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11institutetext:  11email<EMAIL_ADDRESS>

1 Graduate School of Informatics, Nagoya University, Japan
2 School of Environment and Society, Tokyo Institute of Technology, Japan

# Characterizing the roles of bots during the COVID-19 infodemic on Twitter

Wentao Xu1 Kazutoshi Sasahara2

(Received: date / Accepted: date)

###### Abstract

An infodemic is an emerging phenomenon caused by an overabundance of
information online. This proliferation of information makes it difficult for
the public to distinguish trustworthy news and credible information from
untrustworthy sites and non-credible sources. The perils of an infodemic
debuted with the outbreak of the COVID-19 pandemic and bots (i.e., automated
accounts controlled by a set of algorithms) that are suspected of spreading
the infodemic. Although previous research has revealed that bots played a
central role in spreading misinformation during major political events, how
bots behaved during the infodemic is unclear. In this paper, we examined the
roles of bots in the case of the COVID-19 infodemic and the diffusion of non-
credible information such as “5G” and “Bill Gates” conspiracy theories and
content related to “Trump” and “WHO” by analyzing retweet networks and
retweeted items. We show the segregated topology of their retweet networks,
which indicates that right-wing self-media accounts and conspiracy theorists
may lead to this opinion cleavage, while malicious bots might favor
amplification of the diffusion of non-credible information. Although the basic
influence of information diffusion could be larger in human users than bots,
the effects of bots are non-negligible under an infodemic situation.

###### Keywords:

Bot COVID-19 Conspiracy theory Infodemic Misinfomation Social media

##  1 Introduction

“We’re fighting an infodemic. Fake news spreads faster and more easily than
this virus, and is just as dangerous…” said the World Health Organization
(WHO) Director-General on February 15, 2020 WHO20200915 . Prior to this
comment, a large amount of misinformation about the new coronavirus emerged on
popular social networking sites (SNSs) and these began to play a major role in
the diffusion of misinformation. According to Brennen , in terms of
information sources, top-down misinformation from politicians, celebrities,
and other prominent public figures accounted for 69% of total social media
engagement. Additionally, approximately 60% of the COVID-19 related
information was reconfigured, twisted, recontextualized and reworked on
Twitter, while 38% of the misinformation was completely fabricated. The fact-
checking organization, Politifact also pointed out that trustful and mostly
trustful news about the coronavirus comprised only up to 10% of the total
information Politifact  being disseminated. SNS users tend to connect to like-
minded users, which is known as the “birds-of-a-feather phenomenon” or
homophily Menczer2004 ; Redner1998 . Meanwhile, users also tend to follow
influencers and celebrities on SNS who present themselves as information hubs.
Thus, when an influencer posts misinformation, his or her followers, often
with no doubts about the content, tend to believe it and share the post in a
homophilic social network.

In addition to human users, previous studies have shown that bots (i.e.,
automated accounts controlled by a set of algorithms) play a key role in
propagating misinformation.
MessiasJohnnatanLucasSchmidtRicardoAugustoRabeloOliveira  discussed how bots
engaged with humans to increase their influence. A well-known case in politics
is the 2016 US presidential election, during which bots were used to widely
diffuse misinformation AlessandroBessi ; Varol2017 . It was estimated that of
all tweeted links to popular websites, 66% were shared by bots
PewResearchCenter . Ferrara_2020  analyzed 43.3 million English tweets and
found the preferences of bots and humans in tweet contents by analyzing
n-grams and hashtags. Furthermore, it was suggested that bots may play a
critical role in driving the viral spread of content from low-credibility
sources and may be able to amplify misinformation Shao2018 .
doi:10.1080/10584609.2018.1526238  found that social bots increased from 7.1%
to 9.9% when examining German Twitter followers’ prevalence and activities of
seven German parties before the 2017 electoral campaigns. Bots were also used
for spreading and amplifying vaccine misinformation Broniatowski2018a .
Research about the COVID-19 infodemic has discovered the information flow
pathways between humans and bots Gallotti2020 . A study of social media
manipulation in the 2020 US presidential election characterized the
differences between right and left-leaning bot behavior
Ferrara_Chang_Chen_Muric_Patel_2020 . In addition, bot activity is believed to
be linked to hate toxicity in denser and more isolated local communities in
both the US and the Philippines Uyheng2020 .

There are several methods for bot characterization and identification.
Botometer is a well-known tool for automatically detecting bots based on
supervised machine learning models DBLP:journals/corr/DavisVFFM16 ;
Sayyadiharikandeh2020DetectionON . Botometer examines six classes of features
including profile, friends, social network, temporal activity patterns,
language, and sentiment which is further categorized into approximately 1,200
features for a Twitter account. This tool computes a “bot score” for each user
that ranges from 0.0 to 1.0. The higher the score, the higher the probability
that the user is a bot. Botometer is a state-of-the-art tool for identifying
bots, and a series of studies have used this tool to quantify their online
behaviors of bots Shao2018 ; Vosoughi1146 . We applied Botometer in our study
therefore, to discriminate between bots and humans.

Given this context, an important research question is how bots behaved in the
spread of misinformation during the COVID-19 infodemic. To study this, we
focused on retweets on Twitter. Retweet is an information sharing behavior by
which any user can share messages immediately with his or her followers. A
retweet can be both a productive communicative tool and a selfish act of
attention seekers Boyd2010TweetTR . Cha2010MeasuringUI  found that an popular
tweet has either an interesting context or it is produced (retweeted) by an
influencer. 10.1007/978-3-642-23808-6_2  pointed out that a user’s high
popularity does not necessarily imply a high level of influence and vice-
versa, indicating that an influencer’s popularity and influence are weakly
correlated. However, Hong2011 ; Suh2010 ; 10.1145/1772690.1772751  considered
that a user’s contextual information (e.g., social network topology, tweet
content, URLs) affects the retweet behavior. In this paper, we used retweets
to address information-sharing behavior, shedding light on how COVID-19
misinformation is shared in an information ecosystem where bots live.

Misinformation is classified into several types and conspiracy theory is one
type Douglas2019 . The negative effect of a conspiracy theory is to elicit
emotions including avoidance, fear, anger, aggression and further result in
irrational behaviors Douglas2019 . As an example, the 5G conspiracy theory was
reported on January 22 by a local Belgian newspaper, reporting that a local
doctor claimed that 5G might be linked to the coronavirus TEMPERTON . In the
UK, 5G cell-phone masts came under arson attacks due to this conspiracy theory
Hamilton . Another version of this conspiracy theory claims that 5G alters
people’s immune systems and changes DNA structures, thus increasing people’s
susceptibility to contracting the coronavirus DisinfoLab ; Ketchell . Another
popular conspiracy theory targeted Bill Gates, co-founder of Microsoft
Corporation, and claimed that Gates supported implanting tracking chips in
people under the pretext of a mandatory coronavirus vaccination JackGoodman2 ;
Jr. . US political groups were reported as showing a significant partisan bias
regarding this conspiracy yahoo20200910 ; compared to the left wing, the right
wing was more inclined to believe in this conspiracy.

To investigate the spreading of misinformation during the COVID-19 infodemic,
we focused on the above-mentioned conspiracy theories related to 5G and Bill
Gates as mentioned above. For comparison, we also focused on other topics,
such as “WHO” and “Trump” (the 45th US president). These keywords were
selected because health and political misinformation flourished during the
COVID-19 infodemic. Moreover, recent research revealed that Trump was the
largest driver of COVID-19 misinformation Evanega2020 .

In this paper, we first characterize the credible and non-credible bots around
the four topics in the retweet networks. We then compare the retweet
activities as well as other features in the four topics considered. Our
results may help us understand how bots played a role during the COVID-19
infodemic and provide insights into a mitigation strategy.

##  2 Data and Methods

###  2.1 Data collection and preprocessing

We used Twitter as a data source to characterize the COVID-19 infodemic. We
collected 279,538,960 English tweets from February 20 to May 31 by querying
COVID-19 related keywords: “corona virus”; “coronavirus”; “covid19”;
“2019-nCoV”; “SARS-CoV-2”; and “wuhanpneumonia” using Twitter’s standard
search API. We focused on four topics in our analyses: “WHO”; “Trump”; “Bill
Gates”; and “5G” as noted. Extracting tweets regarding these topics resulted
in a total of 37,219,979 tweets, of which 23,1515,441 (82.8%) were retweets.
The breakdown of this dataset is shown in Table 1. Some users in our dataset
could be considered malicious and were suspended based on Twitter’s spam
policy during the period between the date we collected the tweets and the date
we computed the corresponding bot scores; These users were, thus, not included
in our analyses.

