which is apparently not reliable
enough for trustworthy analysis. Recent studies revealed that a user’s
vaccination attitudes correlate with those of their neighbours in social
networks, e.g., friends and friends of friends. For example, online social
network users with negative attitudes often have social relations with users
of positive attitudes (Johnson et al., 2020; Morales et al., 2021). Inspired
by such studies, we integrate the recent posts of a user’s social network
neighbours in our framework to help detect the user’s genuine attitude and
reduce the impact of sarcasm.

To train and test models instantiated from our framework, we collect 9,135,393
tweets from Twitter generated by 69,936 users, and create the first annotated
dataset of 18,246 tweets manually labelled with affective vaccination stances
(e.g., positive and negative). In addition to the experimental evaluation, we
draw the temporal evolution of vaccination attitudes extracted from our
collected tweets. We cross-validate with published social studies and manually
analyse the popular social events occurring around significant changes of
vaccine hesitancy levels. All the validation results successfully illustrate
the effectiveness of our framework, as well as the power of social media as a
data source to grasp public vaccine hesitancy in practice in near real time.

Newsagents, governments, healthcare professionals and even anti-vaccine
activists use social media to spread news, knowledge and suggestions to
persuade or dissuade people from getting vaccinated (Van Bavel et al., 2020).
To showcase the practical use of our post-based attitude learning framework,
to the best of our knowledge, we are the first to demonstrate that the
information that users perceive from social media can be used as predictors of
their vaccine hesitancy changes.

Our contributions. Our contributions are as follows:

  * •

We propose a framework to extract vaccination stances from textual social
media posts. Our framework integrates recent posts of a user’s social network
neighbours to help reduce the interference of linguistic features, e.g.,
sarcasm and irony.

  * •

We design models instantiating our framework. Based on our annotated dataset
from Twitter, the best model can increase the performance of attitude
extraction by up to 23% compared to state-of-the-art text-only models.

  * •

Using the model with the best performance, we track the evolution of
vaccination attitudes. The utility of the extracted vaccination attitudes is
further validated by the consistency with published statistics and explainable
significant fluctuations of vaccine hesitancy in terms of social events such
as wide propagation of misinformation and negative news.

  * •

We show a practical use of our framework by validating the possibility to
predict a user’s vaccination hesitancy changes with the information he/she
perceives from social media.

Through this paper, we (re-)establish the power of social media as a
complementary data source in public health surveillance in spite of its
inherent biases. Specifically, when exploited properly, it can provide
healthcare bodies with useful information to guide or support their decision-
making processes.

##  2\. Related Work

Vaccine hesitancy is believed to be a major cause of stagnant vaccine coverage
and contributor to vaccine program failure (Dubé et al., 2013). In spite of
the lack of a unified definition, one widely accepted representation of
vaccine hesitancy is a continuum, ranging from complete rejection of vaccines
to varying degrees of scepticism (Wilson et al., 2014). In this section, we
concentrate on the vaccine hesitancy studies after the onset of the COVID-19
pandemic. A considerable amount of literature has been published investigating
the state of vaccine hesitancy and the influential factors in different
regions (Khubchandani et al., 2021; Sherman et al., 2021) for specific groups
of people such as healthcare employees (Biswas et al., 2021; Grech et al.,
2020), immigrants (Alabdulla et al., 2021) and college students (Barello et
al., 2020). Although surveys are still the most adopted method to collect
sampled populations’ attitudes or stances toward vaccination (Al-Amer et al.,
2021), some recent works leverage social media as a new dimension (Johnson et
al., 2020; Gunaratne et al., 2019). Compared to self-reporting questionnaires,
social media data are cost-effective to access, and more importantly, allow
analysis over large populations which was not previously feasible (Budd et
al., 2020; Aiello et al., 2020; Johnson et al., 2020).

The methods extracting vaccination attitudes from social media fall into two
categories: community-based and post-based. Cossard et al. (2020) found pro-
and anti-vaccine users naturally cluster into communities and calculated
community partitions of various communication graphs to infer users’
vaccination stances. Johnson et al. (2020) made use of the topics of fan pages
(similar to discussion groups) on Facebook to approximate users’ attitudes,
and analysed the communities formed by 100 million users across the world in
terms of their vaccination attitudes. Post-based methods benefit from the
various types of information encoded in social media posts such as texts,
labels and pictures. Gunaratne et al. (2019) relied on the hashtags in tweets
to approximate the vaccination attitudes in tweets. Sentiment analysis, as
part of natural language processing (NLP), aims to derive the subjective
opinions expressed in texts. The introduction of deep learning leads to more
powerful models that can process posts at the sentence or paragraph levels
such as word2vec (Mikolov et al., 2013) and BERT (Devlin et al., 2019). The
sentiments extracted from texts have been used as references to study vaccine
hesitancy (Gbashi et al., 2021). For instance, Gbashi et al. (2021) detected
the opinions of media towards vaccines in Africa through Twitter and Google
news.

Discussion. The community-based methods cannot capture the fast development of
public vaccination attitudes due to the relatively stable connections between
users. Moreover, community memberships are effective for analysis on the level
of populations but fail to accurately derive individual users’ attitudes. The
post-based methods in previous studies are not specifically designed and
trained for COVID-19 vaccines. As a result, they cannot capture the special
linguistic characteristics of the online discourses during the COVID-19
pandemic. This is partially because of the lack of social media posts which
are related to COVID-19 vaccines and annotated with vaccination stances. In
this paper, we propose a framework that can not only benefit from state-of-
the-art post-based methods, but also deal with the interference of linguistic
features such as sarcasm and irony in discourses related to COVID-19
vaccination. We also create the first annotated dataset of tweets which can
facilitate developing future models on subjective opinion extraction.

##  3\. Extracting Vaccination Attitudes from Social Media Posts

###  3.1. Problem definition

Extracting the vaccination stance of a social media post can be technically
formulated as a natural language processing (NLP) task (Li et al., 2020b),
i.e., classifying texts according to given class labels. In this paper, we
focus on the affective stance towards COVID-19 vaccination. Thus the set of
labels is $\mathcal{L}=\\{{\it NE},{\it PO},{\it NG}\\}$ where ${\it NE}$,
${\it PO}$ and ${\it NG}$ correspond to neutral, positive and negative,
respectively. The basic idea of text classification in NLP is first to
calculate a representation of the given text which summarises its linguistic
features and then output the most likely class label. Classification methods
differ from each other in terms of the formats of text representation and the
mapping from representations to class labels. Text classification confronts
the challenge that the attitude or emotion expressed by the same words varies
according to the context. For instance, the figurative usage of symptom words
can fool the keyword-based classification methods and significantly
deteriorate the precision of health mention classification (Biddle et al.,
2020). Sentiments of the short texts with symptom words are thus introduced as
auxiliary information to deal with this figurative interference.

As discussed previously, we notice that during the COVID-19 pandemic, Twitter
users tend to use sarcastic or ironic expressions to describe their
disagreements with those with different vaccination stances. Inspired by
previous works such as (Biddle et al., 2020), given a post, we aim to benefit
from the originator’s social relations as well as their past posts to help
reduce or eliminate the interference of sarcasm and irony.

We use $\mathcal{G}=(\mathcal{V},\mathcal{E})$ to represent the social graph
recording the social relations between users where $\mathcal{V}$ is the set of
nodes and $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ is the set of edges
between nodes. A node $v\in V$ corresponds to a social media user and an edge
$(v,v^{\prime})$ indicates the existence of a relationship between two users
$v$ and $v^{\prime}$. Note that we ignore the direction of relationships in
this paper to take into account all the neighbours of a user, e.g., both
followers and followees on Twitter. Thus, $(v,v^{\prime})\in\mathcal{E}$
implies $(v^{\prime},v)\in\mathcal{E}$. We abuse the terms _user_ and _node_
in the following discussion when it is clear from the context. Let
$\mathcal{N}_{i}^{k}$ be the set of neighbours of node $v_{i}$ within $k$
hops, i.e., $\\{v\mid d_{\mathcal{G}}(v,v_{i})\leq k\\}$ where
$d_{\mathcal{G}}(v,v_{i})$ is the shortest distance between $v$ and $v_{i}$ in
the graph $\mathcal{G}$. Note that node $v_{i}$ is also in
$\mathcal{N}_{i}^{k}$ as $d_{\mathcal{G}}(v_{i},v_{i})=0$. We use $x_{i}^{t}$
to denote the textual message posted by user $v_{i}$ at time $t$. We use
$\mathcal{M}^{<t}_{i}$ to denote the list of posts originated by user $v_{i}$
before time $t$ chronologically ordered by their post time, and
$\mathcal{M}^{<t}_{\mathcal{N}^{k}_{i}}$ to represent the set of post lists of
the neighbours of user $v_{i}$ within $k$ hops.

Our COVID-19 vaccination attitude classification problem can be defined as
calculating the probability distribution of all labels in $\mathcal{L}$. The
final vaccination stance of $x_{i}^{T}$, i.e., $f(x_{i}^{t})$, is determined
by the label with the largest probability. Formally, we have

$f(x_{i}^{t})=\operatorname*{argmax}_{{\it stance}\in\mathcal{L}}\Pr({\it
stance}\mid x_{i}^{t},\mathcal{G},\mathcal{M}^{<t}_{\mathcal{N}^{k}_{i}}).$

Figure 1. An illustration of our attitude classification framework and model.

###  3.2. A vaccination attitude learning framework

To solve the classification problem formulated previously, we propose a
framework which takes advantage of the recent success of adopting deep
learning in NLP and graph analysis such as text embedding and graph neural
networks (GNNs).

