# Harmful Conspiracies in Temporal Interaction Networks: Understanding the
Dynamics of Digital Wildfires through Phase Transitions

Kaspara Skovli Gåsvær  Department of Physics and Center for Computing in
Science Education, University of Oslo, 0316 Oslo, Norway  these authors
contributed equally to this work  Pedro G. Lind  Department of Computer
Science, OsloMet–Oslo Metropolitan University, N-0130 Oslo, Norway  Artificial
Intelligence Lab, Oslo Metropolitan University, N-0166 Oslo, Norway
NordSTAR–Nordic Center for Sustainable and Trustworthy AI Research,
Pilestredet 52, N-0166 Oslo, Norway  Johannes Langguth  Simula Research
Laboratory, HPC, Oslo, 0164, Norway
Morten Hjorth-Jensen  Department of Physics and Center for Computing in
Science Education, University of Oslo, 0316 Oslo, Norway  Department of
Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State
University, East Lansing, 48824 MI, USA  Michael Kreil  Bayerischer Rundfunk,
München, 80335, Germany  Daniel Thilo Schroeder  Department of Computer
Science, OsloMet–Oslo Metropolitan University, N-0130 Oslo, Norway  SINTEF,
Sustainable Communication Technologies, Oslo, 0373 Norway  these authors
contributed equally to this work<EMAIL_ADDRESS>
###### Abstract

Shortly after the first COVID-19 cases became apparent in December 2020,
rumors spread on social media suggesting a connection between the virus and
the 5G radiation emanating from the recently deployed telecommunications
network. In the course of the following weeks, this idea gained increasing
popularity, and various alleged explanations for how such a connection
manifests emerged. Ultimately, after being amplified by prominent conspiracy
theorists, a series of arson attacks on telecommunication equipment follows,
concluding with the kidnapping of telecommunication technicians in Peru. In
this paper, we study the spread of content related to a conspiracy theory with
harmful consequences, a so-called digital wildfire. In particular, we
investigate the 5G and COVID-19 misinformation event on Twitter before,
during, and after its peak in April and May 2020. For this purpose, we examine
the community dynamics in complex temporal interaction networks underlying
Twitter user activity. We assess the evolution of such digital wildfires by
appropriately defining the temporal dynamics of communication in communities
within social networks. We show that, for this specific misinformation event,
the number of interactions of the users participating in a digital wildfire,
as well as the size of the engaged communities, both follow a power-law
distribution. Moreover, our research elucidates the possibility of quantifying
the phases of a digital wildfire, as per established literature. We identify
one such phase as a critical transition, marked by a shift from sporadic
tweets to a global spread event, highlighting the dramatic scaling of
misinformation propagation. Additionally, we argue that the driving forces
behind this observed transition are attributed to influential users, who act
as catalysts, accelerating the spread of misinformation. Lastly, our data
suggest that the characteristics of such events may be predictable, at least
in some instances. From this data, we hypothesize that monitoring minor peaks
in user interactions, which precede the critical phase culminating in real-
world consequences, could serve as an early warning system, aiding in the
prediction and potentially the mitigation of digital wildfires.

###### keywords:

fake news, complex networks, twitter, phase transition, digital wildfire

## Introduction and Background

Before the advent of the internet, people primarily relied on print media and
radio as their main sources of news. During that era, information
dissemination was characterized by a clear separation between the source and
the consumer. The flow of information was unidirectional and slow-paced, and
it was common practice to trust journalists and newspapers as authoritative
and reliable sources of information.

The inception of the internet [1]on January 1, 1983, marked a turning point in
the way we exchange and receive information. Initially, the Internet was
primarily populated by users with expertise in science, technology,
engineering, and mathematics. At this stage, discussions were restricted to a
small audience, and traditional broadcasting media outlets still held sway.
However, a significant shift occurred in the late 1990s with the introduction
of the first online social networks like Bolt.com [2] or SixDegrees.com [3],
which opened the gates for diverse individuals to participate in a broader
online discourse. With the ensuing absence of a clear separation between
information sources and consumers, the issue of the trustworthiness of the
sources started to become more pressing.

The reliability of news agencies today can vary widely depending on the
country and agency in question. Nevertheless, news agencies are generally held
to a higher level of accountability than OSNs. While fact-checking of social
media content may occur in rare cases, it typically only happens after posts
potentially reach a large audience [4], and even then, the desired effect
often fails to materialize [5]. Despite these efforts, a remaining challenge
for fact-checking is the extreme amount of data available online. Although
estimates for the amount of produced data vary depending on the source and the
definition of ’data’, it is estimated that the amount of data produced
globally has been growing exponentially in recent years. According to a report
by Seagate and IDC from 2020 [6], the global datasphere - which includes all
the data created, captured, and replicated in a year - was projected to reach
175 zettabytes by 2025. Even though OSN data is only a smaller part of this,
with this amount of information, it is impossible to manually fact-check;
thus, misinformation often spreads unnoticed, especially on social media. We
argue that the fact that (1) anyone, regardless of their qualifications, can
post about anything online, (2) the resulting sheer amount of unchecked
misinformation, and (3) the lack of accountability imposed on the providers of
OSNs, turns the sea of online information into a maze; tricky to navigate even
if aware of these challenges, and potentially dangerous if not.

As a result, misinformation travels at a speed never previously seen, possibly
resulting in severe real-world implications [7, 8, 9, 10]. According to the
World Economic Forum [11], these phenomena are called Digital Wildfires and
defined as the rapid spread of information or rumors, amplified by the power
of social media, which can create significant societal and economic damage in
a short amount of time. The term DW we use in this paper extends this
definition by adding a temporal dimension. Here, DWs begin with the first
social media post that addresses the potentially fast-spreading topic and end
after real-world consequences occur. Langguth et al. [12] have shown that the
topic leading to real-world consequences may continue to be discussed even
after these consequences occur. Even more, a new DW may emerge around the same
complex of topics in a different context. However, for this article, we
refrain from such an extended definition and stick to defining the lifecycle
of a DW from the first social media post until after the real-world
consequences.