According to a list of non-credible websites released on MisinfoMe
111https://misinfo.me and a list of non-credible news website domains released
in Zimdars2016 , 893 responsive websites from the total of 1,143 domains were
collected and used as the non-credible domain list. We also examined a list of
trustworthy rated media released by NELA-GT-2018  and retrieved 30 (all
responsive) credible media domains. In addition, we added major science
journals _Nature_ 222https://www.nature.com and _Science_
333https://www.sciencemag.org/ as credible domains. Thus, we obtained a total
of 32 credible domains and used these as the credible domain list. Based on
the credible and non-credible domain lists, each tweet was labelled as
“credible” if the tweet included a URL in the credible domain list, and as
“non-credible” if the tweet included a URL in the non-credible domain list.
Then, given a topic, each user was labelled as “credible” if the user
retweeted credible tweets exclusively and “non-credible” if the user retweeted
non-credible tweets exclusively. In other words, non-credible users are those
who posted with URLs from the non-credible domain list at least once but never
posted with URLs from the credible domain list. Credible users were similarly
defined. Note that a user’s label can change from topic to topic. For
instance, a user is labeled “credible” in the WHO topic if the user retweets
credible domains, exclusively in that topic, even if the user retweets non-
credible domains in other topics.

Furthermore, we used the Botometer API V3 to compute bot scores of users. In
this study, we used the CAP (complete automation probability) value instead of
the raw bot score. Given a screen name of a Twitter account, the Botometer
returns a CAP value ranging from 0.0 to 1.0. After multiple trials of
analyses, We set 0.54 as the threshold in our analyses. This means that a user
was considered to be a bot if the CAP value was larger than or equal to 0.54;
otherwise, the user was considered to be human.

Using the above discriminative conditions for credible/non-credible and
bot/human, we classified users into five types for analyses: credible humans
(CH), non-credible humans (NH), credible bots (CB), non-credible bots (NB),
and the other.

Table 1: Overview of COVID-19 tweets by topic |  | Unique
---
users (U)
Unique Users with Bot score (US) |  Percentage (US/S) |  # Tweets | # Retweets
WHO | 88,719 | 73,704 | 83.1 | 128,016 | 46,650
Trump | 1,125,251 | 947,694 | 84.2 | 5,631,459 | 2,322,036
5G | 67,523 | 55,315 | 81.9 | 97,638 | 31,814
Bill Gates | 94,584 | 77,896 | 82.3 | 138,042 | 75,885

###  2.2 Verification of user classes

The classification results of users in the previous section (i.e., CH, CB, NH,
NB) affect the results of our analyses. We confirmed the reliability of the
classification using manual verification as described next.

For bot classification, we randomly selected 200 accounts from our dataset,
with 100 labeled as bots and 100 labeled as humans. We provided two human
workers with these account names and asked them to independently read profiles
and tweets by each user and then classify the user as a bot or human. Then we
checked the classification consistency by computing Cohen’s kappa coefficient.
The resulting $\kappa=0.68$ indicates substantial agreement between the two
workers, indicating that the bot score threshold used is sufficiently
reliable. Note that according to 10.2307/2529310 , Cohen’s kappa value is
interpreted as follows: 0.0-0.2 for slight agreement; 0.2-0.4 for fair
agreement; 0.4-0.6 for moderate agreement; 0.6-0.8 for substantial agreement;
and 0.8-1.0 for near perfect agreement.

A similar verification process was conducted for credible/non-credible
classification with two additional workers. They evaluated 100 randomly
selected accounts, with 50 labeled as credible and 50 labeled as non-credible,
by reading their tweets and profiles, and then classified them as credible or
non-credible users. After that, Cohen’s kappa was computed and resulted in
$\kappa=0.70$, indicating a significant match. In this way, credible/non-
credible classification based on the above-mentioned criteria is, likewise,
sufficiently reliable.

###  2.3 Retweet behavior analysis

To examine information spreading patterns by topic, we constructed a retweet
network for each topic, in which nodes represent users and a directed edge was
made from the source to the target if a target user is retweeted by a source
user. The structural features were quantified using these retweet networks.
For visibility, we illustrated the retweet network by including only bots and
selected accounts (as anchors) by using the software Gephi ICWSM09154 , with
the graph layout algorithm ForceAtlas2 Jacomy2014 . We highlighted the
selected users with a large indegree that includes famous politicians, well-
known mainstream media, and right-wing media from the top 40 indegree users.
Moreover, we compared temporal patterns of retweet activities by topic among
four types of users (i.e, CH, CB, NH, NB).

Furthermore, we quantified how bots interacted with media and celebrities
using the aggregated retweet network ($n=211$) based on the bots that appeared
across the topics. For this, we first identified 19 credible media and
celebrity accounts (CM and CC), and 12 non-credible counterparts (NM and NC)
in reference to their user profiles, their Wikipedia articles, and the
official websites, Media Bias / Fact Check, 444https://mediabiasfactcheck.com
etc. Then, we classified media and celebrity accounts as: (a) those retweeted
by non-credible bots only; (b) those retweeted by both credible and non-
credible bots; and (c) those retweeted by credible bots only. The network
visualization process was the same as above.

###  2.4 Retweeted contents analysis

We also looked at differences in the contents of the retweeted URLs in each
topic. Retweets in our COVID-19 dataset did not contain a sufficient amount of
texts, instead using hyperlinks or URLs to online articles. Thus, we focused
on tweets including URLs to online articles and collected these articles
retweeted by credible/non-credible humans and bots, separately. We
characterized the articles based on terms (nouns), with their importance
measured by the TF-IDF score. For this analysis, we selected the top 30 terms
from credible users and the top 30 terms from non-credible users, and then
merged them without duplicates from the collected articles.

TF-IDF stands for term frequency–inverse document frequency and is commonly
used in natural language processing (NLP). TF-IDF is calculated as follows:

$TF\times IDF,$ (1)

where $TF$ is the number of a given term (noun). We used the following formula
for IDF:

$IDF=\ln\frac{N}{d},$ (2)

where $N$ represents the total number of documents, and $d$ represents the
number of documents that include the term.

To compare important terms used in articles retweeted by credible and non-
credible users, we summarized TF-IDF values by using the laterality index
($LI$) Sasahara2019a , defined as follows:

$LI=\frac{C-NC}{C+NC},LI\in[-1,1],$ (3)

where $C$ is the TF-IDF score for terms used in articles retweeted by credible
users and $NC$ is used for terms used in articles retweeted by non-credible
users. $LI$ compares the importance of a term between credible sites and non-
credible sites. A negative $LI$ indicates that the term is characteristic of
non-credible sites; a positive $LI$ indicates that the term is characteristic
of credible sites; $LI=0$ indicates that the term is equally important in both
sites.

##  3 Results

###  3.1 Segregated structures of retweet networks

We looked at the retweet interactions between humans and bots for each topic
using the preprocessed COVID-19 tweets. The resulting retweet networks are
shown in Fig. 1. It is notable that segregated structures emerged in all the
topics considered, with dense connections inside and sparse connections in
between.

In the “WHO” network ($n=88,719$), 3.8% of labelled users are non-credible
users while 19.7% are credible bots. The credible group contained official
media accounts such as “@washingtonpost”, “@ABC”, “@Reuters”, “@CNN”and
“@BBCWorld” and were separated from the non-credible group containing
“@DailyCaller”, “@gatewaypundit” and “@KimStrassel”(Fig. 1(a)). We found that
non-credible bots were appearing around the US conservative columnist
“KimStrassel” (Kimberley Strassel) as well as “@DailyCaller” (an American
right-wing misinformation website) and “@gatewaypundit” (an American far-right
website publishing misleading news). These results imply that the non-credible
bots might be attempting to interact with politically right-leaning users to
increase these users’ exposures to negative information. Although the WHO
itself is a neutral topic, partisan asymmetry was visible during the COVID-19
infodemic.

Previous research has found that the retweet network of the 2010 US midterm
elections showed typical “left” and “right” segregated groups Conover2012 .
Thus, we examined whether the “Trump” retweet network shares the similar
features. Fig. 1(b) shows the Trump network ($n=1,125,251$) where 3.2% of the
labelled users are non-credible bots while 23.5% are credible bots. Here
“@HillaryClinton” (Hillary Clinton) and “@JoeBiden” (Joe Biden), representing
the progressives clustered together, were distant from the conservative
cluster with “@realDonaldTrump” (Donald Trump). The political echo chamber was
reobserved in 2020 in the context of the COVID-19 infodemic. A notable finding
is that “@realDonaldTrump” was mostly retweeted by non-credible bots (shown in
red), whereas “@HillaryClinton” and “@JoeBiden” were less so.