Figure 1 depicts an overview of our framework by labelling its three main
components in different background colours: i) a text-level information
embedding module, ii) a GNN-enhanced module, and iii) a classification module.
The first module is used to learn the linguistic representation of the
targeted post while the second module summarises the linguistic features of
the recent messages posted by the user’s neighbours. We concatenate the
outputs of these two modules as the input of the classification module. GNN
(Kipf and Welling, 2017) has shown its advantage in transforming graph
information, including structures and attributes of nodes and edges in both
academia and industry. Intuitively, it employs a message passing scheme to
integrate the information of a node’s neighbourhood as the representation of
the node. The calculated embedding is then used for various downstream
applications such as node classification and link prediction. Variants of GNN
differ from each other in terms of the implementation of their message passing
schemes.

Our framework can be instantiated by assembling various methods that can
achieve the corresponding tasks of the modules. In the following, we present
how we implement every module of the framework and justify our choices.

###  3.3. Our model

Text-level information embedding. In order to mine meaningful information from
a limited number of annotated posts, recent solutions leverage pre-trained NLP
transformers to calculate the embedding for short texts (Li et al., 2020a).
NLP transformers have been empirically evaluated in (Zhao et al., 2021) where
RoBERTa (Liu et al., 2019) is shown to outperform the competing models. Due to
its high effectiveness, we adopt RoBERTa to learn text representations in our
model. The model takes a textual post, e.g., $x_{i}^{t}$, as input, and
outputs a sentiment-oriented representation vector $z^{{\it
text}}_{i,t}\in\mathbb{R}^{d}$ where $d$ is the pre-defined dimension of the
vector. RoBERTa will be fine-tuned with the posts in the training set.

GNN-enhanced module. Given the target post $x_{i}^{t}$, we utilise this module
to capture the linguistic features of the recent messages posted by $v_{i}$’s
friends within $k$ hops before $t$. The definition of “recent” is flexible and
depends on application scenarios. In this paper, we select the last $\lambda$
tweets before $t$ as a user’s recent tweets. Its output will be subsequently
used as complementary contextual information to further ameliorate
classification performance. Therefore, the input of this module consists of
the social network $\mathcal{G}$ and the historical messages of user $v_{i}$’s
$k$-hop neighbours, i.e., $\mathcal{M}_{\mathcal{N}^{k}_{i}}^{<t}$. Note that
the post originator’s recent messages are also considered as $v_{i}$ is
included in $\mathcal{N}_{i}^{k}$. The output will be a text-level embedding
vector that can be intuitively interpreted as a summary of useful features of
friends’ recent discourse.

We take two successive steps to calculate the output text-level representation
vector $z_{i,t}$. We first integrate the recent posts of each user in
$v_{i}$’s $k$-hop neighbourhood as a summary of his/her vaccination discourse.
In the second step, we propose a new GNN-based model named by H2GAT to
aggregate the discourse of all $v_{i}$’s neighbours into the social text
encoding vector.

STEP 1: Text-level encoding. For each user $v_{j}\in\mathcal{N}_{i}^{k}$, we
use his/her last $\lambda$ posts in $\mathcal{M}_{j}^{<t}$, denoted by the
list $(x^{t_{\lambda}}_{j},x^{t_{\lambda-1}}_{j},\ldots,x^{t_{1}}_{j})$ where
$t_{m}$ ($1\leq m\leq\lambda$) is the time stamp of $v_{j}$’s last $m$-th
post. We apply the pre-trained RoBERTa model and then obtain the corresponding
list of text-level representations, i.e., $(z^{{\it
text}}_{j,t_{\lambda}},z^{{\it text}}_{j,t_{\lambda-1}},\ldots,z^{{\it
text}}_{j,t_{1}})$. There are many ways to integrate $v_{j}$’s past text-level
representations while distinguishing their various temporal importance, e.g.,
Hawkes (Hawkes, 1971) and GRU (Cho et al., 2014). In our implementation, we
adopt the dynamic-aware position encoding by assigning a fixed importance
factor to each past post according to its position in the list. This method is
simple but more effective than other competing ones in our experiments (as
shown in Section 5). Formally, the integrated text-level representations of
user $v_{j}$ is calculated as follows:

$z^{{\it hist}}_{j,t}=\sum_{m\leq\lambda}\alpha_{m}\cdot z^{{\it
text}}_{j,t_{m}}$

where $\alpha_{m}$ ($1\leq m\leq\lambda$) is trainable and describes the
positional relation between the past posts. Note that $k$ and $\lambda$ are
predefined hyper-parameters that should be tuned manually.

STEP 2: H2GAT. It is pointed out that the heterophily phenomenon widely exists
among online social network users (Pandit et al., 2007). This phenomenon also
exists in vaccination-related discourses as the attitudes and linguistic
features can be significantly different between users. Considering the
_heterophily_ of vaccination discussion in a user’s neighbourhood, we adopt
and extend a recent GNN-based method called H2GCN (Zhu et al., 2020). The same
as other GNNs, it also has multiple layers, the $\ell$-th of which can be
formulated as follows:

$\begin{array}[]{ll}H^{\ell}_{i}={\sf Combine}(\\{{\sf
Aggregate}\\{H^{\ell-1}_{j}:j\in\widehat{\mathcal{N}}^{k^{\prime}}_{i}\\}:k^{\prime}\in\\{1,\dots,k\\}\\})\end{array}$

where $\widehat{\mathcal{N}}^{k^{\prime}}_{i}$ represents node $v_{i}$’s
$k^{\prime}$-order neighbours, i.e., nodes that have an exact distance of
$k^{\prime}$ from $v_{i}$ in $\mathcal{G}$. Formally,
$\widehat{\mathcal{N}}^{k^{\prime}}_{i}=\\{v_{j}\mid
d_{\mathcal{G}}(v_{j},v_{i})=k^{\prime}\\}$. Note the difference of
$\widehat{\mathcal{N}}^{k^{\prime}}_{i}$ from $\mathcal{N}^{k^{\prime}}_{i}$.
As $\mathcal{G}$ is connected, when $k^{\prime}>0$, we have
$\widehat{\mathcal{N}}^{k^{\prime}}_{i}\subset\mathcal{N}^{k^{\prime}}_{i}$.
The initial $H_{j}^{0}$ for each $v_{j}\in\mathcal{V}$ is set to $z^{{\it
hist}}_{j,t}$.

Note that different from (Zhu et al., 2020) which adopts the ${\sf Aggregate}$
function of GCN (Kipf and Welling, 2017), we use GAT (Velickovic et al., 2018)
for better performance. The output representation of $v_{i}$, denoted by
$z^{\it social}_{i,t}$, is calculated by combining node representations of all
layers:

$z^{\it social}_{i,t}={\sf
Combine}(\\{H^{0}_{i},H^{1}_{i},\dots,H^{k}_{i}\\}).$

Many ways exist to implement the function ${\sf Combine}$. We adopt the one in
H2GCN (Zhu et al., 2020) in our model which outputs the concatenation of all
inputs.

By concatenating $z^{\it social}_{i,t}$ and $z^{{\it text}}_{i,t}$, we obtain
the text-level representation vector for classification. Formally,

$z_{i,t}=z^{\it social}_{i,t}\;\|\;z^{{\it text}}_{i,t}.$

Attitude classification. We implement a simple two-layer structure to conduct
the classification. Note the output vector has a length of $|\mathcal{L}|$.
Recall $\mathcal{L}$ is the set of class labels. The first layer applies the
ReLU function to each element in $z_{i,t}$ and at the second layer, we use a
linear regression where $W\in\mathbb{R}^{2d\times|\mathcal{L}|}$ and
$b\in\mathbb{R}^{|\mathcal{L}|}$ is the bias vector. The softmax function is
applied to calculate the probability distribution over the class labels.
Formally, the distribution $p$ is calculated as follows:

$p={\sf softmax}\left({\sf ReLU}(z_{i,t})\cdot W+b\right).$

The class label with the largest probability will be chosen as the output. We
use CrossEntropyLoss as the objective function to train the entire model.

##  4\. Data Curation

To train and evaluate our vaccination attitude learning framework, we collect
a dataset from Twitter focusing on four adjacent Western European countries:
Germany, France, Luxembourg and Belgium. Reasons to select these countries
include their importance to European economy and their similar and almost
synchronous pandemic management policies. They can also well represent the
first group of countries receiving and administering COVID-19 vaccines. Due to
the lack of publicly available data with vaccination stances, we create the
first public set of annotated tweets for training and testing our framework.
The statistics about our dataset are summarised in Table 1. This dataset and
our annotation are publicly available.111The download link is hidden due to
the double-blind review policy.

Table 1. Dataset statistics.

Social network | #node | France | 35,081
---|---|---|---
Germany | 16,304
Belgium | 15,647
Luxembourg | 2,904
69,936
#edge | 8,909,985
avg. degree | 127.40
Tweets | #tweet | France | 5,925,354
Germany | 2,669,875
Belgium | 530,885
Luxembourg | 9,279
9,135,393
#tweet/user | 130.62
Annotated tweets | #tweet | 18,246
#user | 4,157

###  4.1. Data collection and preprocessing

Our dataset consists of two types of data: i) a social network composed of
active Twitter users, and ii) the tweets of selected users related to COVID-19
vaccine or vaccination. By ‘active users’, we mean users that are active in
vaccination-related discussion and frequently interact with others.