To effectively combat the rise of DWs, it is vital to establish automated
systems capable of early misinformation detection, thereby allowing for prompt
interventions. Despite the wealth of research dedicated to the automated
detection of misinformation, categorizable into approaches such as linguistic-
based [13], visual-based [14], user-based [15], post-based [16, 17], and
network-based detection [18, 19], a comprehensive strategy targeting DWs
specifically remains undeveloped. Thus, before devising effective automated
systems, it is imperative to delve deeper into understanding the mechanisms
and dynamics fostering the proliferation of DWs. Acquiring such knowledge lays
a crucial foundation for crafting robust and precise algorithms vital for
early detection and prevention.

In this article, we aim for a generic approach exploiting not only the content
but rather the underlying interactions of a particular DW, namely the 5G and
COVID-19 misinformation event [20], within the OSN Twitter to gain knowledge
about the properties and dynamics of the spread of DWs on a societal scale.
Specifically, we investigate the evolution of the temporal networks induced by
the interactions between Twitter users to uncover the emergence of DW.

Previous research has shown that investigating only the diffusion pattern of
this kind of misinformation on an individual, per social media post, -basis is
not promising at all [21, 22]. Even more, it seems like we can only understand
a DW when examining the entirety of information cascades associated with it
[23]. In this paper, we address the following question: Given the interaction
data of an entire DW from the online social network Twitter, can we explain
its dynamics and temporal evolution on a societal scale by using complex and
temporal networks? The specific temporal network we study originates from
interactions between Twitter users connected to the 5G and COVID-19
misinformation event, a series of tweets claiming a link between the COVID-19
virus and 5G technology that lead to a DW. This DW reached its peak around
April $2020$ (see Figure 1), resulting, among other things, in the destruction
of 5G-related telecommunication equipment and the harassment as well as the
kidnapping of telecommunication workers.

We undertake a examination of the temporal evolution of interaction networks —
precisely focusing on the dissemination of misinformation surrounding the 5G
and COVID-19 event as it unfolded on Twitter. Individual information cascades
on this platform predominantly manifest through threads of tweets and
retweets. Understanding that DWs are essentially conglomerates of numerous
such cascades, we venture to scrutinize the evolution of the DW through a lens
encompassing a significantly comprehensive set of related cascades — a
holistic approach facilitated by our comprehensive dataset that, to the best
of our knowledge, stands as the most expansive in network-based study of DWs.
Utilizing community detection methodologies [24, 25, 26], we delve into
discerning the dynamics. Furthermore, by evaluating both the centrality and
activity of the vertices, we aim to pinpoint the roles and repercussions of
group and individual activities in steering the temporal trajectory of the
network.

Our objective is to investigate dynamics to describe and predict the evolution
of complex temporal networks related to the spread of DWs. The insights gained
from this study can aid in the early detection and prevention of
misinformation events before they escalate into DWs and end with real-world
consequences.

In the following, we list our contributions to the understanding of the 5G and
Covid-19 DW. First, we demonstrate that the DW displays phase transition
behavior, highlighting the need to approach this phenomenon from a complex
systems and network science perspective. Second, our investigation of
community dynamics reveals a synchronization of communities towards the peak
of the DW, which corresponds to the time when real-world consequences occur.
Third, we identify a small group of influential users who are crucial in
driving the conversation on a large scale, drawing in a significant number of
new users. Finally, our analysis of the largest cluster shows that it is
unstable and characterized by oscillations of partly contradictory narratives.

Previous work addresses similar questions and problems. Vosoughi et al. [23]
investigate the dissemination of true and false news on Twitter, as detailed
in their 2018 study. The authors analyze a dataset encompassing approximately
126 thousand news stories tweeted by around three million users, assembled
into rumor cascades. Their findings underscore that false news stories
propagate significantly faster, reach deeper, and spread more broadly than
true stories. Furthermore, false information was found to be retweeted more
frequently than true news. Intriguingly, true stories take approximately six
times longer to reach a similar audience size of 1,500 users compared to their
false counterparts.

Starbird [27] analyzes Twitter data related to eight mass shooting events from
2013 to 2016, identifying alternative narratives that emerged on the platform.
These alternative narratives often contradict mainstream media reports and
suggest that the shootings were false flag operations or hoaxes. The study
uncovers that alternative media sources play a central role in the production
and dissemination of these narratives, acting as key amplifiers of
misinformation. The research also highlights the interconnected nature of the
alternative media ecosystem, with multiple alternative media sources cross-
promoting each other’s content and reinforcing the alternative narratives.
This interconnectedness contributes to the spread of misinformation and
fosters distrust in mainstream media sources.

Del Vicario et al. [28] conducted a study investigating the spreading of
misinformation on social media, with a specific focus on Facebook. Their
research identifies similar consumption patterns among users who prefer
scientific news and conspiracy theories. However, the patterns of information
spread, or "cascade dynamics," shows differences. The authors discover that
users tend to form "echo chambers," polarized, homogenous clusters where they
share content that aligns with their beliefs. Furthermore, the authors
introduce a data-driven model that successfully mimicked these dynamics,
reinforcing that homogeneity and polarization are key determinants of content
spread.

In their study, Friggeri et al. [29] examine the dynamics of rumor propagation
on Facebook. They find that rumors, irrespective of their veracity, spread
deeply through social networks, with true rumors generating larger cascades.
Their propagation continues even after debunking, indicating that users might
overlook or ignore debunking comments. Furthermore, they observe that the
popularity of rumors is bursty, with humor sometimes serving as an antidote to
rumor propagation. However, despite these findings, the authors acknowledge
potential biases in their sample collection and analysis.