(a) WHO

(b) Trump

(c) 5G

(d) Bill Gates

Figure 1: Retweet networks related to “WHO”,“Trump”, “5G”, and “Bill Gates”.
Red nodes indicate non-credible bots; green nodes indicate credible bots.
Edges represent retweets among bots, and between bots and popular accounts
(labelled).

As far as “5G” is concerned, two separate groups were observed again in the
retweet network ($n=67,523$), in which 1.62% of the labelled users are non-
credible bots and 8.82% are credible bots (Fig. 1(c)). One side of the network
includes “@davidicke” (David Icke) and “@davidkurten” (David Kurten). The
former is reported to be a conspiracy theorist, and the latter is currently a
member of the UK Independence Party (right-wing populist party) and has been
since 2016 BBCUKIP ; Skynews ; UropeanConservative . They were the two most-
retweeted users in the 5G conspiracy topic. By contrast, the mainstream
British media “@BBCWorld” and “@WHO” were located on the other side of the
network in Fig. 1(c). More non-credible bots were involved on the side of
“@davidicke”, while there were more credible bots on the other side. Although
“5G” was considered as a popular conspiracy theory in the early days of
COVID-19 pandemic, a larger number of non-credible bots was not observed in
comparison with other topics.

“Bill Gates” is another conspiracy theory topic as mentioned earlier. In
regard to this topic, 5.95% of the labelled users in the retweet network
($n=94,584$) are non-credible bots while 18.0% are credible bots (Fig. 1(d)).
Again, we can find the “@davidicke” and “@DailyCaller”, which are occupied by
non-credible bots, on the right side, while a cluster on the left is covered
by credible bots with legitimate media including “@business” and “@thehill”.
Moreover, “@davidicke” was observed in both “5G” and “Bill Gates” conspiracy
topics; this account was suspended by Twitter and is no longer accessible. It
can be observed that there are no visible links among these bots in the “Bill
Gates” network. This is because these bots and labelled accounts did not have
mutual retweets; for example, “@DailyCaller” was retweeted by 336 humans, 294
of whom were non-credible humans.

Then, we quantified indegrees (the numbers of retweeted posts by different
users, used as a measure for engagement) as a function of the bot score (CAP).
The resulting scatter plots are shown in Fig. 2. The complementary cumulative
distribution Function (CCDF) of indegrees is shown in the supplementary
information (SI). The majority of users are obviously credible humans. It
turns out that the indegrees tend to be inversely proportional to the bot
score and, on average, indegrees for humans are larger than those for bots in
all the topics. Compared with humans, bots were, therefore less engaging in
retweets in general. However, average indegrees of non-credible bots are
higher than those of credible bots ($t$-test, $p=0.00047$).

Figure 2: Degrees vs. Bot Score (CAP) in “WHO”, “Trump”, “5G” and “Bill Gates”
topics. The red dashed line is the threshold for bots/humans classification
(CAP=0.54).

There are several outliers of non-credible bots that have larger indegrees in
Fig. 2. For example, “ @JoyaMia00”(NB) is an MAGA (Make America Great Again)
user and a Trump supporter; “@soleil82639” (NB) retweets posts in at least
three languages including English, Japanese, and French; These posts are
related to history and politics as well as COVID-19. In addition,
“@freesandstorm” (CB) is always retweeting climate and environment-related
posts (“@BillEsteem” (CB) is closed.) (Fig. 2a). In the Trump topic,
“@DallasJames428” (NB) and “@Rparkerscience” (NB) are suspended.
“@HarryPotterMAGE” (CB) is retweeting political tweets while “@KMGGaryde” (CB)
is closed (Fig. 2b); In the “5G” topic, “@iotworlds” (CB) is closed. From the
page of “@guidaautonoma” (CB) we can say that the users is enthusiastic about
autonomous car; “@orion_pentict” (NB) retweets many anti-Biden posts, while
“@KJovic” is an anti-lockdown supporter (Fig. 2c). In the Bill Gates topic,
“@dyro874” (CB) is suspended; “@DrTony44” is now retweeting posts not related
to COVID-19 but to entertainment; “@covid19_alert” (NB) is retweeting COVID-19
posts; and “@ttocs35” is now suspended (Fig. 2d). These non-credible bots were
actually retweeted as many times as those of human accounts and, thus, were as
equally influential as humans. We confirmed that some outlier bots were
actually suspended by Twitter and are no longer active. These results indicate
that, although the number of retweets by non-credible humans could be larger
than the number by non-credible bots, the effects of the latter are still non-
negligible because of the existence of the outliers as well as the parasitic
nature of non-credible bots, which we will discuss later.

Overall, two segregated networks of information-spreading emerged in all the
topics considered. It turns out that one side of the dense connected
components was propagating credible sources mostly by credible bots. By
contrast, the other component was diffusing non-credible information, mainly
amplified by non-credible bots.

###  3.2 Parasitic natures of bots

Next, we examined how bots selectively amplified voices from media and
celebrity accounts in the aggregated retweet networks mentioned in Sec. 2.3.
Fig. 3 shows the results by bot type. The top-five most-retweeted accounts
were plotted alongside.

In Fig. 3(a), we can clearly see that non-credible bots are parasitic on far-
right mass media accounts such as “@DailyCaller”($k_{in}=84$) in terms of
their retweets, although the indegrees of the other four are much smaller.
Note that “@DailyPostNGR”is a Nigerian local newspaper; “@michaeljohns” is an
American conservative commentator, speechwriter, and policy analyst; also,
“@nascarred14” is now suspended.

Looking at the accounts retweeted by both credible and non-credible bots in
Fig.3(b), it turns out that most non-credible bots are also parasitic on
right-leaning celebrity and media accounts, including “@DonaldJTrumpJr”
($k_{in}=61$). His most popular article is reposted from the “DailyCaller”
website, saying that “Chinese Government Finally Acknowledges Underreporting
Coronavirus Cases”. Other accounts include “@TrumpWarRoom” (“The office of
Donald J Trump”) 555https://https://donaldjtrump.com, “@seanhannity”(an
American talk show program presenter and conservative political commentator.)
and “@yogagenie” (suspended). By contrast, most credible bots are
preferentially amplified voices from legitimate media accounts, such as
“@washingtonpost” ($k_{in}=205$). Other accounts include “@thehill”,
“@CNNPolitics” and “@Independent”. A marked exception is “@realDonaldTrump”
whose posts were largely shared not only by non-credible bots ($k_{in}=29$)
but also credible bots ($k_{in}=199$).

Fig.3(c) shows that major celebrities and media were selectively retweeted
only by credible bots, regardless of their political attitude. For instance,
“@Kimstrassel” (American conservative writer, $k_{in}=155$) and
“@HillaryClinton” (US politician, $k_{in}=267$). The other accounts include
“@guardian” (British newspaper), “@JoeBiden” and “@BillKristol” (American
neoconservative political analyst).

These results further support that non-credible bots cannot be ignored in the
context of the COVID-19 infodemic due to their parasitic natures toward
popular right-leaning accounts.

(a)

(b)

(c)

Figure 3: Celebrities and media accounts retweeted by (a) non-credible bots
only; (b) both credible and non-credible bots; (c) credible bots only. Green
denotes celebrity and orange denotes media. Accounts with black labels were
all suspended by Twitter (as of Aug. 7, 2021). The size of a node is
proportional to its indegee.

###  3.3 Temporal patterns of retweets in humans and bots

We assumed that non-credible bots’ behaviors are correlated with non-credible
humans rather than credible accounts, because the intention of non-credible
bots would be to amplify the spread of misinformation including conspiracy
theories. Thus, we quantified temporal patterns of retweet behaviors in humans
and bots. For comparisons among credible/non-credible humans and bots, we
scaled daily retweet counts between 0 and 1, respectively. Fig. 4 shows daily
retweet series by humans and bots for each topic, in which the patterns of
retweets increase following similar trends.

Figure 4: Retweet count series (scaled 0-1) generated for bots and humans in
four topics.

To statistically confirm this observation, we measured the Pearson correlation
coefficient of temporal oscillations of retweets generated by these users. The
results are summarized in Table 2. This reveals that retweet actions by non-
credible bots correlated with those by non-credible humans to a much higher
degree than by credible humans in all of the topics. The above assumption is
therefore partially supported. We further consider this assumption in the next
section by looking at commonality in retweets generated by bots and humans.