Step 1. Social graph construction. We start with identifying the Twitter users
in our targeted region who actively participated in vaccine-related discourse.
Instead of directly searching tweets by keywords, we refer to a publicly
available dataset which contains the IDs of COVID-19 related tweets (Chen et
al., 2020). We extract the tweet IDs spanning between January 22, 2020 and
March 15, 2021 covering the beginning of the vaccination campaign. Through
these IDs, we download the corresponding tweets. Each downloaded tweet is
associated with meta-information which includes the location of the
originator, either self-provided by the originator or attached by the device’s
positioning services such as GPS. Due to the ambiguity of the self-reported
locations, we use the geocoding APIs, Geopy and ArcGis Geocoding to regularise
their formats. For example, a user input location _Moselle_ is transformed to
a preciser and machine-parsable location: _Mosselle, Lorraine, France_. Based
on the regularised locations, we filter the downloaded tweets and remove those
posted by users out of the region. In total, we obtain 990,448 tweets from
767,583 users.

To find the users with frequent interaction with other Twitter users, we
construct a re-tweeting weighted graph. An edge is created between two users
if one user retweeted a tweet from the other user or mentioned him/her. The
edge weight is assigned as the number of mentions or retweets between them. We
remove all edges with weights smaller than two and calculate the largest
weakly connected component of the graph which consists of 72,960 active users.
As retweeting or mentioning a user does not mean these two users have a
following relationship, we crawl the remaining users’ followers, with which a
graph is constructed with the remaining users and the relationships between
them as edges. In the end, we take the largest weakly connected component of
the resulted graph as the final social graph. This graph consists of 69,936
nodes and 8,909,985 edges. On average, each user has 127.4 followers. This
indicates our selected users are sufficiently active on Twitter.

Step 2. Vaccine-related tweet collection. In this step, we crawl the tweets
originated or retweeted by the users in the social graph. Note that we are
only interested in the tweets related to COVID-19 vaccination. We use a list
of keywords to filter out the irrelevant ones. The keywords should be general
enough to cover tweets in German, French and English. With our observation and
several trials, we select the keywords containing the following strings: ‘
_vax_ ’, ‘ _vaccin_ ’, ‘ _covidvic_ ’, ‘ _impfstoff_ ’, ‘ _vacin_ ’, ‘
_vacuna_ ’ and ‘ _impfung_ ’. We use the Twitter Academic Research API to
download tweets. Due to the limitations of the API, each request can only
download a maximum of 500 tweets. To be efficient and ensure the coverage
ratio, we create a download request for every user in each month. In total, we
collect 1,626,472 tweets, and each user has about 130 tweets on average. We
clean the downloaded tweets by removing mentions of other users with ‘@’,
quoted hyperlinks and ‘RT’.

Table 2. Annotation labels and examples.

Label |  Examples (translated to English)
---|---
positive |  vaccines: why there are no long-term side effects.
We have a new weapon against the virus: the vaccine. Hold together, again.
negative |  this nurse gets covid-19 vaccine. then she talks to media how great it is. then passes out. watch!
Corona mass vaccination aka disability manufacturing
neutral |  How safe is the Covid 19 vaccine for people with diabetes?
If there is a covid vaccine, what will you do?
positive but dissatisfied with government management |  It’s bad enough for individuals to refuse #COVID19 #vaccines for themselves. But forcing a mass vax site to shut down, knowing it means vaccines may go to waste, is criminal. Call it pandemicide.

###  4.2. Data annotation

According to the best of our knowledge, no datasets of social media posts are
publicly available with users’ COVID-19 vaccination attitudes annotated. As a
result, we select a subset of our downloaded tweets and manually attach them
with attitude labels. In this paper, we focus on users’ affective stances
towards COVID-19 vaccination which are _positive_ , _negative_ and _neutral_.
Table 2 lists examples of the attitude labels. After a closer check of the
tweets, we notice a relatively large number of tweets in which users express
their dissatisfaction or complaints about governments’ COVID-19 management
policies but possess positive attitudes towards vaccination. Take the last row
in Table 2 for an example. It contains a few negative words such as ‘bad’,
‘criminal’ and ‘waste’ but the originator explicitly expresses his/her support
for vaccination. As such tweets deliver a negative emotion, if we do not
separate them from those with negative vaccination attitudes, NLP models will
be confused and their classification accuracy will be deteriorated. Therefore,
we add a label ${\it PD}$ indicating ‘ _positive but dissatisfied with
government management_ ’. Then the set of labels $\mathcal{L}$, used in the
rest of the paper, consists of four labels, i.e., $\\{{\it PO},{\it NG},{\it
NE},{\it PD}\\}$.

We select tweets to be annotated with the purpose to cover all the users in
our dataset. We order our downloaded tweets in a list according to their
numbers of times being re-tweeted in descending order. We then iteratively
remove the most transmitted tweet from the list and put it into the list of
selected tweets until every user in our dataset originally posted or retweeted
at least one selected message. After this step, we select 18,246 tweets
originated from 4,157 users.

As our tweets are multilingual, we hire 10 bachelor students who can speak at
least two of the three most used languages (i.e., German, French and English).
One author of this paper acts as the coordinator and trains all the hired
annotators by explaining the semantics of all labels with examples. To ensure
all annotators hold the same understanding of the labels, they are asked to
annotate 200 tweets. The coordinator checks their annotation and communicates
to them with extra explanation when necessary. We conduct three rounds to make
sure each tweet’s label is double-validated. In the first round, an annotator
annotates all the selected tweets. In the second round, each of the rest 9
annotators is assigned randomly 2,000 tweets and validates the annotation. In
the last round, the coordinator goes through all validated annotations. In
Figure 2, we show the distributions of the tweets according to their labels.

Figure 2. Distribution of vaccination attitude labels.

Annotator agreement. We leverage three widely accepted measurements to
evaluate the inter-annotator reliability for each label: Average Observed
Agreement (AOA) (Fleiss et al., 2013), Fleiss’ kappa (Fleiss et al., 2013),
and Krippendorff’s Alpha (Krippendorff, 1970). The values of all the three
measurements range from 0 to 1, where 0 indicates complete disagreement and 1
indicates absolute agreement. Table 3 summarises the inter-annotator agreement
for each annotation label. We can see that for labels PO, NG and NE, all the
three measurements produce scores larger than 0.73, indicating an outstanding
agreement level. The annotators’ agreement on PD falls drastically compared to
other labels, but still remains moderate according to the ranking criteria of
the Fleiss’ Kappa measurement. This can be explained by our difficulties
during annotation in dealing with the special linguist features of PD tweets,
i.e., frequently used negative terms or sarcastic expression.

Table 3. Inter-annotator agreement (PO: Positive, NG: Negative, NE: Neutral,
PD: Positive but dissatisfaction).

Label | AOA | Feliss’ kappa | Krippen-dorf’s Alpha
---|---|---|---
PO | 0.72 | 0.73 | 0.73
NG | 0.82 | 0.88 | 0.88
NE | 0.74 | 0.78 | 0.77
PD | 0.61 | 0.63 | 0.62

##  5\. Experimental evaluation

Evaluation setup. We set up an evaluation pipeline following the approach for
traditional supervised classification (Minaee et al., 2021). Specifically, we
split labelled tweets into training (80%), validation (10%) and testing (10%)
sets. The models are optimised with the training set, and the validation set
is used to tune hyper-parameters. The model performance is evaluated on the
testing set.

Hyperparameter settings. We train our model for 400 epochs and use Adam
(Kingma and Ba, 2015) for optimisation with the learning rate of ${10}^{-5}$
and weight decay of $5\text{\times}{10}^{-4}$. For the text encoder, i.e.,
RoBERTa, we adopt the implementation XLM-RoBERTa (Ou and Li, 2020) and follow
their default settings where the maximum string length, i.e., parameter $d$,
is 128. For our GNN-enhanced module, we set the embedding dimension as $64$.
The neighbourhood order $k$ which is also the number of layers and the number
of historical tweets $\lambda$ are important to ensure representation quality.
Therefore, we conduct an empirical study to analyse the influence of these two
key hyper-parameters to ensure the final performance. In Figure 3, we present
the classification accuracy with different values of $k$ (on the left) and
$\lambda$ (on the right). We observe that these two hyper-parameters indeed
significantly influence classification accuracy. Our model arrives at the best
performance with $k=2$ and $\lambda=3$.

Figure 3. Parameter tuning for $k$ and $\lambda$.

Experimental results. We compare our model with other possible implementations
of our proposed vaccination attitude learning framework. In order to
distinguish these models, we name them with two parts concatenated with ‘+’.
The first part tells the adopted GNN model while the second part gives the
method handling the temporal importance of past tweets. As all models use
RoBERTa for text encoding, we do not explicitly put it in the model names. We
present their performances in Table 4.

We have four major observations that justify the effectiveness of our
implementation. First, the consideration of friends’ vaccination discourse
increases the performance. The text-only classification model with RoBERTa
only has an average accuracy of 0.65 while the other models, which are
implemented with the GNN-enhanced module, achieve at least an accuracy above
$0.70$. Second, the vaccination discourse between friends on Twitter is
actually heterophily and the choice of heterophily-aware GNN models, i.e.,
H2GCN and our H2GAT, can further significantly improve the performance. The
next four models below RoBERTa in Table 4 have the same settings except for
the GNN methods. Both the application of H2GCN and our H2GAT achieve an
increase of about 0.04 compared to the models with GCN (Kipf and Welling,
2017) and GAT (Velickovic et al., 2018). Third, the consideration of the
temporal importance of past tweets leads to another up to 0.06 improvement. We
consider four methods to combine a user’s last $\lambda$ tweets: MEAN, GRU,
Hawkes and PE (short for positioning encoding). The method denoted by MEAN
simply averages the text-level encodings. The positioning encoding method
adopted in our model generates the best performance. Last, our extended H2GAT
model outperforms the original H2GCN. Our implementation, i.e., H2GAT+PE,
finally improves the text-only RoBERTa model by 23% in terms of accuracy.