Figure 1: Most significant events that occurred during the course of the
COVID-19 and 5G Misinformation event that transpired online in 2020.

Langguth et al.’s study [12] expanded on the concept of DWs, focusing on the
event we aim to research in this study, the misinformation linking 5G
technology with the COVID-19 pandemic. They trace the origin of this rumor and
reveal how it grew across social media platforms. The study encounters that
even contradictory narratives could strengthen DWs, and that the role of
commercially-influenced videos is often underestimated in Twitter-only
analyses. The authors suggest several countermeasures, including focusing on
the financial motivations behind the spread of misinformation in general and
DWs in particular and promoting international cooperation in research on DWs.
However, the analyses in this study are more qualitative in nature and do not
include a structural analysis of the underlying communication in networks.

## COVID-19 and 5G Conspiracy Theories as a Specific Digital Wildfire Case

As the COVID-19 pandemic swept across the globe in early 2020, a proliferation
of tweets emerged linking the virus’s origins to 5G wireless technology.
Initially confined to a small and insignificant number, the volume of such
tweets surged exponentially throughout April 2020, culminating in a series of
arson attacks on 5G towers in multiple countries, including the United Kingdom
[30], Nigeria [31], and Canada [32]. As mentioned in the previous section,
formally, such fast-growing dissemination of online misinformation leading to
real-world implications is known as a Digital Wildfire (DW) and ranked as a
top global risk by the World Economic Forum [11].

In the following, we introduce the chronology of the 5G and COVID-19
misinformation event delineated into three distinct phases, a classification
grounded in a qualitative evaluation of the DW: pre-real world events, during-
real world events, and post-real world events. The demarcation of these phases
serves as a framework for unraveling the intricate dynamics at play. Notably,
the event persists to this day, warranting continued scrutiny. However, this
study is confined to exploring the events until May 2020. For a more detailed
overview of the DW under investigation, including developments stretching to
late 2022, we refer to the adept assessment provided by Langguth et al. [33].

### Pre-real world event:

With the first tweet collected in early January, we observed a slow growth in
daily tweets, insinuating a connection between COVID-19 and 5G throughout
January and February 2020. In addition, we note the gradual uptick in traction
for such content on platforms beyond Twitter, including notable activity on
YouTube. Pinpointing the exact inception point, however, presents a challenge
due to the presence of multiple sub-narratives that are arguably leading to
the DW. Furthermore, we use Twitter data only, leaving open the possibility
that discussions regarding the event were initiated on a platform other than
Twitter. However, when investigating the early tweets, we discover an entire
spectrum of conspiracy narratives claiming a causality between 5G radiation
and the coronavirus. Even though these narratives seem to be as diverse as the
individuals spreading them, they share the idea that the 5G technology is
dangerous, can hurt people, and thus should not be implemented. For detailed
descriptions and tweet samples for subnarratives, we point to the datasets
published by Konstantin et al. [34] and Schroeder et al. [21]. At this point,
we would like to draw your attention to the fact that before the end of
January 2020, only 685 tweets and 1,081 retweets containing both keywords
referencing COVID-19 and 5G appeared on Twitter. This comparatively small
number has led to the decision that we limit ourselves to the period from the
first of February for the purposes of this study.

### During-real world event:

In March, when the pandemic began to take a foothold in Europe, the spread of
tweets also picked up its pace, resulting in four times as many tweets from
late March to early April. Consequently, the first series of arson attacks
happened in the UK, the Netherlands, and New Zealand during the weekend of
April 3, 2020. Multiple more followed in the week after, and later some
occurred in Canada as well. By July 2, 2020, there were reports of 273 cases
of clashes between people who believed in some version of the conspiracy, as
well as 121 reports on arson and other types of destruction [35], including
the detainment of 8 telecommunication workers in Peru.

### Post-real world event:

In late April of 2020, Twitter banned material and users promoting attacks on
5G infrastructure, and the spreading of content related to the connection
seemed to halt. However, even as late as the first quarter of 2021, suspected
cases of arson in Africa and Canada [36, 37] started to occur.

Figure 2: (Left) Illustration of the concept of temporal slices and (Right)
accumulative slices. Temporal slices do not contain the vertices and edges of
previous slices, while accumulative slices contain all vertices and edges from
the previous slices.

In this paper, we recognize the phase characterized as "during-real world
events" as indicative of a phase transition phenomenon, a concept borrowed
from the field of statistical physics [38, 39]. This classification draws
parallels with transitions seen in critical phenomena such as percolation, a
well-studied concept in physics. As we delve deeper, it will be evident that
the evolution of the DW during the COVID-19 and 5G misinformation event
exhibits characteristics akin to the onset of a percolation threshold,
signifying a critical phase in the information dissemination process.

## Data Collection and Preprocessing of Massive Twitter Datasets

Since Twitter’s Terms of Service prohibit storing large datasets, we choose a
streaming-based approach, first introduced in Schroeder et al. [40]. We keep
only in-stream-anonymized user IDs, the corresponding timestamps, and texts to
create the tweet-retweet-user mapping. Moreover, we do neither store nor
process any other information. The data collection took place using a custom
build framework for Twitter graph analysis [41] and a custom scraping strategy
[42].

Since Twitter’s search API, at that time, only returned tweets that were not
older than two weeks, it is necessary to collect data preventively while
hoping that this collection then, at the time of a DW, contains the relevant
tweets. In the following, we describe exactly this procedure.

Between December 2019 and May 2020, we amassed a total of 6,286,886,977
COVID-19-related tweets, retweets, replies, and quotes (referred to as
statuses) leveraging the keywords outlined in Appendix A and using Twitter’s
search API. It is pertinent to note that querying the Twitter search API
frequently yields duplicate entries. To ensure the robustness of our dataset,
we meticulously identified and removed these duplicates in the initial phase
of our data processing. This filtration process resulted in a refined dataset
comprising 2,570,581,178 unique statuses, which formed the basis for our
subsequent analysis.