Table 2: Pearson correlation coefficients of retweets between NB and NH and between NB and CH. Topic | Type | Correlation Coefficient | P-value
---|---|---|---
WHO | NB & NH | 0.84 | 6.05E-29
NB & CH | 0.15 | 0.1250527
Trump | NB & NH | 0.96 | 4.65E-61
NB & CH | 0.82 | 1.81E-26
5G | NB & NH | 0.45 | 1.31E-06
NB & CH | 0.32 | 0.001031208
Bill Gates | NB & NH | 0.91 | 1.46E-41
NB & CH | 0.04 | 0.712220673

###  3.4 Commonality in retweets by humans and bots

Finally, we examined terms (nouns), domains (URLs), and users that commonly
appeared in retweets generated by humans and bots. Consider the 5G topic, for
example; Fig. 5 compares term importance (measured by TF-IDF) on the
5G-related articles retweeted by bots and humans. As seen in the red bars of
Fig. 5 (a) and (b), the non-credible bots selectively retweeted articles that
include China-related terms such as “wuhan”, “china”, “chinese”, and the non-
credible humans had the same tendency. Sample articles include the following:

_“The China Coronavirus 5G Connection is a very important factor when trying
to comprehend the coronavirus (formerly abbreviated 2019-nCoV, now COVID-19)
outbreak.”_ 666https://worldtruth.tv

Such articles aimed to connect “china” with coronavirus subjectively and
highlighted that Wuhan was a test city chosen for China’s 5G rollout. By
contrast, the blue bars in Fig. 5 (a) and (b) show that the credible bots and
humans retweeted articles that include the word “conspiracy.” This suggests
that both credible bots and humans tended to retweet articles designed to warn
the readers about the conspiracy theory. Sample articles include:

_“One theory claims that the novel coronavirus originated in Wuhan because the
Chinese city had recently been rolling out 5G. It’s now supposedly spread to
other cities that are also using 5G. These false conspiracy theories neglect
to mention that a highly contagious virus would naturally spread more in
densely populated cities with access to 5G, and that the coronavirus pandemic
has hit countries like Iran and Japan where 5G isn’t in use yet.”_
777https://www.theverge.com

Thus, the 5G conspiracy theory went viral inside of an echo chamber consisting
of non-credible accounts while shutting down contrasting voices from a
credible cluster.

The same linguistic analysis was conducted for the “WHO”, “Trump”, and “Bill
Gates” topics, and resulted in the similar linguistic features (see Figs. 2a-f
in SI). That is, the articles including China-related terms were
preferentially shared by both non-credible bots and humans in the Trump topic
and others. These consistent patterns suggest that non-credible humans were
diffusing China-related mis/disinformation and conspiracy theories, amplified
by non-credible bots with political inclinations.

We also calculated the shared terms ratio in each topic. The non-credible bots
and humans shared 57%, 90%, 50%, and 30% of terms in the retweeted articles
for “WHO”, “Trump”, “5G”, and “Bill Gates” topics, respectively (cf. credible
bots and humans shared 73%, 93%, 70%, and 40% of terms, respectively). This
indicates that the non-credible bots and humans are topically similar, and
therefore, one may not be able to easily distinguish humans or bots based on
the observation of a few tweets. These results, again, suggest that the non-
credible humans were diffusing China-related misinformation and misleading
claims, and that the non-credible bots were probably amplifying their effects.
We will discuss the implications of this later.

(a)

(b)

Figure 5: Term importance in retweeted articles in the “5G” topic: (a) non-
credible bots vs. credible bots; (b) non-credible humans vs. credible humans.
Red bars indicate word importance for non-credible users, whereas blue bars
indicate credible users. Table 3: Popular retweeted domains and users in the
5G topic. (a) Top 10 domains retweeted by credible humans (CH), credible bots
(CB), non-credible humans (NH), and non-credible bots (NB). (b) Top 10 users
retweeted by CH, CB, NH, and NB. Green denotes credible domains or users; red
denotes non-credible domains or users; blue denotes unknown domains or users.

(a) Top 10 domains retweeted by | NB | CB | NH | CH
---|---|---|---|---
1 | beforeitsnews.com | bbc.co.uk | worldtruth.tv | theguardian.com
2 | dailypost.ng | theguardian.com | express.co.uk | bbc.co.uk
3 | worldtruth.tv | bbc.com | infowars.com | bbc.com
4 | zerohedge.com | reuters.com | beforeitsnews.com | theverge.com
5 | dailyrecord.co.uk | theverge.com | humansarefree.com | businessinsider.com
6 | today.ng | cnn.com | neonnettle.com | cnn.com
7 | infowars.com | businessinsider.com | thelastamericanvagabond.com | reuters.com
8 | banned.video | nytimes.com | dailycaller.com | ft.com
9 | thetruthaboutcancer.com | newsweek.com | paulcraigroberts.org | vox.com
10 | rt.com | vox.com | thesun.co.uk | nytimes.com

(b) Top 10 users retweeted by | NB | CB | NH | CH
---|---|---|---|---
1 | @shinethelight17 | @Reuters | @WorldTruthTV | @guardian
2 | @DailyPostNGR |  @guidaautonoma | @BILDERBERG_GP | @rooshv
3 | @Laurel700 | @Exchange5g | @davidicke | @guardiannews
4 | @davidicke | @rooshv | @shinethelight17 | @verge
5 | @freezerohedge | @HaroldSinnott | @TornadoNewsLink | @Omojuwa
6 | @NigeriaNewsdesk | @verge | @boblister_poole | @Reuters
7 | @BANNEDdotVIDEO |  @ipfconline1 | @DailyPostNGR | @davidicke
8 | @WorldTruthTV | @Shirastweet | @BANNEDdotVIDEO | @Exchange5g
9 | @owhy3 | @nuskitconsultan | @davidkurten | @davidkurten
10 |  | @guardian | @buttscornershop | @ruskin147

Table 4: Common domains and users retweeted by non-credible users (NB and NH)
and by credible users (CB and CH)

(a) Domains Topic | NB $\cap$ NH | CB $\cap$ CH
---|---|---
WHO | 6 | 8
Trump | 10 | 8
5G | 3 | 9
Bill Gates | 6 | 8

(b) Users Topic | NB $\cap$ NH | CB $\cap$ CH
---|---|---
WHO | 8 | 6
Trump | 7 | 7
5G | 5 | 4
Bill Gates | 7 | 7

Furthermore, we found that both non-credible bots and humans exhibit high
commonality in retweeted domains (URLs) and retweeted users. Again, take the
5G topic as an example, the non-credible bots and humans shared many popular
domains as well as users (Table 3). The same analyses were also conducted for
other topics. For each user type, Table 4 summarizes the domains and users
that commonly appeared in retweets in all the topics. The non-credible bots
shared many in common with respect to the top-10 retweeted domains and users,
indicating the similarity in retweet behaviors between non-credible bots and
humans (as well as credible bots and humans). These findings further support
the assumption that non-credible bots were following non-credible humans
rather than credible accounts.

##  4 Discussion

In this paper, we investigated the roles of bots by analyzing retweet
networks, temporal patterns of retweets as well as retweeted contents during
the COVID-19 infodemic. For analysis, we focused on misinformation and
conspiracy-theory-related topics, such as “WHO”, “Trump”, “5G”, and “Bill
Gates”. Our analyses provided two major findings: the segregated retweet
networks of bots and their temporal and topical similarities. We revisit the
findings and discuss their implications here.

First, we found that the retweet networks exhibited segregation and parasitic
natures, suggesting two types of voices or echo chambers in all the topics
(Fig. 1). One represents mainstream media and public institutions and the
other represents right-wing (self-)media and celebrities. The echo chamber
structures may amplify mis/disinformation from non-credible sources while
hindering the dissemination of information from legitimate sources. According
to the indegrees (Fig. 2), the basic influence of retweets by non-credible
humans can be much larger than that by non-credible bots. Thus, one might
think that bots did not play as important a role during the COVID-19 infodemic
as they did in previous political events, including the 2016 US presidential
election. However, we cannot simply make this a definitive conclusion. Rather,
the clustering of non-credible bots may reflect a partisan asymmetry and that
non-credible bots follow non-credible humans. In particular, non-credible bots
were parasitic on the far-right media and celebrity accounts, selectively
amplifying their voices (Fig. 3). We speculate that non-credible bots targeted
at such influential accounts because they needed to get amplification from
prestigious accounts with social capital and high follower accounts and,
thereby, extend their reach and engagement. The parasitic role of non-credible
bots is for them to interact with accounts that might give them a chance to
have their narratives or views amplified into the public sphere.