Table 4. Model performance.

Model | Precision | Recall | F1 | Accuracy
---|---|---|---|---
RoBERTa | $0.6758\scriptstyle\pm 0.0218$ | $0.5848\scriptstyle\pm 0.0232$ | $0.6249\scriptstyle\pm 0.0210$ | $0.6517\scriptstyle\pm 0.0348$
GCN+MEAN | $0.6936\scriptstyle\pm 0.0101$ | $0.6932\scriptstyle\pm 0.0153$ | $0.6890\scriptstyle\pm 0.0113$ | $0.7033\scriptstyle\pm 0.0087$
GAT+MEAN | $0.7001\scriptstyle\pm 0.0091$ | $0.7002\scriptstyle\pm 0.0102$ | $0.6983\scriptstyle\pm 0.0078$ | $0.7096\scriptstyle\pm 0.0099$
H2GCN+MEAN | $0.7387\scriptstyle\pm 0.0021$ | $0.7144\scriptstyle\pm 0.0081$ | $0.7286\scriptstyle\pm 0.0027$ | $0.7412\scriptstyle\pm 0.0015$
H2GAT+MEAN | $0.7361\scriptstyle\pm 0.0038$ | $0.7371\scriptstyle\pm 0.0007$ | $0.7331\scriptstyle\pm 0.0019$ | $0.7461\scriptstyle\pm 0.0008$
H2GCN+GRU | $0.7813\scriptstyle\pm 0.0025$ | $0.7794\scriptstyle\pm 0.0048$ | $0.7712\scriptstyle\pm 0.0027$ | $0.7829\scriptstyle\pm 0.0010$
H2GAT+GRU | $0.7937\scriptstyle\pm 0.0036$ | $0.7968\scriptstyle\pm 0.0021$ | $0.7927\scriptstyle\pm 0.0011$ | $0.8009\scriptstyle\pm 0.0008$
H2GCN+Hawkes | $0.7843\scriptstyle\pm 0.0016$ | $0.7760\scriptstyle\pm 0.0038$ | $0.7699\scriptstyle\pm 0.0047$ | $0.7831\scriptstyle\pm 0.0025$
H2GAT+Hawkes | $0.7946\scriptstyle\pm 0.0036$ | $0.7922\scriptstyle\pm 0.0010$ | $0.7903\scriptstyle\pm 0.0019$ | $0.7988\scriptstyle\pm 0.0023$
H2GCN+PE | $0.7859\scriptstyle\pm 0.0031$ | $0.7813\scriptstyle\pm 0.0045$ | $0.7792\scriptstyle\pm 0.0022$ | $0.7889\scriptstyle\pm 0.0028$
H2GAT+PE | $\textbf{0.7948}\scriptstyle\pm 0.0051$ | $\textbf{0.7954}\scriptstyle\pm 0.0029$ | $\textbf{0.7936}\scriptstyle\pm 0.0020$ | $\textbf{0.8030}\scriptstyle\pm 0.0017$

Empirical complexity analysis. As the RoBERTa model is pre-trained, the models
instantiated from our framework have the same complexity as the adopted GNN
models. In our experiments, we conduct the training on a server with Xeon E5
CPU and Tesla V100 GPU. On average, the training time for RoBERTA is about
$115$ seconds for an epoch while $52.5$ seconds are needed for an epoch in
training the GNN-enhanced module and the classification module. What is more
important is the running time of the models when processing a tweet. This will
determine the practical utility of our framework in tracking public
vaccination attitude in real time. We run four parallel instances of our model
H2GAT+PE on the server. On average, it takes 24.68 seconds for every 1,000
tweets, which means more than 3.5 million can be processed a day. For the
regions we target at, we collect in total 9 million vaccination-related tweets
over two years. This implies our model is sufficiently efficient for
processing posts on a daily basis.

Cross-validation. In addition to experimental evaluation, we also make use of
published social studies to cross-validate our model’s effectiveness. Lazarus
et al. conducted a survey in June 2020, and estimated that the vaccine
acceptance rates in France and Germany are 58.9% and 64.5%, respectively
(Lazarus et al., 2021). After applying our model to classify the tweets in the
same period, we find the percentages of tweets with positive vaccination
attitudes of these two countries are 42.27% and 53.12%, which are similar and
retain the relative difference between the two countries. This implies that
posts on Twitter can be used as a reference to fast grasp the vaccine
hesitancy levels when surveys are not available.

Vaccine hesitancy tracking and manual analysis. We draw the temporal evolution
of the percentage of tweets for each selected label in Figure 4 on a daily
basis starting from November 8, 2020. Based on previous research reporting
that the content of tweets is highly correlated with real-world situations
(Puri et al., 2020), we make a hypothesis that real-world events may
contribute to the fluctuating proportion of tweets with different vaccination
stances. In vaccine hesitancy monitoring, special attention should be paid to
the fluctuations of negative attitudes. We take three time points that
correspond to turning points of the curve of label NE as examples and discuss
the potential causes. Among them, two correspond to apex points where negative
tweets reach local maximum percentages and one corresponds to a base point
with local minimum negative tweets. We first plot word clouds in Figure 5 to
identify the most frequently used keywords in the week around the selected
points. Then we search these keywords on the Internet to identify the events
that may contribute to the changes.

The first apex occurred around January 16, 2021. We notice that this surge of
negative tweets attributes to the propagation of a large volume of
misinformation. Take the two most dominant pieces of misinformation as
examples. One said that on January 14, the Norwegian Medicines Agency reported
that a total of 29 people had suffered side effects, 13 of which were fatal.
The other was about the death of an Indian healthcare worker after receiving
COVID-19 vaccines. The second peak happened around February 15, 2021. One
piece of negative news was reported that AstraZeneca vaccines were stopped
from administration after many health workers of Morlaix hospital in France
suffered from side effects. This news subsequently led to anti-vaccination
discussions. The base point occurred between the two peaks around February 3,
2021. From Figure 5(c), we find the dominant positive news that Russia started
to offer other countries such as Pakistan with its vaccines.

From the above discussion, we can see our model can enable the use of social
media data to track on a daily basis the changes of vaccination attitudes, and
capture the impact of social events on public vaccine hesitancy. This may
finally help the governments identify the right time to take intervention
actions.

Figure 4. Temporal distribution of tweets with different vaccination attitude
labels.

##  6\. Use case: Predicting vaccination hesitancy changes

In this section, we illustrate a use of our vaccination attitude learning
framework. Specifically, we analyse the role of the vaccination information
widely spread across Twitter in affecting users’ attitudes towards
vaccination. Considering the comprehensiveness of vaccination discourses, we
classify the most popular vaccination-related tweets into _themes_ that may
correlate with vaccine hesitancy. Based on users’ perceived information in
these themes, we succeed in forecasting their vaccination attitude changes
with classic machine learning models.

(a) Apex 1 (01/14–01/21) (b) Apex 2 (02/10–02/17) (c) The base (01/30–02/06)

Figure 5. Word clouds of tweets around selected points.

###  6.1. Period selection and theme labelling

The participation of vaccination discourses fluctuates over time along with
the occurrence of social events related to COVID-19 vaccines. We select two
time periods after the start of COVID-19 vaccination campaign, in which the
volume of tweets experiences significant surges compared to adjacent periods.
The first period lasts for 25 days spanning from December 27, 2020 to January
20, 2021 while the second period lasts for 15 days between January 25 and
February 8, 2021. These two periods involve 161,611 original tweets posted in
total among which 25,449 are retweeted at least once. The total number of
times of being retweeted adds up to 242,129. We encounter two challenges to
continue our analysis of the impact made by diffused information: the
comprehensiveness and large volume of propagated tweets. Due to the huge
volume of tweets propagated over Twitter, it is not plausible to consider all
of them. Previous studies show that tweets’ influence follows the power-law
distribution and 80% of the impacts come from 20% of the most widely spread
tweets (Ediger et al., 2010). Inspired by this result, we leverage the top 25%
most widely propagated tweets in every period to approximately represent the
themes expressed in the diffused information. In total, we select 501 original
tweets that are retweeted 78,891 times from 72.16% of the users. To deal with
the comprehensiveness, with a careful examination of the selected tweets, we
categorise them into themes that are considered to be responsible for the
changes of vaccination attitudes. We refer to previous studies (Bianco et al.,
2019; Della Polla et al., 2021), especially the Parent Attitudes about
Childhood Vaccines (PACV) survey (Opel et al., 2011) and the WHO Vaccine
Hesitancy Matrix (Organization, 2014), and identify 11 themes that are
relevant and can cover the propagated tweets (see Table 6 in Appendix for
explanation and examples). We ask two of the 10 hired annotators to manually
annotate the selected tweets with their corresponding themes. The Cohen’s
Kappa coefficient $k=0.82$ implies a high rate of agreement between them.

###  6.2. Predictability of vaccine hesitancy changes

Handling retweets and quotations. In addition to original posts, retweets and
quotations also take up a large proportion of a user’s historical posts. For
quotations, a user added some comments which may express opposite opinions to
that of the quoted one. Therefore, we only use the texts users upload as valid
posts encoding users’ vaccination stances. Although retweets cannot fully
represent a user’s own opinion, the behaviour of retweeting itself indicates
some sort of agreement with the ideas expressed in the message retweeted (Dunn
et al., 2015). Based on this idea, we take retweets into account when
calculating an individual user’s vaccine hesitancy. The same approach is also
adopted in the vaccination attitude tracking discussed in the previous
section.