Next, we filter for those tweets that mention “ 5G ” and “ 5g ”. We do not
remove the whitespaces because doing so produces too many false positives
completely unrelated to 5G. Moreover, we include alternative spellings such as
“ 5-G ”, although the number of tweets containing these is negligible. All
keywords are listed in Appendix B. After applying the second filter, $364,325$
Covid-19-related and 5G-related tweets remain. As a concluding procedure, we
exclude any statuses not originating from the specific timeframe of February
1, 2020, up to and inclusive of May 11, 2020. Hence, our study spans a precise
duration of 100 days, beginning on February 1, 2020, and ending on May 11,
2020.

The enrichment phase commences with the curated dataset derived from the
preceding filtering phase. Central to this phase is the intricate process of
Twitter thread completion. To elucidate, a Twitter thread is a cohesive series
of interconnected tweets stemming from an inaugural tweet and encapsulating
all ensuing replies and quoted tweets to foster a consolidated conversation.
Such threads offer a structured vantage point, enabling a comprehensive
insight into the contextual dynamics enveloping the primary tweet. During this
phase, individual threads pertaining to a specified tweet are queried to
facilitate the incorporation of statuses surpassing the two-week retrieval
constraint imposed by the Twitter search API. Consequently, this method
permits the inclusion of tweets devoid of the keywords delineated in Appendix
A and B. It warrants mention that despite endeavors to augment the dataset by
resurrecting more dialogues linked to the DW, the potential for thread
incompletion persists, attributed to the inherent limitations of the Twitter
API, which confines queries to parent statuses within a thread exclusively.
Notwithstanding this limitation, we contend that the augmentation process,
albeit yielding incomplete threads, substantially enhances the dataset by
infusing it with valuable context.

Figure 3: Three-part depiction of Network $G_{\downarrow}$. The left subfigure
presents the global degree distribution, illustrating the range of degree
centralities. The central subfigure displays the distribution of cluster sizes
across all accumulative slices, with clusters identified via the Leiden
algorithm [43]. The rightmost subfigure explores the cluster size distribution
across all temporal slices ($\Delta t=4h$). Both the middle and rightmost
figures are created by counting the occurrence of clusters with size $C$ in
every slice before averaging the number of occurrences by the number of slices
in the experiment. Notably, both degree and cluster sizes exhibit a power law
distribution, mirroring patterns found in other social networks.

Following the enrichment phase, we devoted substantial time and resources to
delve extensively into the investigation of the resulting dataset, which has
fostered a rich bedrock for a plethora of scientific publications. A testament
to the rigorous endeavors undertaken in this period is our meticulous labeling
effort encompassing more than 9,688 tweets to ascertain whether they were
indeed a segment of the 5G COVID-19 misinformation event. This scrupulous
initiative bifurcated into two distinct datasets: one housing the tweets, and
the other cataloging as many as 3,492 individual tweet-retweet cascades, also
labeled to demarcate if a cascade was intertwined with the misinformation
event — the datasets are referenced as WICO-Text and WICO-Graph, detailed in
publications by Pogorelov et al.[44], and Schroeder et al.[34], respectively.

Following the enrichment phase, we engaged in a detailed analysis of the
resulting dataset, a process that fostered the foundation for multiple
scientific publications. One significant step in this process was the
meticulous labeling of over 9,688 tweets to determine their association with
the 5G COVID-19 misinformation event. This effort yielded two distinct
datasets: one archiving the tweets and another detailing 3,492 individual
tweet-retweet cascades, each labeled to indicate association with the
misinformation event. These datasets, denominated as WICO-Text and WICO-Graph,
are discussed in depth in works by Pogorelov et al.[44] and Schroeder et
al.[34], respectively. To further leverage these datasets, we organised a
MediaEval Benchmark Challenge task, wherein both datasets were subdivided into
testing and training sets, and distributed to an initial pool of 15 groups.
These groups embarked on developing distinct classifiers capable of
differentiating between tweets and cascades genuinely associated with the
misinformation event. The entire endeavor is documented in detail in [18].

## From Temporal Interactions to Interaction Networks

Given the filtered and enriched dataset, we now extract user interaction by
counting contacts between each pair of users. We define $Z_{u}=\text{set of
users}$ and $Z_{s}=\text{set of statuses}$. A contact between two users is
defined as

$\text{any user }j\text{ interacting with any user }i\text{ through user
}j\text{ either retweeting, replying, or quoting user }i\text{.}$ (1)

The set of contacts induces a symmetric adjacency matrix $A$ with $A_{ij}=1$
to label an existing contact between users $i$ and $j$ and $0$ if such contact
does not exist. By keeping track of the number of retweets, replies, and
quotes between users, we are able to build the directed and weighted network
of interactions. Since in this paper, we focus on assessing the size of
connected users, we consider, for simplicity, the contacts as unweighted and
undirected edges, forming the interaction network and its communities.
Furthermore, we call such a network temporal interaction network when contacts
have timestamps allowing for only considering excerpts of an interaction
network within an arbitrary time window. More precisely, we build temporal
interaction networks based on the adjacency matrix $A$, defining the
underlying graph $G_{\downarrow}$ as the temporal graph containing the
entirety of vertices and edges, i.e.

$G_{\downarrow}=(V_{\downarrow},E_{\downarrow}),$ (2)

where $V_{\downarrow}=\\{v_{i}\\}_{i=0}^{N}$ is the total set of vertices
labeling the users and $E_{\downarrow}=\\{(u,v,t)|u,v\in
V_{\downarrow},t_{0}<t\leq t_{0}+nT\\}$, with $T$ the size of the time window
during which connections composing the graph occur, $t_{0}$ is the initial
time and $n$ labels the time-window. Notice that both $G_{\downarrow}$ and
$E_{\downarrow}$ are functions of the time-window $n$.