Second, our results show that retweet actions and contents were highly
correlated between non-credible bots and humans for each topic (Figs. 4 and
5). We found that non-credible bots preferentially distributed China-related
terms (e.g., “wuhan”,“china”, “chinese”) from non-credible sites and users in
the 5G topic (Fig. 5 and Table 3) and the other topics (Figs. 2a-f in SI).
These consistent patterns may reflect that many (although not all) non-
credible humans were diffusing China-related misleading claims and the non-
credible bots were amplifying their effects. It is said that Trump’s
fabrication of the term “China virus” led to racism targeted at Chinese and
Asian Americans further fueled Asian hate and Sinophobia (anti-Chinese
sentiment) in Western countries. Therefore, we suspect that bots were abused
for spreading such propaganda during the early COVID-19 infodemic. Although
Twitter suspended many malicious accounts during the COVID-19 infodemic,
sophisticated bots remained active and selectively parasitic on the partisan
clusters. Therefore, we conclude that the role of non-credible bots is non-
negligible in the COVID-19 infodemic.

This evidence and these considerations cause us to rethink the weaponized use
of bots for online manipulation and political interventions. This calls for
the necessity of continuously monitoring the information ecosystem of bots.
This is especially important to detect their coordinated acts, although we did
not find evidence of such events in the current settings. However, this could
still represent a credible future threat that would likely have a negative
societal impact. As the WHO mentioned, an infodemic is a “second disease” that
emerged alongside with COVID-19, and it is important to take immediate action
to address this infodemic. As demonstrated in this study, social media
analysis is important in order to acquire an overview of the infodemic and
gain insights into a mitigation strategy.

Our study has several limitations that need to be addressed in future
research. Since Twitter suspends any accounts that it considers as
“malicious”, we were unable to obtain a comprehensive picture of users’
interactive behaviors in this study. We also had limited information about the
sources of credible and non-credible domains (URLs), which require frequent
updates; thus, all of the URLs could not be labelled in our analyses. The
availability problem of a credible/non-credible domain list requires a
collective effort to solve. Despite these limitations, this study furthers our
understanding of the roles of bots in misinformation propagation during an
infodemic in the midst of a world-wide healthcare crisis, and re-emphasizes
the need to develop an efficient method to address behaviors of malicious
bots.

## Conflict of interest

The authors declare that they have no conflict of interest.

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# Ricci–Bourguignon Almost Solitons with Special Potential on Sasaki-like
Almost Contact Complex Riemannian Manifolds

Mancho Manev 1,2 1 Department of Algebra and Geometry; Faculty of Mathematics
and Informatics; University of Plovdiv Paisii Hilendarski; 24 Tzar Asen St;
4000 Plovdiv, Bulgaria<EMAIL_ADDRESS>2 Department of Medical Physics
and Biophysics; Faculty of Pharmacy; Medical University of Plovdiv; 15A Vasil
Aprilov Blvd; 4002 Plovdiv, Bulgaria<EMAIL_ADDRESS>
###### Abstract.

Almost contact complex Riemannian manifolds, known also as almost contact
B-metric manifolds, are equipped with a pair of pseudo-Riemannian metrics that
are mutually associated to each other using the tensor structure. Here we
consider a special class of these manifolds, those of the Sasaki-like type.
They have an interesting geometric interpretation: the complex cone of such a
manifold is a holomorphic complex Riemannian manifold (also called a
Kähler–Norden manifold). The basic metric on the considered manifold is
specialized here as a soliton, i. e. has an additional curvature property such
that the metric is a self-similar solution of an intrinsic geometric flow.
Almost solitons are more general objects than solitons because they use
functions rather than constants as coefficients in the defining condition. A
$\beta$-Ricci-Bourguignon-like almost soliton ($\beta$ is a real constant) is
defined using the pair of metrics. The introduced soliton is a generalization
of some well-known (almost) solitons (such as those of Ricci, Schouten,
Einstein), which in principle arise from a single metric rather than a pair of
metrics. The soliton potential is chosen to be pointwise collinear to the Reeb
vector field, or the Lie derivative of any B-metric along the potential to be
the same metric multiplied by a function. The resulting manifolds equipped
with the introduced almost solitons are characterized geometrically.
Appropriate examples for two types of almost solitons are constructed and the
properties obtained in the theoretical part are confirmed.

###### Key words and phrases:

Ricci–Bourguignon; almost contact B-metric manifold; almost contact complex
Riemannian manifold; Sasaki-like manifold; vertical potential; conformal
potential

###### 2010 Mathematics Subject Classification:

53C25; 53D15; 53C50; 53C44; 53D35; 70G45

##  1\. Introduction

The notion of _Ricci-Bourguignon flow_ was introduced by J. P. Bourguignon in
1981 [1]. A time-dependent family of (pseudo-)Riemannian metrics $g(t)$
considered on a smooth manifold $\mathcal{M}$ is said to evolve through Ricci-
Bourguignon flow if $g(t)$ satisfies the following evolution equation

$\frac{\partial}{\partial t}g=-2\left(\rho-\beta\tau g\right),\qquad
g(0)=g_{0},$

where $\beta$ is a real constant, $\rho(t)$ and $\tau(t)$ are the Ricci tensor
and the scalar curvature regarding $g(t)$, respectively.

This flow is an intrinsic geometric flow on $\mathcal{M}$, whose fixed points
or self-similar solutions are its solitons. The _Ricci-Bourguignon soliton_
(in short RB soliton) is described by the following equation [2, 3]

(1) $\rho+\frac{1}{2}\mathcal{L}_{\vartheta}g+(\lambda+\beta\tau)g=0,$

where $\mathcal{L}_{\vartheta}g$ denotes the Lie derivative of $g$ along the
vector field $\vartheta$ called the soliton potential, and $\lambda$ is the
soliton constant. Briefly, we denote this soliton by
$(g,\beta;\vartheta,\lambda)$. In the case that $\lambda$ is a differential
function on $\mathcal{M}$, the solution is called a _RB almost soliton_ [3]. A
RB soliton is called expanding if $\lambda>0$, steady if $\lambda=0$ and
shrinking if $\lambda<0$. In case that the soliton potential $\vartheta$ is a
Killing vector field, i.e. $\mathcal{L}_{\vartheta}g=0$, the RB soliton is
called trivial.

This family of geometric flows contains, the famous Ricci flow for $\beta=0$,
the Einstein flow for $\beta=\frac{1}{2}$, the traceless Ricci flow for
$\beta=\frac{1}{m}$ and the Schouten flow for $\beta=\frac{1}{2(m-1)}$, where
$m$ is the dimension of the manifold [5, 4].

For this reason, we consider it more correct to say _$\beta$ -RB solitons_ and
_$\beta$ -RB almost solitons_, respectively.

Other recent studies on Ricci-Bourguignon solitons have been done in [6, 7, 8,
9, 10].

##  2\. accR manifolds

Let us consider a $(2n+1)$-dimensional smooth manifold $\mathcal{M}$, equipped
with an almost contact structure $(\varphi,\xi,\eta)$ and a B-metric $g$. It
is called an _almost contact B-metric manifold_ or _almost contact complex
Riemannian_ (abbr.  _accR_) _manifold_ and it is denoted by
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$.
In detail, $\varphi$ is an endomorphism of the tangent bundle $T\mathcal{M}$,
$\xi$ is a Reeb vector field, $\eta$ is its dual contact 1-form and $g$ is a
pseudo-Riemannian metric of signature $(n+1,n)$ such that

(2)
$\begin{array}[]{c}\varphi\xi=0,\qquad\varphi^{2}=-\iota+\eta\otimes\xi,\qquad\eta\circ\varphi=0,\qquad\eta(\xi)=1,\\\\[4.0pt]
g(\varphi x,\varphi y)=-g(x,y)+\eta(x)\eta(y),\end{array}$

where $\iota$ denotes the identity on $\Gamma(T\mathcal{M})$ [11].

In the latter equality and further, $x$, $y$, $z$ will stand for arbitrary
elements of $\Gamma(T\mathcal{M})$ or vectors in the tangent space
$T_{p}\mathcal{M}$ of $\mathcal{M}$ at an arbitrary point $p$ in
$\mathcal{M}$.

The following equations are immediate consequences of (2)

(3) $g(\varphi x,y)=g(x,\varphi y),\qquad g(x,\xi)=\eta(x),\qquad
g(\xi,\xi)=1,\qquad\eta(\nabla_{x}\xi)=0,$

where $\nabla$ denotes the Levi-Civita connection of $g$.