Quantifying individual vaccine hesitancy. We measure the vaccine hesitancy of
an individual user according to the tweets posted or retweeted by the user in
a time interval. Formally, it is calculated as:
$\frac{N_{p}(v)-N_{n}(v)}{N_{p}(v)+N_{n}(v)}$ where $N_{p}(v)$ denotes the
number of posts with positive vaccination attitudes of user $v$ during the
selected interval, and $N_{n}(v)$ is the corresponding number of tweets with
negative attitudes. Considering our purpose being idea validation, we do not
distinguish the various significance of original posts and retweets.

For each selected period, we use the tweets posted 14 days before and after
the period to evaluate individual users’ hesitancy levels and see how they
change. In order to ensure the reliability, we only consider the users who
posted or retweeted at least 3 tweets. If a user’s vaccine hesitancy
experiences a change smaller than $0.05$, we consider the user’s attitude
_unchanged_ , otherwise, _increased_ or _decreased_ depending on the change
direction.

Table 5. Model performances for attitude change prediction.

Model | Precision | Recall | F1 | Accuracy
---|---|---|---|---
SVM | 0.7374 | 0.7382 | 0.7345 | 0.7477
Naive Bayes | 0.6468 | 0.6559 | 0.6427 | 0.6658
Random Forest | 0.6811 | 0.6838 | 0.6795 | 0.6958
XGBoost | 0.7232 | 0.7229 | 0.7198 | 0.7342
GBDT | 0.7533 | 0.7516 | 0.7498 | 0.7603

Modelling perceived information. A Twitter user perceives information from the
tweets retweeted or originally posted by his/her direct friends. As our focus
is the information widely diffused on Twitter, we use a vector
$I_{u}=(c_{1},c_{2},\ldots,c_{m})$ to approximately represent a user $u$’s
perceived information where $c_{i}$ is the number of popular tweets a user
receives from followers in $i$-th theme. As we have $11$ themes, $m=11$ in our
analysis.

Model evaluation. We make use of various standard machine learning methods to
predict the change of a user $u$’s vaccination attitudes with the input of
$I_{u}$. The methods consist of SVM with rbf kernel ($C=1$, $\gamma=0.1$),
Naive Bayes ($\alpha=1$), Random Forest ($100$ trees with maximum tree depth
of $5$), XGBoost ($100$ trees with maximum tree depth of $4$) and GBDT ($100$
trees with maximum tree depth of $5$). Table 5 shows the performance of these
methods. All numbers are averaged over 5 training sessions. We can see all the
methods can achieve reasonably good prediction performance and GBDT
outperforms the rest models with an accuracy of $0.76$. When we consider
additional factors such as users’ vaccine hesitancy levels before the periods,
the accuracy can be improved to $0.86$.

Discussion. These results show that we can make accurate predictions with
users’ perceived popular information. Since we have empirically illustrated
the plausibility to use social media posts to track public vaccination
attitudes, the results imply that the diffused information on social media
like Twitter could be used as indicators to forecast the changes of vaccine
hesitancy levels. As repeated many times, although such predictions cannot
achieve the same level of trust as social surveys, they provide decision-
makers with a method to quickly understand and get prepared for the potential
damage of certain misinformation or compare different vaccine hesitancy
intervention strategies over social media.

##  7\. Conclusion and Discussion

In this paper, we proposed a deep learning framework to learn vaccination
attitudes from social media textual posts. Although vaccination attitudes
extracted from social media cannot be as accurate and reliable as conventional
social surveys, our framework allows for continuously tracking the fast
development of public vaccination attitudes and capturing the changes that
deserve specific attention in time. By leveraging friends’ vaccination
discourse as contextual information, our model successfully reduces the
interference of linguistic features such as sarcasm and irony. Our model
instantiated from the framework improves the state-of-the-art text-only method
by up to 23% in terms of accuracy according to our manually annotated dataset.
With cross-validation with published statistics and manually analysis, we
further validated the effectiveness of the model to capture public vaccine
hesitancy in real life. After identifying 11 themes from widely diffused
information on Twitter, with the help of our model, we validated the
predictability of users’ vaccine hesitancy changes by the information they
perceived from social media. This showed a potential use of our model in
practice. Through this paper, we established again the power of social media
data in supplementing public health surveillance, especially in combating
infectious virus like COVID-19.

Limitations and future work. We have three main limitations to address in
future. First, in our work we have primarily focused on Twitter which
potentially induces bias in our data and analysis. Thus, it is important to
extend our work to other social media platforms such as Facebook and
Instagram, and cross-validate our results. Second, we only analysed users’
affective vaccination stances (i.e., positive, negative and neutral), which
can only be used as an indicator of users’ intention of getting vaccinated. It
will be interesting to look deeper into users’ tweets for a longer time and
identify underlying determinants that lead to vaccination acceptance. Third,
we used only the top 25% most widely spread tweets as representatives to
extract the themes of diffused information partly limited by manual
annotation. Some information in certain themes may be missed. As an
interesting future work, we will develop effective NLP models to learn
different tweet themes automatically.

Ethical considerations. This work is based completely on public data and does
not contain private information of individuals. Our dataset is built in
accordance with the FAIR data principles (Wilkinson et al., 2016) and Twitter
Developer Agreement and Policy and related policies. Our release of the
dataset is also compliant with General Data Protection Regulation (GDPR). To
conclude, we have no ethical violation in the collection and interpretation of
data in our study.