## Assessing the Evolution of Interaction Networks

To examine the network dynamics in a temporal way, we slice $G_{\downarrow}$.
Thus, a slice is a subgraph of $G_{\downarrow}$ and the set including all
slices for the observing period is $S=\quantity{G(V_{\downarrow},E_{i}),i\in
L,E_{i}\in E_{\downarrow}}$ where $L$ is the number of slices. Although a set
of slices is temporal, each slice is a static "snap-shot" of a time period in
the interaction network. In the following, we present two distinct types of
slices: accumulative slices and contact slices (see Figure 2). The rationale
behind developing multiple slice types is the potential to extract diverse
information from each one.

In the context of network analysis, pure temporal slices offer valuable
insights into the overall temporal evolution of the system. However, their
limitation lies in their inability to trace clusters across multiple slices
because of the removal of nonactive vertices in subsequent time periods.
Alternatively, accumulative slices provide the advantage of cluster tracking
but come with the trade-off of rapidly increasing in size, posing a challenge
in handling them effectively.

We define temporal slices (see Figure 2 left) and divide the interaction
network into slices of sub-graphs based only on the timestamps, and in a non-
accumulative manner. Moreover, we remind the reader that edges are contacts,
e.g., retweets or comments, and thus associated with a timestamp. We divide
our set of edges into intervals, e.g., a day, a week, which we call $\Delta
t=(t^{s},t^{e})$ so that we get $L$ slices in total and the entire time
interval where we collected our dataset $T=t^{e}_{L}-t^{s}_{1}$. This results
in a temporal graph $\mathcal{G}=(V_{\downarrow},E_{1},...,E_{L})$. Now each
slice $s_{i}\in S$ contains all edges added to the network in the time period
$\Delta t_{i}$ with

$S=\quantity{G(V_{\downarrow},E_{i}),i\in L},$ (3)

and $\bigcup_{i=1}^{L}s_{i}=G_{\downarrow}$.

We define accumulative slices (see Figure 2 right) as the set of all contacts
made in the time interval $\quantity[0,t_{i}],\quad 0<t_{i}<t_{i+1}\leq T$,
where $T$ is the data acquisition time window. This means that the slice
$s_{i+1}$ contains all contacts in $s_{i}$ plus all other contacts made in the
time interval $\quantity[t_{i},t_{i+1}]$.

$s_{i+1}=\quantity{u,v\subset V_{\downarrow}:\exists(u,v,t)\in
E_{\downarrow}\wedge t\in\quantity[0,t_{i+1}]}=s_{i}\cup\quantity{u,v\subset
V_{\downarrow}:\exists(u,v,t)\in E_{\downarrow}\wedge
t\in\quantity[t_{i},t_{i+1}]}.$

Furthermore, we define the distance between two subsequent timestamps,
$t_{i+1}-t_{i}$, as $\Delta t$, which is equal for all $i$. The last slice of
the experiment is, by definition, the entire underlying graph,
$G_{\downarrow}$.

## Degree Centrality as a Proxy for User Activity

Centrality measures give insight into which users or vertices contribute the
most to the flow of information. As we are dealing with an undirected network,
we cannot determine whether a vertex is highly active, e.g., comments on other
statuses with a high frequency or is made highly active by others, e.g., many
other statuses are responses to a status. Thus, we define vertex activity as
the number of contacts a user experiences. In other words, this is the number
of edges connected to a given vertex and thus equal to the degree centrality
defined as

$\mathcal{C}_{\text{deg}}(i)=\sum_{i,j\in\mathcal{N}(i)}e_{j},$ (4)

where $\mathcal{N}(i)$ is the neighborhood of the vertex $i$ defined as

$N(v)=\quantity{u\subset V_{\downarrow},u\neq v:\exists(u,v,t)\in
E_{\downarrow}}.$ (5)

Degree centrality can be calculated for each slice $s_{i}$ or the entire
network $G\downarrow$. Later, we use this measure to explore the correlation
between the overall vertex activity of a group and other properties of the
system, such as the size of the largest clusters and the number of overall
contacts.

## Power Law Distribution of User Contacts and Community Sizes

Many empirical studies have shown that social networks exhibit power-law
degree- and community-size distributions. The phenomenon occurs across complex
networks ranging from social [45, 46] to informational [47, 48] and biological
networks [49, 50]. Here, power law distributions arise for various reasons,
such as preferential attachment, i.e., new nodes are more likely to connect to
already well-connected nodes; growth, i.e., networks expand over time; and
homophily, i.e., similar nodes tend to connect with each other.

Figure 4:  (Left) Total number of users and established contacts in each time
“slice” and (Right) the corresponding accumulative slices. Contacts encompass
retweets, quotes, and comments, along with the count of distinct users per
time segment. The 100-day investigation span, from February 1, 2020, to May
11, 2020, separates into $\Delta t=24h$ accumulative slices (shown on the
right) and $\Delta t=4h$ temporal slices (shown on the left). Both charts
clearly demonstrate a phase transition between slices 360 to 390 and slices 61
to 66, respectively. Additionally, potential predictors emerge between slices
270 and 280 on the right chart.

For the internet itself [45] as well as for social networks [51], including
Twitter [40], it has been shown that both connectivity in general and the
number of communication contacts in particular follow power-law distributions.
Figure 3 shows that this also applies to communication within the DW under
investigation. As with general communication in social networks, there are
many users with few contacts to others as well as few users with many contacts
to others. Figure 3 depicts $G_{\downarrow}$’s global degree distribution,
illustrating the range of degree centralities, the community sizes across all
accumulative slices, and the community size distribution across all temporal
slices. For this, we investigate the community structure underlying our
interaction network using the Leiden algorithm [43] with standard
configurations for the resolution parameter.