The associated metric $\tilde{g}$ of $g$ on $\mathcal{M}$ is also a B-metric
and it is defined by

(4) $\tilde{g}(x,y)=g(x,\varphi y)+\eta(x)\eta(y).$

The Ganchev–Mihova–Gribachev classification of the investigated manifolds,
given in [11], consists of eleven basic classes $\mathcal{F}_{i}$,
$i\in\\{1,2,\dots,11\\}$, determined by conditions for the (0,3)-tensor $F$
defined by

(5) $F(x,y,z)=g\bigl{(}\left(\nabla_{x}\varphi\right)y,z\bigr{)}.$

It has the following basic properties:

(6) $\displaystyle F(x,y,z)=F(x,z,y)=F(x,\varphi y,\varphi
z)+\eta(y)F(x,\xi,z)+\eta(z)F(x,y,\xi),$ (7) $\displaystyle F(x,\varphi
y,\xi)=(\nabla_{x}\eta)y=g(\nabla_{x}\xi,y).$

###  2.1. Sasaki-like accR manifolds

An interesting class of accR manifolds was introduced in [12] by the condition
that the complex cone of such a manifold is a Kähler-Norden manifold. They are
called _Sasaki-like manifolds_ and are defined by the condition

(8) $F(x,y,z)=g(\varphi x,\varphi y)\eta(z)+g(\varphi x,\varphi z)\eta(y).$

The class of Sasaki-like manifolds is contained in the basic class
$\mathcal{F}_{4}$ of the Ganchev–Mihova–Gribachev classification, not
intersecting with the special class $\mathcal{F}_{0}$ of cosymplectic accR
manifolds defined by $F=0$.

Moreover, the following identities are valid for this type of accR manifolds
[12]

(9) $\nabla_{x}\xi=-\varphi
x,\qquad\rho(x,\xi)=2n\,\eta(x),\qquad\rho(\xi,\xi)=2n,$

where $\rho$ stands for the Ricci tensor for $g$.

Let $\tau$ and $\tilde{\tau}$ be the scalar curvatures with respect to $g$ and
$\tilde{g}$, respectively, and let $\tau^{*}$ be the associated quantity of
$\tau$ regarding $\varphi$, defined by $\tau^{*}=g^{ij}\rho(e_{i},\varphi
e_{j})$. Then, for a Sasaki-like manifold we have

(10) $\tilde{\tau}=-\tau^{*}+2n.$

###  2.2. Einstein-like accR manifolds

In [13], it is introduce the following notion. An accR manifold
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is said to be _Einstein-like_ if its Ricci tensor $\rho$ satisfies

(11) $\begin{array}[]{l}\rho=a\,g+b\,\tilde{g}+c\,\eta\otimes\eta\end{array}$

for some triplet of constants $(a,b,c)$. In particular, when $b=0$ and
$b=c=0$, the manifold is called an _$\eta$ -Einstein manifold_ and an
_Einstein manifold_ , respectively. If $a$, $b$, $c$ in (11) are functions on
$\mathcal{M}$, then the manifold is called _almost Einstein-like_ , _almost
$\eta$-Einstein_ and _almost Einstein_ , respectively [14].

Consequences of (11) are the following $\tau=(2n+1)a+b+c$ and $\tau^{*}=-2nb$.

##  3\. $\beta$-RB almost solitons

A generalization of the known RB soliton on a manifold with an additional
1-form $\eta$ is an _$\eta$ -Ricci-Bourguignon soliton_ defined following (1)
by

(12)
$\rho+\frac{1}{2}\mathcal{L}_{\vartheta}g+(\lambda+\beta\tau)g+\mu\eta\otimes\eta=0,$

where $\mu$ is also a constant [6]. Obviously, an $\eta$-Ricci-Bourguignon
soliton with $\mu=0$ is a RB soliton. Again, in the case where $\lambda$ and
$\mu$ are functions on the manifold, almost solitons of the corresponding kind
are said to be given.

In the present paper, we study an accR manifold. Having two B-metrics $g$ and
$\tilde{g}$ related to each other with respect to the structure of such a
manifold gives us reason to introduce a more natural generalization of the
$\beta$-RB soliton than (12). In addition, we also have the structure 1-form
$\eta$ so that $\eta\otimes\eta$ is included in both B-metrics $g$ and
$\tilde{g}$ as their restriction on the vertical distribution
$\operatorname{span}(\xi)$.

######  Definition 3.1.

An accR manifold
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is called a _$\beta$ -Ricci-Bourguignon-like soliton_ (in short _$\beta$ -RB-
like soliton_) with potential vector field $\vartheta$ if its Ricci tensor
$\rho$ satisfies the following condition for a pair of constants
$(\lambda,\tilde{\lambda})$

(13)
$\rho+\frac{1}{2}\mathcal{L}_{\vartheta}g+\frac{1}{2}\mathcal{L}_{\vartheta}\tilde{g}+(\lambda+\beta\tau)g+(\tilde{\lambda}+\beta\tilde{\tau})\tilde{g}=0,$

where $\tilde{\tau}$ is the scalar curvature of the manifold with respect to
$\tilde{g}$ and the corresponding Levi-Civita connection $\tilde{\nabla}$. If
$(\lambda,\tilde{\lambda})$ is a pair of functions on $\mathcal{M}$ satisfying
(13), then
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is called _$\beta$ -Ricci-Bourguignon-like almost soliton_ (in short _$\beta$
-RB-like almost soliton_).

By taking the trace in (13) with respect to $g$, we get

$[1+(2n+1)\beta]\tau+\beta\tilde{\tau}+\operatorname{div}_{g}\vartheta+\frac{1}{2}\operatorname{tr}_{g}\left(\mathcal{L}_{\vartheta}\tilde{g}\right)+(2n+1)\lambda+\tilde{\lambda}=0$

by means of the formula
$\operatorname{div}_{g}\vartheta=\frac{1}{2}\operatorname{tr}_{g}\left(\mathcal{L}_{\vartheta}g\right)=g^{ij}g\left(\nabla_{e_{i}}\vartheta,e_{j}\right)$.
The expression of the trace in the above equality is the following

$\frac{1}{2}\operatorname{tr}_{g}\left(\mathcal{L}_{\vartheta}\tilde{g}\right)=g^{ij}\tilde{g}\left(\tilde{\nabla}_{e_{i}}\vartheta,e_{j}\right).$

###  3.1. The potential is conformal vector field

Recall that a vector field, e.g. the potential $\vartheta$, on $\mathcal{M}$
is called a _conformal vector field with respect to $g$_ if there exists a
function $\psi$ on $\mathcal{M}$ such that [3]

$\mathcal{L}_{\vartheta}g=2\psi g.$

The conformal vector field is nontrivial if $\psi\neq 0$. If $\psi=0$, then
$\vartheta$ is called a _Killing vector field_ with respect to $g$.

Similarly, $\vartheta$ is called a _conformal vector field with respect to
$\tilde{g}$_ if there exists a function $\tilde{\psi}$ on $\mathcal{M}$ such
that

$\mathcal{L}_{\vartheta}\tilde{g}=2\tilde{\psi}\tilde{g}.$

Depending on whether $\tilde{\psi}$ is nonzero or zero, we have a vector field
$\vartheta$ that is nontrivial scalar or Killing, respectively.

######  Theorem 3.2.

Let
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
be a $(2n+1)$-dimensional Sasaki-like accR manifold that is a $\beta$-RB-like
almost soliton with a pair of soliton functions $(\lambda,\tilde{\lambda})$
and conformal potential vector field $\vartheta$ with potential functions
$\psi$ and $\tilde{\psi}$ respect to $g$ and $\tilde{g}$, respectively. Then
the manifold is Einstein-like and has the following Ricci tensor:

(14)
$\rho=\left(\frac{\tau}{2n}-1\right)g+\left(\frac{\tilde{\tau}}{2n}-1\right)\tilde{g}$

and the following property is valid

(15) $\tau+\tilde{\tau}=4n(n+1).$

If $\beta\neq-\frac{1}{2n}$, then the scalar curvatures with respect to $g$
and $\tilde{g}$ can be expressed separately as

(16)
$\tau=-\frac{2n}{1+2n\beta}(\psi+\lambda-1),\qquad\tilde{\tau}=-\frac{2n}{1+2n\beta}(\tilde{\psi}+\tilde{\lambda}-1)$

and the following condition for the used functions is valid

(17) $\psi+\lambda+\tilde{\psi}+\tilde{\lambda}+2n[1+2(n+1)\beta]=0.$

If $\beta=-\frac{1}{2n}$, then the following properties hold

(18) $\psi+\lambda=1,\qquad\tilde{\psi}+\tilde{\lambda}=1.$

###### Proof.

Under these circumstances regarding the considered manifold, due to (13) its
Ricci tensor for $g$ has the following form

(19)
$\rho=-(\psi+\lambda+\beta\tau)g-(\tilde{\psi}+\tilde{\lambda}+\beta\tilde{\tau})\tilde{g}.$

Taking the trace of the last expression, we obtain that the scalar curvatures
of
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
with respect to $g$ and $\tilde{g}$ are related as follows

(20)
$[1+(2n+1)\beta]\tau+\beta\tilde{\tau}+(2n+1)(\psi+\lambda)+\tilde{\psi}+\tilde{\lambda}=0.$

Using (19), we take the appropriate trace to obtain $\tau^{*}$ on the left-
hand side and the resulting relation to $\tilde{\tau}$ is

(21) $\tau^{*}=2n(\tilde{\psi}+\tilde{\lambda}+\beta\tilde{\tau}).$

By virtue of (10) for a Sasaki-like accR manifold and (21), we obtain the
expression of the scalar curvature with respect to $\tilde{g}$ as follows

(22) $\tilde{\tau}=-2n\frac{\tilde{\psi}+\tilde{\lambda}-1}{1+2n\beta}.$

The last formula is true for the case $\beta\neq-\frac{1}{2n}$.