## References

  * (1)
  * Aiello et al. (2020) Allison E Aiello, Audrey Renson, and Paul N Zivich. 2020\.  Social media- and internet-based disease surveillance for public health.  _Annual Review of Public Health_ 41, 1 (2020), 101–118.
  * Al-Amer et al. (2021) Rasmieh Al-Amer, Della Maneze, Bronwyn Everett, Jed Montayre, Amy R Villarosa, Entisar Dwekat, and Yenna Salamonson. 2021.  COVID-19 vaccination intention in the first year of the pandemic: A systematic review.  _Clinical Nursing_ 31, 1-2 (2021), 62–86.
  * Alabdulla et al. (2021) Majid Alabdulla, Shuja Mohd Reagu, Abdullatif Al-Khal, Marwa Elzain, and Roland M Jones. 2021\.  COVID-19 vaccine hesitancy and attitudes in Qatar: A national cross-sectional survey of a migrant-majority population.  _Influenza and Other Respiratory Viruses_ 15, 3 (2021), 361–370.
  * Andrews et al. (2022) Nick Andrews, Julia Stowe, Freja Kirsebom, Samuel Toffa, Ruchira Sachdeva, Charlotte Gower, Mary Ramsay, and Jamie Lopez Berna. 2022\.  Effectiveness of COVID-19 booster vaccines against COVID-19 related symptoms, hospitalisation and death in England.  _Nature Medicine_ (2022).
  * Barello et al. (2020) Serena Barello, Tiziana Nania, Federica Dellafiore, Guendalina Graffigna, and Rosario Caruso. 2020\.  Vaccine hesitancy among university students in Italy during the COVID-19 pandemic.  _European Journal of Epidemiology_ 35, 8 (2020), 781–783.
  * Bianco et al. (2019) Aida Bianco, Valentina Mascaro, Rossella Zucco, and Maria Pavia. 2019.  Parent perspectives on childhood vaccination: How to deal with vaccine hesitancy and refusal?  _Vaccine_ 37, 7 (2019), 984–990.
  * Biddle et al. (2020) Rhys Biddle, Aditya Joshi, Shaowu Liu, Cécile Paris, and Guandong Xu. 2020.  Leveraging sentiment distributions to distinguish figurative from literal health reports on Twitter. In _Proc. 2020 International Conference on World Wide Web (WWW)_. ACM, 1217–1227.
  * Biswas et al. (2021) Nirbachita Biswas, Toheeb Mustapha, Jagdish Khubchandani, and James H Price. 2021.  The nature and extent of COVID-19 vaccination hesitancy in healthcare workers.  _Journal of Community Health_ (2021), 1–8.
  * Budd et al. (2020) Jobie Budd, Benjamin S. Miller, and Erin M. Manning et al. 2020\.  Digital technologies in the public-health response to COVID-19.  _Nature Medicine_ 26, 8 (2020), 1183–1192.
  * Cascini et al. (2021) Fidelia Cascini, Ana Pantovic, Yazan Al-Ajlouni, Giovanna Failla, and Walter Ricciardi. 2021\.  Attitudes, acceptance and hesitancy among the general population worldwide to receive the COVID-19 vaccines and their contributing factors: A systematic review.  _EClinicalMedicine_ 40 (2021), 101113.
  * Chen et al. (2020) Emily Chen, Kristina Lerman, and Emilio Ferrara. 2020\.  Tracking social media discourse about the COVID-19 pandemic: Development of a public coronavirus Twitter data set.  _JMIR Public Health and Surveillance_ 6, 2 (2020), e19273.
  * Cho et al. (2014) KyungHyun Cho, Bart van Merrienboer, Dzmitry Bahdanau, and Yoshua Bengio. 2014.  On the properties of neural machine translation: Encoder-decoder approaches.  _CoRR_ abs/1409.1259 (2014).  arXiv:1409.1259
  * Cossard et al. (2020) Alessandro Cossard, Gianmarco De Francisci Morales, Kyriaki Kalimeri, Yelena Mejova, Daniela Paolotti, and Michele Starnini. 2020\.  Falling into the echo chamber: The Italian vaccination debate on Twitter. In _Proc. 2020 International AAAI Conference on Web and Social Media (ICWSM)_ , Vol. 14. The AAAI Press, 130–140.
  * Della Polla et al. (2021) Giorgia Della Polla, Francesca Licata, Silvia Angelillo, Concetta Paola Pelullo, and Aida Bianco. 2021.  Characteristics of healthcare workers vaccinated against influenza in the era of COVID-19.  _Vaccines_ 9, 7 (2021), 695.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of deep bidirectional transformers for language understanding. In _Proc. 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (NAACL-HLT)_. ACL, 4171–4186.
  * Dubé et al. (2013) Eve Dubé, Caroline Laberge, Maryse Guay, Paul Bramadat, Réal Roy, and Julie A Bettinger. 2013\.  Vaccine hesitancy: An overview.  _Human vaccines & immunotherapeutics_ 9, 8 (2013), 1763–1773.
  * Dunn et al. (2015) Adam G Dunn, Julie Leask, Xujuan Zhou, Kenneth D Mandl, and Enrico Coiera. 2015.  Associations between exposure to and expression of negative opinions about human papillomavirus vaccines on social media: An observational study.  _Journal of Medical Internet Research_ 17, 6 (2015), e144.
  * Ediger et al. (2010) David Ediger, Karl Jiang, Jason Riedy, David A Bader, Courtney Corley, Rob Farber, and William N Reynolds. 2010.  Massive social network analysis: Mining Twitter for social good. In _Proc. 2010 International Conference on Parallel Processing (ICPP)_. IEEE, 583–593.
  * Fleiss et al. (2013) Joseph L Fleiss, Bruce Levin, and Myunghee Cho Paik. 2013\.  _Statistical methods for rates and proportions_.  John Wiley & Sons.
  * Gbashi et al. (2021) Sefater Gbashi, Oluwafemi Ayodeji Adebo, Wesley Doorsamy, and Patrick Berka Njobeh. 2021\.  Systematic delineation of media polarity on COVID-19 vaccines in Africa: Computational linguistic modeling study.  _JMIR Medical Informatics_ 9, 3 (2021), e22916.
  * Grech et al. (2020) Victor Grech, Charmaine Gauci, and Steve Agius. 2020\.  Vaccine hesitancy among Maltese healthcare workers toward influenza and novel COVID-19 vaccination.  _Early Human Development_ (2020).
  * Gunaratne et al. (2019) Keith Gunaratne, Eric A Coomes, and Hourmazd Haghbayan. 2019\.  Temporal trends in anti-vaccine discourse on Twitter.  _Vaccine_ 37, 35 (2019), 4867–4871.
  * Hawkes (1971) Alan G. Hawkes. 1971\.  Spectra of some self-exciting and mutually exciting point processes.  _Biometrika_ 58, 1 (1971), 83–90.
  * Johnson et al. (2020) Neil F Johnson, Nicolas Velásquez, Nicholas Johnson Restrepo, Rhys Leahy, Nicholas Gabriel, Sara El Oud, Minzhang Zheng, Pedro Manrique, Stefan Wuchty, and Yonatan Lupu. 2020.  The online competition between pro- and anti-vaccination views.  _Nature_ 582, 7811 (2020), 230–233.
  * Khubchandani et al. (2021) Jagdish Khubchandani, Sushil Sharma, James H Price, Michael J Wiblishauser, Manoj Sharma, and Fern J Webb. 2021.  COVID-19 vaccination hesitancy in the United States: A rapid national assessment.  _Journal of Community Health_ 46, 2 (2021), 270–277.
  * Kingma and Ba (2015) Diederik P. Kingma and Jimmy Ba. 2015.  Adam: A method for stochastic optimization. In _Proc. 2015 International Conference on Learning Representations (ICLR)_. OpenReview.net.
  * Kipf and Welling (2017) Thomas N. Kipf and Max Welling. 2017.  Semi-supervised classification with graph convolutional networks. In _Proc. 2017 International Conference on Learning Representations (ICLR)_. OpenReview.net.
  * Krippendorff (1970) Klaus Krippendorff. 1970\.  Estimating the reliability, systematic error and random error of interval data.  _Educational and Psychological Measurement_ 30, 1 (1970), 61–70.
  * Lazarus et al. (2021) Jeffrey V Lazarus, Scott C Ratzan, Adam Palayew, Lawrence O Gostin, Heidi J Larson, Kenneth Rabin, Spencer Kimball, and Ayman El-Mohandes. 2021.  A global survey of potential acceptance of a COVID-19 vaccine.  _Nature Medicine_ 27, 2 (2021), 225–228.
  * Li et al. (2020b) Qian Li, Hao Peng, Jianxin Li, Congying Xia, Renyu Yang, Lichao Sun, Philip S. Yu, and Lifang He. 2020b.  A survey on text classification: From shallow to deep learning.  _CoRR_ abs/2008.00364 (2020).
  * Li et al. (2020a) Xinlong Li, Xingyu Fu, Guangluan Xu, Yang Yang, Jiuniu Wang, Li Jin, Qing Liu, and Tianyuan Xiang. 2020a.  Enhancing BERT Representation With Context-Aware Embedding for Aspect-Based Sentiment Analysis.  _IEEE Access_ 8 (2020), 46868–46876.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  RoBERTa: A Robustly optimized BERT pretraining approach.  _CoRR_ abs/1907.11692 (2019).
  * Mikolov et al. (2013) Tomás Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. 2013\.  Efficient estimation of word representations in vector space. In _Proc. 2013 International Conference on Learning Representations (ICLR)_. OpenReview.net.
  * Minaee et al. (2021) Shervin Minaee, Nal Kalchbrenner, Erik Cambria, Narjes Nikzad, Meysam Chenaghlu, and Jianfeng Gao. 2021.  Deep learning–based text classification: A comprehensive review.  _Comput. Surveys_ 54, 3 (2021), 62:1–62:40.
  * Morales et al. (2021) Gianmarco De Francisci Morales, Corrado Monti, and Michele Starnini. 2021\.  No echo in the chambers of political interactions on Reddit.  _Scientific Reports_ 11, 1 (2021), 1–12.
  * Opel et al. (2011) Douglas J Opel, James A Taylor, Rita Mangione-Smith, Cam Solomon, Chuan Zhao, Sheryl Catz, and Diane Martin. 2011.  Validity and reliability of a survey to identify vaccine-hesitant parents.  _Vaccine_ 29, 38 (2011), 6598–6605.
  * Organization (2014) World Health Organization. 2014\.  Report of the SAGE working group on Vaccine Hesitancy.  Available onhttp://www.who.int/immunization/sage/meetings/2014/october/SAGE_working_group_revised_report_vaccine_hesitancy.
  * Ou and Li (2020) Xiaozhi Ou and Hongling Li. 2020.  YNU_OXZ @ HaSpeeDe 2 and AMI : XLM-RoBERTa with ordered neurons LSTM for classification task at EVALITA 2020. In _Proc. 2020 Evaluation Campaign of Natural Language Processing and Speech Tools for Italian (EVALITA)_ , Vol. 2765. CEUR-WS.org.
  * Pandit et al. (2007) Shashank Pandit, Duen Horng Chau, Samuel Wang, and Christos Faloutsos. 2007.  Netprobe: A fast and scalable system for fraud detection in online auction networks. In _Proc. 2007 International Conference on World Wide Web (WWW)_. ACM, 201–210.
  * Puri et al. (2020) Neha Puri, Eric A Coomes, Hourmazd Haghbayan, and Keith Gunaratne. 2020.  Social media and vaccine hesitancy: New updates for the era of COVID-19 and globalized infectious diseases.  _Human vaccines & immunotherapeutics_ 16, 11 (2020), 2586–2593.
  * Sherman et al. (2021) Susan M Sherman, Louise E Smith, Julius Sim, Richard Amlôt, Megan Cutts, Hannah Dasch, G James Rubin, and Nick Sevdalis. 2021\.  COVID-19 vaccination intention in the UK: Results from the COVID-19 vaccination acceptability study (CoVAccS), a nationally representative cross-sectional survey.  _Human Vaccines & Immunotherapeutics_ 17, 6 (2021), 1612–1621.
  * Van Bavel et al. (2020) Jay J Van Bavel, Katherine Baicker, Paulo S Boggio, Valerio Capraro, Aleksandra Cichocka, Mina Cikara, Molly J Crockett, Alia J Crum, Karen M Douglas, James N Druckman, et al. 2020\.  Using social and behavioural science to support COVID-19 pandemic response.  _Nature Human Behaviour_ 4, 5 (2020), 460–471.
  * Velickovic et al. (2018) Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. 2018.  Graph Attention Networks. In _Proc. 2018 International Conference on Learning Representations (ICLR)_. OpenReview.net.
  * Wilkinson et al. (2016) Mark D Wilkinson, Michel Dumontier, IJsbrand Jan Aalbersberg, Gabrielle Appleton, Myles Axton, Arie Baak, Niklas Blomberg, Jan-Willem Boiten, Luiz Bonino da Silva Santos, Philip E Bourne, et al. 2016\.  The FAIR Guiding Principles for scientific data management and stewardship.  _Scientific Data_ 3, 1 (2016), 1–9.
  * Wilson et al. (2014) Kumanan Wilson, Katherine Atkinson, and Shelley Deeks. 2014\.  Opportunities for utilizing new technologies to increase vaccine confidence.  _Expert Review of Vaccines_ 13, 8 (2014), 969–977.
  * Zhao et al. (2021) Lingyun Zhao, Lin Li, Xinhao Zheng, and Jianwei Zhang. 2021\.  A BERT based sentiment analysis and key entity detection approach for online financial texts. In _Proc. 2021 IEEE International Conference on Computer Supported Cooperative Work in Design (CSCWD)_. IEEE, 1233–1238.
  * Zhu et al. (2020) Jiong Zhu, Yujun Yan, Lingxiao Zhao, Mark Heimann, Leman Akoglu, and Danai Koutra. 2020\.  Beyond homophily in graph neural networks: Current limitations and effective designs. In _Proc. 2020 Annual Conference on Neural Information Processing Systems (NeurIPS)_ , Vol. 33. NeurIPS, virtual, 7793–7804.