The power-law degree distributions denote the existence of few hubs, i.e.,
Twitter users with numerous connections or interactions. Simultaneously,
numerous nodes exhibit fewer connections. In the context of DWs, this
distribution suggests that these hubs play a critical role in driving the
dynamics of interactions within the DW. A minor proportion of users can
significantly impact the course of interactions, substantiating their role in
the progression of DWs.

Community size distributions following a power law indicate a composition of
multiple small communities alongside a few large ones within the DW. While
clusters of users exhibit intensive interaction amongst themselves, the
network’s major interaction activity concentrates within a handful of large
communities, for which we later show that conversation within these
communities is mostly driven by influential users.

Power law distributions, in degree and community size, bring forth numerous
implications for DWs. The resilience to random node failures associated with
power-law degree distributions implies that deactivating or removing a random
user might not disrupt the spread of the wildfire. The highly connected hubs
maintain the continuum of interaction. Furthermore, we argue that hubs and
large communities significantly influence the conversation direction,
narrative shape, and information spread within the DW while network structure
facilitates the rapid and broad diffusion of information, ideas, or behaviors,
especially if instigated or promoted by the hubs or large communities.

Figure 5: The annd function at three different stages: (Left) before the phase
transition (before slice 360), (Middle) during the phase transition (between
slices 360 and 390) and (Right) after phase transition (after slice 390).

## Defining the Life Cycle via Phase Transition

A phase transition is a well-established concept in physics [38, 39], with
examples such as the transition from a solid to liquid state (melting) or from
liquid to gas (evaporation) or the transition from a set of small disconnected
groups of individual to a large set of individuals, spanning throughout the
entire society. This latter phase transition is usually called transition to
percolation [39], and occurs often within the realm of complex systems and
network science, usually associated with the emergence of new structures,
functionalities, or patterns. Classic examples of phase transitions in complex
systems are the emergence of a giant connected component in a random network
[52], or the abrupt transition from free-flowing traffic to a traffic jam, a
scenario often referred to as a ’phantom traffic jam’ [53]. Moreover, phase
transitions also serve as effective metaphors for sudden changes in collective
human behaviors, particularly in the digital realm [54]. A specific idea or
movement might transition from being recognized by a limited number of
individuals to achieving widespread recognition or even reaching viral status.

Qualitatively, a phase transition can be investigated by identifying a
parameter whose changes can drive the systems from one phase to another and by
keeping track of some observable which characterizes the phase. In the case of
liquid-to-gas transition, the parameter is, of course, temperature, and the
observable is the density of water. In the case of transition to percolation,
the parameter is a sort of probability that pairs of individuals have in
establishing one contact, and the observable is the size of the largest group
of individuals (cluster). In this section, we argue that the concept of phase
transitions is integral to our understanding of DW. In particular, we use it
to describe the critical moment when the propagation of misinformation
abruptly accelerates, shifting from a slow and steady pace to a rapid,
wildfire-like spread. At the same time, the phase transition quantitatively
derives the three phases of the DW under investigation; see our discussions of
Figure 4.

The 5G COVID-19 DW we analyze in this paper exhibits phase transition
characteristics in its spread and contact pattern, suggesting that this
phenomenon might apply to other DWs as well. Figure 4 on the left side
presents temporal slices, specifically time slices containing only temporary
interaction networks for the period from February 1, 2020, to May 11, 2020.
These slices capture both the number of contacts as described in Equation 1
and the visualization of different user counts. Further, a slice (on the left)
encompasses the interaction network composed of tweets, retweets, comments,
and quotes in time intervals of four hours each. The number of contacts and
different users reveals phase transition characteristics between slices 360
and 390, corresponding to the period from April 1 to April 6, 2020. This
period precisely precedes and follows the first arson attacks. These
observations make it plausible to argue that the categorization of DWs into
three phases, namely before, during, and after real-world consequences, as
qualitatively proposed by Langguth et al. [12], can also be assessed
quantitatively.

Figure 5 shows further evidence that the three stages of a phase transition in
time occur during the 5G COVID-19 DW. While overall, all three phases appear
dominantly disassortative, the average nearest neighbor degree, as well as its
variance, significantly vary among these phases. Moreover, Figure 5 shows the
average nearest neighbor degree dependence on the node degree in a
qualitatively different way. Before and after the transition, there is a clear
degree-range separation: for $k\lesssim 400$ the nearest neighbor degree is
typically larger than for $k\gtrsim 400$. At the transition, the nearest
neighbor degree seems to distribute more closely to a power-law. This variance
could represent an exciting area for future research and a potential unique
feature of DWs.

Figure 6: Evolution of user clustering in the analyzed DW from February 1,
2020, to May 10, 2020, represented in time slices on the x-axis. The left
subplot illustrates the situation with accumulative slices and a $\Delta
t=24h$. It shows a clear and linear increase in the relative size of the
largest cluster. This implies a growing participation in the 5G-COVID
conspiracy narrative, with more users becoming part of the largest cluster
over time. However, the fraction of users who are part of the top 10% of all
clusters does not exhibit such strong growth. The subplot on the right,
displaying temporal slices with a $\Delta t=4h$, shows a similar trend. It is
noteworthy that the identity of the largest cluster changes over time, i.e.,
the largest cluster at a given time does not necessarily contain the same
group of users as the largest cluster at a later time.

## Confluence of Coalescing Narratives

In the study of community dynamics over time, it is crucial to recognize that
a community, in this sense, represents more than just a group of users
interconnected through high volumes of interaction. In our network, an edge
represents an undirected contact, indicating a larger discourse occurring
within a particular timeframe. We refer to this discourse as a narrative. This
terminology draws from the findings of Langguth et al. [12], who suggest that
DWs, particularly the 5G COVID-19 DW, initially combine followers from various
conspiracy narratives. There wasn’t one specific source for the DW. Instead,
various conspiracy theorist groups had already been discussing anti-
vaccination and anti-radiation theories prior to the COVID-19 pandemic, which
then provided an opportunity for these narratives to merge.