Combining (20) and (22), we get the following form of the scalar curvature
with respect to $g$

(23)
$\tau=-\frac{1}{1+(2n+1)\beta}\left\\{(2n+1)(\psi+\lambda)+1+\frac{\tilde{\psi}+\tilde{\lambda}-1}{1+2n\beta}\right\\},$

where $\beta\neq-\frac{1}{2n+1}$. Otherwise, for $\beta=-\frac{1}{2n+1}$ the
identity in (17) is valid for this value of $\beta$.

On the other hand, a consequence of (19) and the value of $\rho(\xi,\xi)$ from
(9) for the Sasaki-like case gives the following relation

(24)
$\beta\tau+\beta\tilde{\tau}+\psi+\lambda+\tilde{\psi}+\tilde{\lambda}+2n=0.$

Let us check what follows in the particular case $\beta=0$. Then, the last
relation implies

$\psi+\lambda+\tilde{\psi}+\tilde{\lambda}+2n=0,$

which we apply in (23) and (22) to specialize them in the following form

(25)
$\tau=-2n(\psi+\lambda-1),\qquad\tilde{\tau}=-2n(\tilde{\psi}+\tilde{\lambda}-1).$

Therefore, the identity in (15) holds, as do the formulas in (16). Thus, we
find that no different results are obtained for $\beta=0$ compared to the case
$\beta\neq-\frac{1}{2n}$.

Let us return to (24) and the solution of the system of equations in (24) and
(20) with respect to $\tau$ and $\tilde{\tau}$ for $\beta\neq 0$ and
$\beta\neq-\frac{1}{2n}$ gives us

(26)
$\tau=-2n\frac{\psi+\lambda-1}{1+2n\beta},\qquad\tilde{\tau}=-\frac{1}{\beta}\left\\{\frac{\psi+\lambda-1}{1+2n\beta}+\tilde{\psi}+\tilde{\lambda}+2n+1\right\\}.$

After that comparing the equalities for $\tau$ in (23) and (26), we obtain
(17). The same relation results from a similar comparison for $\tilde{\tau}$
in (22) and (26). The identity in (17) is a generalization of the
corresponding result in the $\beta=0$ case. In this way, we obtain the
formulas in (16).

Due to (17) the sum of $\tau$ and $\tilde{\tau}$ is a constant that depends
only on the dimension of the manifold as in (15).

For the case $\beta=-\frac{1}{2n}$, the equalities in (15) and (17) imply the
relations in (18). Therefore, (20) and (24) take the form in (15).

In conclusion, using (19) and (16), we obtain the Ricci tensor expression in
(14), which shows that the manifold is Einstein-like since (11) is satisfied
for $a=\frac{\tau}{2n}-1$, $b=\frac{\tilde{\tau}}{2n}-1$ and $c=0$. ∎

####  3.1.1. Example of a $\beta$-RB almost soliton with a conformal potential

As in [12], let us consider a Sasaki-like accR manifold of arbitrary dimension
with an Einstein metric $\bar{g}$. The image of this manifold by a contact
homothetic transformation of the metric given by
$g=p\bar{g}+q\tilde{\bar{g}}+(1-p-q)\eta\otimes\eta$ for $p,q\in\mathbb{R}$,
$(p,q)\neq(0,0)$, is also a Sasaki-like accR manifold. The corresponding Ricci
tensor has the form

$\rho=\frac{2n}{p^{2}+q^{2}}\left\\{pg-q\tilde{g}+\left(p^{2}+q^{2}-p\right)\eta\otimes\eta\right\\}.$

Now we calculate the scalar curvatures with respect to B-metrics $g$ and
$\tilde{g}$ as follows

(27)
$\tau=2n\left\\{1+\frac{2np}{p^{2}+q^{2}}\right\\},\qquad\tilde{\tau}=2n\left\\{1-\frac{2nq}{p^{2}+q^{2}}\right\\}.$

Let $\vartheta$ be a conformal vector field with respect to both B-metrics $g$
and $\tilde{g}$ with functions $\psi$ and $\tilde{\psi}$, respectively. We
construct a $\beta$-RB-like almost soliton on the transformed manifold with
potential $\vartheta$ and a pair of functions $(\lambda,\tilde{\lambda})$
satisfying (13).

By virtue of (15) and (27), we obtain the following condition
$p^{2}+q^{2}-p+q=0$, which has a solution $p=\frac{1}{2}(1+\sqrt{2}\cos{t})$,
$q=-\frac{1}{2}(1-\sqrt{2}\sin{t})$ for $t\in\mathbb{R}$. Then the scalar
curvatures from (27) specialize into the following form for
$t\neq(8l+3)\frac{\pi}{4}$, $l\in\mathbb{Z}$

(28)
$\begin{array}[]{l}\tau=2n\dfrac{(n+1)\sqrt{2}+(2n+1)\cos{t}-\sin{t}}{\sqrt{2}+\cos{t}-\sin{t}},\\\
\tilde{\tau}=2n\dfrac{(n+1)\sqrt{2}+\cos{t}-(2n+1)\sin{t}}{\sqrt{2}+\cos{t}-\sin{t}}.\end{array}$

Then we determine the functions $\psi$, $\tilde{\psi}$, $\lambda$,
$\tilde{\lambda}$ in the case $\beta\neq-\frac{1}{2n}$ as follows

(29)
$\begin{array}[]{l}\psi+\lambda=1-(1+2n\beta)\dfrac{(n+1)\sqrt{2}+(2n+1)\cos{t}-\sin{t}}{\sqrt{2}+\cos{t}-\sin{t}},\\\
\tilde{\psi}+\tilde{\lambda}=1-(1+2n\beta)\dfrac{(n+1)\sqrt{2}+\cos{t}-(2n+1)\sin{t}}{\sqrt{2}+\cos{t}-\sin{t}}.\end{array}$

In the case $\beta=-\frac{1}{2n}$, the equalities in (29) obviously reduce to
those in (18).

We directly verify that (28) and (29) satisfy the expressions in (16) and
(17).

In conclusion, we found that the constructed manifold satisfies the conditions
of Theorem 3.2.

###  3.2. The potential is vertical vector field

Suppose that
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is a Sasaki-like accR manifold admitting a $\beta$-RB-like almost soliton
whose potential vector field $\vartheta$ is pointwise collinear with $\xi$,
i.e. $\vartheta=k\xi$, where $k$ is a differentiable function on
$\mathcal{M}$. It is clear that $k=\eta(\vartheta)$ and therefore $\vartheta$
belongs to the vertical distribution $H^{\bot}=\operatorname{span}(\xi)$,
which is orthogonal to the contact distribution $H=\ker(\eta)$ with respect to
both $g$ and $\tilde{g}$.