Table 6. Diffused information themes and examples.

Theme |  Description |  Example (Translated to English)
---|---|---
Positive news |  Positive news about vaccines & vaccination |  Pfizer/BioNTech’s vaccine would be effective against the new British variant of COVID19.
Negative news |  Negative news about vaccines & vaccination |  Portugal : She dies 2 days after the vaccine (at 41 years old). Her family asks for explanations
Distrust in government management |  Doubt about the trustworthiness of governments or medical institutions, e.g., regarding the daily update of statistics |  They have lied to us so much about masks, chloroquine, contagion in children, that it will be difficult to trust them the day they will tell us about a harmless vaccine.
Dissatisfaction with politics/policies |  Unsatisfactory views of politics/policies, such as ineffective vaccination programs. |  I am opposed to mandatory vaccination because all of the world’s health organizations say that it is not the right way for a vaccine to spread.
Perception of the pharmaceutical industry |  Perception that pharmaceutical manufacturers pursue only economic interests rather than public health interests |  Pfizer’s CEO sold 60 percent of his shares when the Covid vaccine was announced. When the CEO sells, it stinks
Conspiracy |  Content that describes the event as the secret acts of a powerful, malevolent force. |  18 months they’ve been on the vaccine ???? When did they know there would be a Covid 19 ”pandemic” ????
Beliefs, attitudes about health and prevention |  Personal views on vaccines and the immune system, e.g. homeopathy, natural immunity, alternative therapies. |  There is no point in a generalized vaccine for a disease whose mortality is close to 0.05%.
Positive personal expression |  Personal expression of positive attitude towards vaccines |  We have a new weapon against the virus: the vaccine. Hold together, again.
Negative personal expression |  Personal expression of negative attitude towards vaccines |  Why could actually 1.5 billion Chinese get healthy without vaccination, and with us it only works with vaccination…?
Positive information |  Positive expressions about vaccines from healthcare professionals |  #COVID19 #vaccinationHow does an mRNA vaccine work?
Negative information |  Negative expressions about vaccines from healthcare professionals |  My daughter, a nurse at the AP-HP, on the vaccine ”Ah ah ah! They don’t even dream about it, they start with the old ones so that we can attribute the side effects to age”.


# MCPrioQ: A lock-free algorithm for online sparse markov-chains

Jesper Derehag1 and Åke AI Johansson2 1J. Derehag is a senior machine learning
engineer at Ericsson AB, Lindholmspiren 11, 417 56 Göteborg, Sweden
jesper.derehag at ericsson.com2Å. Johansson is a senior machine learning
engineer at Ericsson AB, Lindholmspiren 11, 417 56 Göteborg, Sweden
ake.ai.johansson at ericsson.com

###### Abstract

In high performance systems it is sometimes hard to build very large graphs
that are efficient both with respect to memory and compute. This paper
proposes a data structure called Markov-chain-priority-queue (MCPrioQ), which
is a lock-free sparse markov-chain that enables online and continuous learning
with time-complexity of $O(1)$ for updates and $O(CDF^{-1}(t))$ inference.
MCPrioQ is especially suitable for recommender-systems for lookups of
$n$-items in descending probability order. The concurrent updates are achieved
using hash-tables and atomic instructions and the lookups are achieved through
a novel priority-queue which allows for approximately correct results even
during concurrent updates. The approximatly correct and lock-free property is
maintained by a read-copy-update scheme, but where the semantics have been
slightly updated to allow for swap of elements rather than the traditional
pop-insert scheme.

##  I INTRODUCTION

In recommender-systems or collaborative filtering applications it is
relatively common to produce recommendations that is based on some cumulative-
sum of probabilities. For example, recommend any number of products such that
the probability of finding a product that matches a users preferences is above
a certain threshold. E.g. a 90% probability that a matching item exists in the
list of recommended items. This is also a useful context in for example
telecommunications systems where a users location may be unknown in the
network. As such, a cellular network could be considered as a directed graph
where the base stations would be nodes and the physical movement of a user
through that network are the edges. If the location (node) of a user is
unknown, a telecommunications network would then typically perform a paging
operation, essentially querying all nodes if the user resides within that
nodes area. This could be considered a recommender-system where one wants to
recommend a number of nodes such that the system will find the user with some
threshold-probability [1]. In such a system, it would be highly beneficial to
be able to construct the graph while simultaneously being able to query it,
with inference being optimized for returning a list of items.

##  II MCPrioQ

Figure 1: Pseudo class-diagram for the data structures

The proposed algorithm is a specialized version of a markov-chain where one
wants to sort edges based on some counter. The counter could be anything but
this implementation is optimized for counters that tend to be incremented in
small steps and where increments occur in accordance with the probability of a
edge transition. In other words, it is optimized for the assumption that the
priority queue is already sorted and any increments will unlikely lead to
resorting.

####  II-1 Node lookup tables

src-node and dst-node lookup tables need to be lock-free shared data
structures. In this particular case we assume these to be Read-copy-update
(RCU) hash-tables [2]. Specifically, as we assume that the PriorityQueue is a
doubly linked list, which eventually will need elements to be freed, it is
important that also the hash-tables are RCU based such that the read-side-
critical-sections share the same grace period.

####  II-2 Priority Queue

The more interesting part in Fig. 1 are the Priority Queue as a sorted doubly
linked list, which is central to the contributions of this paper. In this
particular case we will assume that increments of the priority are small and
that the input order is not shuffled. Therefore we argue that the no-inversion
case is the normal case for inserts. Furthermore, we know that the search is
strictly finding max priority items, so random access is not the normal case.
Formally this could be considered a monotone priority queue. A popular choice
for priority queues tends to be a heap which is generally optimized for fast
insert and finding the top most important element, while in the cumulative-
probability case one wants to find a list of items of cumulative probability.
Other popular implementations of priority queues are skip-lists [3] which
would be a suitable choice for cumulative-probability applications as well.
One could argue that we could skip the dst-hash table and only use a skip-
list, but since the edge-list is sorted on the edge transition probability and
not its node-id, it would mean that the skip-connections cannot be used for
any practical benefit. But since we want to order the elements in transition
probability and we can assume that updates are not shuffled, average search
complexity (for updates) are then equal to the edge probability distribution.
A hash-table with $O(1)$ is hard to beat, but practically the choice may not
be that obvious as it would depend on the exact probability distribution and
cache-line availability. For this reason we propose that the priority queue is
a doubly-linked list, and the dst-node hash-table is an optional optimization.

There are a number of different lock-free implementations of doubly-linked-
lists, for ex. [4]. But since we are already using an RCU based hash-table we
might as well also use an RCU based doubly-linked-list. This will allow us to
re-use the read-side-critical-section in the src-node hash-table, and in
principle get insert and remove for free. However, the typical RCU doubly-
linked-list semantics does not allow for swapping nodes while still
maintaining approximately correct order. Specifically one would need to pop-
insert elements where concurrent readers could end up with missing nodes in
between the pop and insert. Instead we propose an extension to the standard
RCU semantics to swap the order of 2 nodes, which allows for an implementation
of lock-free bubble-sorting. The swap operation for 2 adjacent nodes are
described in Fig. 2.

Figure 2: Wait-free bubble-sort of doubly linked list

####  II-3 Counters

In the case of cumulative transition probability, it is implemented as 2
atomic counters, with one indicating the total number of transitions between
two nodes (edge), located as part of the priority queue in Fig.1. There is
also an additional counter keeping track of total transitions for all edges
given a src-node. The transition probability is then calculated as the edge
counter divided by the total number of transitions for the node. By having 2
counters and calculating the probability at inference, there is no need for
updating all edges as the probability distribution changes.

###  II-A Updates

####  II-A1 New edge

Insert the node(s) in the 2 hash-tables and adding an element at the tail of
the priority queue.

####  II-A2 Update of edge

Updates of edges would be the normal case, so a $O(1)$ lookup in src-node plus
an $O(1)$ lookup in dst-node, and a simple atomic increment in the transition
counter inside the priority queue. The normal case would be that they are
already in order, so nothing additional would be done. If the update result
indicates that the previous node has a smaller count than the current node,
those elements should be swapped. Similarly, the previous-previous node needs
to be checked for swapping in a bubble-sort style leading to $O(n)$ time-
complexity in the worst case. However, since increments are small and assuming
non-uniform probability distributions, the normal case would likely be no-swap
and in rare cases a single-swap.

###  II-B Inference

The src-node hash-table of $O(1)$ gives the priority-queue. The priority queue
is then the sorted-list of edges with $O(nt)$ for worst-case scenario of a
uniform transition probability where $t$ is the probability threshold.
Slightly more formally, the complexity is equal to the quantile function of
the probability distribution. Practically, oftentimes the edges follow a Zipf
distribution, and so intuitively you could say its much more likely that you
will only need a small subset of edges early on in the priority queue.

###  II-C Model decay

One way of dealing with massively large graphs that change over time is to add
intentional forgetting, or model decay. As the intended use of this data
structure is to be online and which will exist over extended periods of time
where we dont want to have to retrain the model at some frequency. One way of
dealing with that is to uniformly reduce the transition counts over all edges.
For example multiplying all transition counts with $0.5$, either at some
threshold over the number of total transitions, or maybe more robustly at some
frequency that reflects the probability of graph-topology changes. This will
maintain the probability distribution, but as some transition counts reaches
$0$, that will indicate that edge is no longer used and as such it can be
removed. This will allow the graph to remain roughly up to date, although with
some added convergence delay for the probability distributions.