Figure 6 portrays the evolution of user communities in the analyzed DW from
February 1, 2020, to May 10, 2020. This is represented as time slices on the
$x$-axis. The left subplot, indicating accumulative slices and a $\Delta
t=24h$, shows a clear and linear increase in the relative size of the largest
community. However, the proportion of users belonging to the top 10% of all
communities does not increase as significantly. The subplot on the right,
which represents temporal slices with a $\Delta t=4h$, shows a similar trend.
Importantly, the composition of the largest community changes over time,
meaning the largest community at a given moment may not include the same users
as the largest community at a later moment.

When comparing the growth in relative size of the largest community with that
of the top 10% of all communities, we observe that the latter grows at a
slower rate. This suggests increasing participation in the 5G-COVID conspiracy
narrative, with a growing number of users becoming part of the largest
community over time. Therefore, a trend emerges toward the merging of
different narratives.

Figure 7: Correlation between total contacts and the fraction attributable to
the most active users, gauged by degree centrality. The left subplot reveals
the percentage of vertices in relation to the total number in an accumulative
slice with a $\Delta t=24h$ interval, marking the transition around slice 63.
The changing ratio of connections during this transition suggests that highly
active users interact with a larger set of distinct users compared to those
with lower activity levels. Furthermore, the figure indicates a potential
predictor around slice 45, which appears to be primarily noticeable to the
active users. The right hand plot presents the same relationship for the
temporal slices with $\Delta t=4h$. Here, we observe similar patterns as on
the left. Additionally, the gap becomes apparent between the most active 2% of
users and the less active 5%, 10%, or 20% between slices 350 and 470,
underlining the influential role of a small proportion of highly active users
in shaping the network dynamics.

## Influential Users are Steering the Course of Large-Scale Conversations

A critical aspect of the dynamic evolution of DWs is the role of influential
users. These users, often characterized by a high degree centrality, have the
potential to significantly shape the course of large-scale conversations.
Given the network structure of social media platforms like Twitter, a message
from an influential user can quickly reach a vast audience, potentially
altering the trajectory of an ongoing narrative or sparking a new one.

Figure 7 illustrates the correlation between total contacts and the fraction
attributable to the most active users, gauged by degree centrality. The left
subplot reveals the percentage of vertices in relation to the total number in
an accumulative slice with a $\Delta t=24h$ interval, marking the transition
around slice 63. The changing ratio of connections during this transition
suggests that highly active users interact with a larger set of distinct users
compared to those with lower activity levels. Furthermore, the figure
indicates a potential predictor around slice 45, which appears to be primarily
noticeable to the active users.

The right-hand plot presents the same relationship for the temporal slices
with $\Delta t=4h$. Here, we observe similar patterns as on the left.
Additionally, the gap becomes apparent between the most active 2% of users and
the less active 5%, 10%, or 20% between slices 350 and 470, underlining the
influential role of a small proportion of highly active users in shaping the
network dynamics. Our analysis of the 5G-COVID-19 DW reveals that these
influential users contribute significantly to the formation and growth of the
largest community, as well as to the overall narrative evolution within the
DW. Notably, we find that users with a high degree centrality have a
disproportionate impact on the narrative dynamics, reaching more users
compared to those with lower activity levels. Furthermore, our data shows that
influential users can act as connectors between different narratives, further
fueling the growth of the DW. This finding is consistent with the idea that
the confluence of coalescing narratives is a fundamental characteristic of
DWs.

In the context of mitigating the impact of harmful DWs, our findings
underscore the importance of monitoring the activity of influential users.
Effective strategies could include promoting accurate information through
these users or mitigating their influence if they are spreading harmful
narratives. Our analysis, however, also highlights the complexity of this
task. As the identity of the largest community is not constant over time, so
too are the influential users within it. This fluidity necessitates dynamic
strategies for monitoring and intervention, tuned to the temporal evolution of
the DW.

## Examination of Pre-transition Phenomena and Exploration of Potential
Predictors

In the realm of DWs, understanding the dynamics prior to a transition phase
can yield critical insights into the potential predictors of such large-scale
shifts. Our research illuminates several pre-transition phenomena that suggest
impending changes in the narrative or the community structure. We observe a
small spike visible around slice 260 in Figure 4 (left), which portrays the
time or time slice on the x-axis and the number of users and user contacts on
the y-axis. This peak, preceding the actual transition, could indicate that a
topic is gaining potential to generate wider reach, serving as a visible spark
that signals an upcoming phase shift.

Before the transition, we also observe modifications in user behavior
patterns. Specifically, the most active users begin to expand their range of
contacts. These users, who typically have a high degree centrality, start
interacting with a more extensive set of unique users compared to less active
ones. This observation reinforces the central role of influential users in
guiding the DW into a new phase, a notion we discussed in the previous
section.

We also see changes in the structural properties of the network before the
transition. For instance, the ratio of connections, i.e., the number of
communications with different actors, shifts during the transition. This
suggests the possibility of using network structure changes as predictive
indicators of a DW phase transition. These observations suggest that, with
further study and refinement, we might be able to establish a predictive model
for DW transitions. Such a model would not only enhance our understanding of
DW dynamics but could also provide actionable insights to contain or guide
these narratives. Consequently, our future research will focus on refining
these potential predictors and testing their predictive power across different
contexts.

## Discussion and Conclusion

In this paper, we address the dynamics and temporal evolution of a specific
DW, namely the COVID-19 and 5G misinformation event from early 2020. By
processing large amounts of Twitter data, we trace the course of this DW
beyond the investigation of individual information spreading cascades. Rather,
we examine the entirety of the underlying communication in interaction
networks and thus provide a more holistic view of the dynamics that underlie a
DW. We show that both the average count of user contacts and the average
community sizes over the total duration of the phenomena adhere to a power law
distribution. Thus, underline the intrinsic structure and similarity to other
communication networks.