According to [15], we have the expression
$\left(\mathcal{L}_{\vartheta}g\right)(x,y)=h(x,y)-2kg(x,\varphi y)$ for the
vertical potential $\vartheta$, where the first equality of (9) is used and
the symmetric tensor $h(x,y)=\mathrm{d}{k}(x)\eta(y)+\mathrm{d}{k}(y)\eta(x)$
is denoted for the sake of brevity. Then, by virtue of (4) we get

(30) $\mathcal{L}_{\vartheta}g=h-2k\left(\tilde{g}-\eta\otimes\eta\right).$

Similarly, since for a Sasaki-like accR manifold
$\tilde{\nabla}_{x}\xi=-\varphi x$ is true [16], we have
$\left(\mathcal{L}_{\vartheta}\tilde{g}\right)(x,y)=h(x,y)-2kg(\varphi
x,\varphi y)$ and considering the last equality of (2), the last expression
takes the following form

(31) $\mathcal{L}_{\vartheta}\tilde{g}=h+2k\left(g-\eta\otimes\eta\right).$

######  Theorem 3.3.

Let
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
be a $(2n+1)$-dimensional Sasaki-like accR manifold that is a $\beta$-RB-like
almost soliton with a pair of soliton functions $(\lambda,\tilde{\lambda})$
and vertical potential vector field $\vartheta$ with potential function $k$.
Then the manifold is Einstein-like and has the following Ricci tensor

(32)
$\rho=\left(\frac{\tau}{2n}-1\right)g+\left(\frac{\tilde{\tau}}{2n}-1\right)\tilde{g}-\left\\{\frac{\tau+\tilde{\tau}}{2n}-2(n+1)\right\\}\eta\otimes\eta.$

In the case of $\beta\neq-\frac{1}{2n}$, the scalar curvatures with respect to
$g$ and $\tilde{g}$ are determined by:

(33)
$\tau=-\frac{2n}{1+2n\beta}(\lambda+k-1),\qquad\tilde{\tau}=-\frac{2n}{1+2n\beta}(\tilde{\lambda}-k-1).$

Moreover, the following condition for the used functions is valid

(34)
$\mathrm{d}{k}=\mathrm{d}{k}(\xi)\eta,\qquad\mathrm{d}{k}(\xi)=-\frac{\lambda+\tilde{\lambda}-2}{2(1+2n\beta)}-n-1.$

In the case of $\beta=-\frac{1}{2n}$, the two scalar curvatures satisfy the
following relation

(35) $\tau+\tilde{\tau}=4n\left\\{\mathrm{d}{k}(\xi)+n+1\right\\}$

and the soliton functions are expressed by $k$ as follows

(36) $\lambda=1-k,\qquad\tilde{\lambda}=1+k.$

###### Proof.

Replacing (30) and (31) in (13), we obtain

(37)
$\rho=-(\lambda+\beta\tau+k)g-(\tilde{\lambda}+\beta\tilde{\tau}-k)\tilde{g}-h.$

For a Sasaki-like accR manifold we know the expression of $\rho(x,\xi)$ from
(9). We then compare it with the corresponding consequence of (37) and obtain
$\mathrm{d}{k}=\mathrm{d}{k}(\xi)\eta$, where

(38)
$\mathrm{d}{k}(\xi)=-\frac{1}{2}\left\\{\lambda+\tilde{\lambda}+\beta(\tau+\tilde{\tau})+2n\right\\}.$

Therefore, $k$ is a horizontal constant and $h$ takes the following form

(39)
$h=-\left\\{\lambda+\tilde{\lambda}+\beta(\tau+\tilde{\tau})+2n\right\\}\eta\otimes\eta.$

Then, applying the last equality and (38) in (37), for the Ricci tensor $\rho$
we get

(40)
$\rho=-(\lambda+\beta\tau+k)g-(\tilde{\lambda}+\beta\tilde{\tau}-k)\tilde{g}+\left\\{\lambda+\tilde{\lambda}+\beta(\tau+\tilde{\tau})+2n\right\\}\eta\otimes\eta.$

Taking the appropriate traces in (40), we obtain for $\beta\neq-\frac{1}{2n}$
the first equality in (33) as well as

(41) $\tau^{*}=2n(\tilde{\lambda}+\beta\tilde{\tau}-k).$

Then, bearing in mind (10), the equality in (41) implies for
$\beta\neq-\frac{1}{2n}$ the second equality in (33). Substituting the
expressions of $\tau$ and $\tilde{\tau}$ from (33) into (40), we get (32).

As a consequence of (33) and (38), we express $\tau+\tilde{\tau}$ in two ways.
One is given in (35) and the other is as follows

(42)
$\tau+\tilde{\tau}=-\frac{2n}{1+2n\beta}\left(\lambda+\tilde{\lambda}-2\right).$

Comparing two expressions implies the relation in (34) between the used
functions and the constant $\beta$.

Comparing (40) with (11), it follows that
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is Einstein-like and the coefficients in (11) are the following

(43) $a=\frac{\tau}{2n}-1,\qquad b=\frac{\tilde{\tau}}{2n}-1,\qquad
c=-\frac{\tau+\tilde{\tau}}{2n}+2(n+1).$

In the particular case $\beta=-\frac{1}{2n}$, the dependencies in (36) hold
due to (33). Then, (40) is specialized as in (32). In addition, according to
(38), the equality in (35) also holds. Note that in this case the scalar
curvatures with respect to each of the B-metrics cannot be expressed
separately. ∎

####  3.2.1. Example of a $\beta$-RB almost soliton with a vertical potential

Let us consider an explicit example given as Example 2 in [12]. It concerns a
Sasaki-like accR manifold derived on a Lie group $G$ of dimension $5$, i.e.
for $n=2$, with a basis of left-invariant vector fields
$\\{e_{0},\dots,e_{4}\\}$. The corresponding Lie algebra is defined by the
commutators

(44)
$\begin{array}[]{lll}[e_{0},e_{1}]=pe_{2}+e_{3}+qe_{4},&[e_{0},e_{2}]=-pe_{1}-qe_{3}+e_{4},&\\\\[0.0pt]
[e_{0},e_{3}]=-e_{1}-qe_{2}+pe_{4},&[e_{0},e_{4}]=qe_{1}-e_{2}-pe_{3},&p,q\in\mathbb{R}.\end{array}$

The introduced accR structure is defined as follows

(45)
$\begin{array}[]{l}g(e_{0},e_{0})=g(e_{1},e_{1})=g(e_{2},e_{2})=-g(e_{3},e_{3})=-g(e_{4},e_{4})=1,\\\\[0.0pt]
g(e_{i},e_{j})=0,\quad i,j\in\\{0,1,\dots,4\\},\;i\neq j,\\\\[0.0pt]
\xi=e_{0},\quad\varphi e_{1}=e_{3},\quad\varphi e_{2}=e_{4},\quad\varphi
e_{3}=-e_{1},\quad\varphi e_{4}=-e_{2}.\end{array}$

Then, in [13], the components of the curvature tensor
$R_{ijkl}=R(e_{i},e_{j},e_{k},e_{l})$ and those of the Ricci tensor
$\rho_{ij}=\rho(e_{i},e_{j})$ are calculated. The non-zero of them are
determined by the following equalities and the property
$R_{ijkl}=-R_{jikl}=-R_{ijlk}$:

(46) $\begin{array}[]{l}R_{0110}=R_{0220}=-R_{0330}=-R_{0440}=1,\\\
R_{1234}=R_{1432}=R_{2341}=R_{3412}=1,\\\
R_{1331}=R_{2442}=1,\qquad\rho_{00}=4.\end{array}$

Therefore, its Ricci tensor has the form $\rho=4\eta\otimes\eta$ and the
manifold is $\eta$-Einstein. Hence, the scalar curvature of $g$ is $\tau=4$
and the constructed manifold is $*$-scalar flat, i.e. $\tau^{*}=0$ [17]. Then,
due to (10), we obtain for the scalar curvature of $\tilde{g}$ the value
$\tilde{\tau}=4$.

Using (13), let us construct a $\beta$-RB-like almost soliton on
$(G,\varphi,\xi,\eta,g)$ with vertical potential $\vartheta=k\xi$ for a
constant $\beta$ and a pair of functions $(\lambda,\tilde{\lambda})$.

From (33), we determine the following conditions in the case
$\beta\neq-\frac{1}{2n}$

(47) $\lambda+k=-4\beta,\qquad\tilde{\lambda}-k=-4\beta.$

Then, (34) implies $\mathrm{d}{k}(\xi)=-2$ and $\mathrm{d}{k}=-2\eta$. A
solution of the last equation is e.g. $k=-2t$ assuming $\eta=\mathrm{d}{t}$.
This form of $k$ also satisfies the condition in (35) for the case
$\beta=-\frac{1}{2n}$. This allows us to determine functions
$(\lambda,\tilde{\lambda})$ from (47) and (36) for all values of $\beta$ by

(48)
$\lambda=2\left(t-2\beta\right),\qquad\tilde{\lambda}=-2\left(t+2\beta\right).$

Hence and from (39) we obtain $h=-4\eta\otimes\eta$. As a consequence, (30)
and (31) take the following form

$\mathcal{L}_{\vartheta}g=4t\tilde{g}-4(t+1)\eta\otimes\eta,\qquad\mathcal{L}_{\vartheta}\tilde{g}=-4tg+4(t+1)\eta\otimes\eta.$

Finally, we found that all the findings in Theorem 3.3 are satisfied for the
given example.

## Acknowledgments

The research is partially supported by project FP23-FMI-002 of the Scientific
Research Fund, University of Plovdiv Paisii Hilendarski.

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