## References

  * [1] J. Jeong, D. Roeland, J. Derehag, Å. A. Johansson, V. Umaashankar, G. Sun, and G. Eriksson, “Mobility prediction for 5g core networks,” IEEE Communications Standards Magazine, vol. 5, no. 1, pp. 56–61, 2021.
  * [2] P. E. McKenney and J. D. Slingwine, “Read-copy update: Using execution history to solve concurrency problems,” in Parallel and Distributed Computing and Systems, (Las Vegas, NV), pp. 509–518, October 1998.
  * [3] H. Sundell and P. Tsigas, “Fast and lock-free concurrent priority queues for multi-thread systems,” Journal of Parallel and Distributed Computing, vol. 65, no. 5, pp. 609–627, 2005.
  * [4] H. Sundell and P. Tsigas, “Lock-free deques and doubly linked lists,” Journal of Parallel and Distributed Computing, vol. 68, no. 7, pp. 1008–1020, 2008.


Statement from the author

This paper is wrong. The main idea is probably interesting but the statement
(and obviously, the proof) are wrong. I thank Y. Genzmer for pointing out the
issues which led me to realizing the mistake.

I am leaving the contents as I wrote them in case someone finds them relevant,
even if only for avoiding the same mistakes.

Essentially: the Weierstrass derivation is wrongly computed everywhere.

The simplest example (Genzmer) is a homogeneous cubic $x^{3}+xy^{2}+y^{3}$.

Pedro Fortuny Ayuso, June 5 2021.

# Irreducibility criterion for singular hypersurfaces of $(\mathbb{C}^{n},0)$

P. Fortuny Ayuso  Dpto. Matemáticas, Universidad de Oviedo. Oviedo, Spain.
<EMAIL_ADDRESS>
###### Abstract.

Using vector fields we obtain an irreducibility criterion for hypersurfaces.
It only requires the Weierstrass division.

###### 2020 Mathematics Subject Classification:

32S25, 32S05, 32S15, 14B05, 14J17

##  1\. Introduction and Notation

The problem of deciding whether a plane curve (in implicit form, obviously) is
irreducible is of paramount importance for studying its singularity (see [1,
2, 8, 7, 5] just for several relevant examples, without any aim to
completeness). The usual ways to solve it in characteristic $0$ are by means
of the Newton diagram (and Puiseux series) or using Approximate Roots,
following Abhyankar and Moh. Both ways require “going further” than the first
Puiseux exponent, and a very delicate analysis of the singularity.

In this short note we provide a criterion which, using elementary vector
fields and Weierstrass division decides whether a germ of (reduced)
_hypersurface_ is or not irreducible. The result is based on the fact that
vector fields can have “bad order of tangency” with a hypersurface if and only
if this is reducible, because the contact structure associated to vector
fields is essentially different in the reducible and irreducible cases.

##  2\. Irreducibility criterion for hypersurface singularities

Let $f:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)$ define a reduced germ of
hypersurface $f=0$. For simplicity, we denote $(x,\overline{y})$ a system of
coordinates in $(\mathbb{C}^{n},0)$ such that $f(x,\overline{y})$ is in
Weierstrass form:

(1) $f(x,\overline{y})=x^{k}+\sum_{i=0}^{k-1}F_{i}(y)x^{i}.$

where $k$ is the multiplicity of $f$. From here on, we assume that $k>2$ (the
case $k=2$ is trivial) and $F_{0}(\overline{y})\neq 0$ (the hypersurface does
not contain $x=0$). Let $df$ denote the differential form of $f$ and
$X\in{\mathfrak{X}}(\mathbb{C}^{2},0)$ a germ of holomorphic vector field in
$(\mathbb{C}^{2},0)$.

######  Definition 1.

The _tangency function_ of $X$ with $f(x,\overline{y})=0$ (with respect to the
specific coordinate function) is the remainder $R(x,\overline{y})$ of the
Weierstrass division:

$df(X)=Q(x,\overline{y})f(x,\overline{y})+R(x,\overline{y}).$

The _tangency order_ of $X$ with $f(x,\overline{y})$ _in the $x$-direction_ is
the order of $R(x,\overline{y})$ as a power series in $x$.

The tangency function measures, in some sense, “how” $X$ fails to be tangent
to $f(x,\overline{y})=0$ [6, 3, 4], the tangency order the order of that
“failure”. We shall omit the qualifier “in the $x$-direction” because it is
unnecessary in what follows.

The irreducibility criterion is the following (recall that $x$ does not divide
$f(x,\overline{y})$):

######  Theorem 1.

With the notations and hypotheses above (recall that $f$ is reduced and
$k>2$), then $f(x,\overline{y})=0$ is reducible if and only if there exists an
integer $2\leq r<k$ and a vector field

$X=x^{r}\frac{\partial}{\partial x}$

whose tangency order with $f(x,\overline{y})$ is $0$.

###### Proof.

Assume $f(x,\overline{y})=0$ is reducible and set
$f(x,\overline{y})=f_{1}(x,\overline{y})f_{2}(x,\overline{y})$. As $f$ is in
Weierstrass form, we can assume $f_{1}(x,\overline{y})$ is too and
$f_{2}(x,\overline{y})$ is almost:

$f_{1}(x,\overline{y})=x^{a_{1}}+\sum_{i=0}^{a_{1}-i}F^{1}_{i}(\overline{y})x^{i},\;f_{2}(x,\overline{y})=u(x,\overline{y})\bigg{(}x^{a_{2}}+\sum_{i=0}^{a_{2}-i}F^{2}_{i}(\overline{y})x^{i}\bigg{)}.$

Let $X$ be the vector field

$X=x^{a_{1}+1}\frac{\partial}{\partial x}.$

As $f_{1}(x,\overline{y})$ has degree $a_{1}$ in $x$, performing the
Weierstrass division, we obtain
$x^{a_{1}+1}=xf_{1}(x,\overline{y})+R(x,\overline{y})$ where
$R(x,\overline{y})=-xF^{1}_{0}(y)+x^{2}S(x,\overline{y})$ with
$S(x,\overline{y})$ a holomorphic function. From this:

$df(X)=\left(\left(f_{2}\frac{\partial f_{1}}{\partial x}+f_{1}\frac{\partial
f_{2}}{\partial
x}\right)dx+(\cdots)\right)\left(xf_{1}+R\right)\frac{\partial}{\partial x}$

where the dots denote an irrelevant holomorphic $1$-form. This gives:

(2) $df(X)=xf_{1}f_{2}\frac{\partial f_{1}}{\partial
x}+xf_{1}^{2}\frac{\partial f_{2}}{\partial
x}+x(-F^{1}_{0}(\overline{y})+xS(x,\overline{y}))\left(f_{1}\frac{\partial
f_{2}}{\partial x}+f_{2}\frac{\partial f_{1}}{\partial x}\right).$

The first term is a multiple of $f$. The second one is:

$xf_{1}^{2}\left(u\frac{\partial(f_{2}/u)}{\partial x}+(f_{2}/u)\frac{\partial
u}{\partial
x}\right)=v(x,\overline{y})\left(x^{2a_{1}+a_{2}}+\sum_{i=1}^{2a_{1}+a_{2}-1}h_{i}(\overline{y})x^{i}\right)$

(for a unit $v(x,\overline{y})$). This term is not a multiple of $f$ because
$f$ is reduced. Notice the indices starting at $1$. The last term in (2) has
degree $a_{1}+a_{2}$ in $x$ as a Weierstrass polynomial (except for a unit).
Performing the Weierstrass division, we obtain:

$df(X)=Q(x,\overline{y})f(x,\overline{y})-w(x,\overline{y})F^{1}_{0}(\overline{y}))^{2}F_{0}^{2}(\overline{y})+xT(x,\overline{y})$

for a unit $w(x,\overline{y})$ and some holomorphic function
$T(x,\overline{y})$. Thus, the contact order of $X$ with $f=0$ is $0$.

The reciprocal: If $f(x,\overline{y})$ is irreducible of multiplicity $n$
then, for any $X=x^{r}\frac{\partial}{\partial x}$ with $2\leq r<n$

$df(X)=\left(\frac{\partial f}{\partial
x}dx+\cdots\right)x^{r}\frac{\partial}{\partial x}=x^{r}\frac{\partial
f}{\partial x}$

whose contact order with $f$ is exactly $r-1$ (as $x$ does not divide $f$). ∎

## References

  * [1] S.S. Abhyankar.  Irreducibility criterion for germs of analytic functions of two complex variables.  Adv. in Math, (74):190–257, 1989.
  * [2] S.S. Abhyankar and T.T. Moh.  Newton puiseux expansion and generalized tshianhausen transformation ii.  J. reine und angew Math, (261):29–54, 1973.
  * [3] P. Fortuny Ayuso.  Vector flows and the analytic moduli of singular plane branches.  Rev. R. Acad. Cienc. Exactas. RACSAM, 113(4):4107–4118, 2019.
  * [4] P. Fortuny Ayuso.  Normal forms for hypersurface singularities.  arXiv:2106.00562, 2021.
  * [5] E. R. García Barroso and J. Gwoźdiewicz.  A discriminant criterion of irreducibility.  Kodai. Math. J., (35):403–414, 2012.
  * [6] P. Fortuny Ayuso and J. Ribón.  The action of a plane singular holomorphic flow on a non-invariant branch.  Canadian Journal of Mathematics, 2019.
  * [7] T.C. Kuo and Y.C. Lu.  On analytic function germs of two complex variables.  Topology, (16):299–310, 1977.
  * [8] T.T. Moh.  On approximate roots of a polynomial.  J. reine und angew. Math., (278):301–306, 1975.