Based on the study of these dynamics, we propose a framework that not only
allows for accessing the three stages of a DW quantitatively through the
concept of phase transitions but also allows for the application of well-
established methods to understand DWs through the lens of physics. Physicists
are able to predict phase transitions by identifying drivers. Temperature, for
example, is the driver for the transition water undergoes between gas, liquid
and solid phases. In this work, we identify potential candidates for drivers
in the phase transition of a DWs based on the underlying communication and
thus paving the way towards a DW prediction model.

Thus, we observe a confluence of coalescing narratives by studying the
community dynamics over the course of the DW. We ask the reader to recall that
a community in a time slice of a temporal interaction network is a group of
people, in our case, Twitter users, talking to each other (about a topic)
within a certain time window. In future work, we plan not only to extend this
research to other DWs, but also to understand the mechanism underlying this
confluence, model it, and possibly comprehend it as a building block of a
driver.

Furthermore, we identify a subset of all users involved in the DW that could
be a major driving force for the transition to a global event. This result
suggests that potential drivers can be found in the underlying characteristics
of the communication of this user group. As with coalescing narratives,
further studies need to show that this result also manifests in other DWs.

Our study’s findings have considerable real-world implications. Unraveling
these dynamics can be a potent tool for entities wishing to manage or control
information propagation. For democratic societies, these tools can be
invaluable in identifying and mitigating nascent extremist groups or
misinformation campaigns. On the other hand, they could potentially be
exploited to suppress democratic dissent in totalitarian regimes. Given the
high stakes, an effective response to DWs requires an approach that harmonizes
technological tools with education, media literacy, and an informed public.
Moreover, it’s critical to ensure democratic accountability for those who
wield these powerful tools.

To conclude, this study represents an effort to model and understand the
dynamics of a DW within complex temporal interaction networks, with results
shedding new light on the societal-scale understanding of these phenomena. Our
ultimate objective is to devise methods that can predict and mitigate the
spread of harmful misinformation. We encourage future research to validate and
expand upon the identified patterns and hypotheses, thus ensuring steady
progress toward this paramount objective.

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## Acknowledgements

This work was funded by the Norwegian Research Council project ”Enabling Graph
Neural Networks at Exascale” (#303404). The research presented in this paper
has benefited from the Experimental Infrastructure for Exploration of Exascale
Computing (eX3), which is financially supported by the Research Council of
Norway under contract #270053. This work also received partial funding from
the projects Graph-Massivizer (HE 101093202), enRichMyData (HE 101070284), and
DataCloud (H2020 101016835). In addition, we are indebted to Tore Heide Larsen
from the Simula Research Laboratory for managing the infrastructure.

## Author Contributions

K.S.G., with guidance from D.T.S., conducted the primary analysis and wrote
the corresponding sections of the paper, deeply informed by the research
undertaken during K.S.G.’s master’s thesis. D.T.S. not only supervised
K.S.G.’s master’s thesis but also led the preprocessing, data enrichment, and
took part in both the data collection and manuscript writing. M.K. was
responsible for collecting the initial dataset that served as a pivotal
resource for this study, laying a substantial foundation for the work
presented in this paper. J.L., M.H.J., and P.G.L. were integral to the
formative stages of the project, lending their expertise to carve out the
creative direction of the study. They remained actively involved through the
development of the manuscript, contributing to the writing and meticulously
reviewing the content to ensure a polished, cohesive output. All authors
reviewed and gave approval for the final version of the manuscript.

## Data Availability

In accordance with Twitter’s Terms of Service and to ensure user privacy, we
will not publish or share the underlying Twitter source data collected during
our study. Our methodology involved the storage of in-stream-anonymized user
IDs, timestamps, and the texts necessary to form the tweet-retweet-user
mapping, strictly omitting any other personal information.

However, we have created interaction networks derived from the processed
Twitter data, which elucidates the structural and relational aspects that
informed our analysis, while being compliant with privacy and ethical
guidelines. These interaction networks, which formed the foundation of our
research, encapsulate the connectivity and dynamics pertaining to the specific
period of study from February 1, 2020, to May 10, 2020 and are available under
https://osf.io/vqhet/?view_only=6b67055f78e047349036332f5bab7365. The
corresponding codebase is available under
https://github.com/KasparaGaasvaer/MasterThesis.

## Competing Interests

The authors declare no competing interests.

## Additional Information

Correspondence and requests for materials should be addressed to D.T.S.

## Ethical Approval

The described data collection, anonymization, storage, and use have been
reviewed by the Ethical Committee of the Simula Research Laboratory in Oslo
and are in accordance with the relevant guidelines and regulations. We see no
ethical concerns with the data use or the publication of results based on the
data.

##  Appendix A Keywords for initial Twitter API search

coronavirus10, corinavirus6, corona, coronaoutbreak, coronavirus,
coronavirusde, coronavirusoutbreak, covid, covid19, covid2019, covid_19,
covid-19, wuhancoronavirus, wuhanvirus9, coronavírus, coronavirus7,
coronavirus8, coronavirus9, zerocovid, coronar_allesoeffnen, allesoeffnen,
allesöffnen, coronadeutchland, xj621, machtbueroszu, machtdiebueroszu,
bueroszu, büroszu, diebüroszu, vaccination, vaccine, epidemic, pandemic,
quarantine, quarantined, mutation, wuhan, coronapanik, covidiot.

##  Appendix B Keywords for filtering 5G-related tweets

5G, 5g, 60Hz, #5G

  *[OSNs]: online social network
  *[OSN]: online social network
  *[DW]: Digital Wildfire
  *[DWs]: Digital Wildfire