# A note on the antimagic orientation problems for the Mycielski construction
of graphs

Eranda Dhananjaya Department of Applied Mathematics, National Chung Hsing
University, Taichung 40227, Taiwan<EMAIL_ADDRESS>supported
by MOST-109-2115-M-005 -002. Wei-Tian Li Department of Applied Mathematics,
National Chung Hsing University, Taichung 40227, Taiwan
<EMAIL_ADDRESS>supported by MOST-109-2115-M-005 -002.

###### Abstract

A simple graph $G$ is said to admit an antimagic orientation if there exist an
orientation on the edges of $G$ and a bijection from $E(G)$ to
$\\{1,2,\ldots,|E(G)|\\}$ such that the vertex sums of vertices are pairwise
distinct, where the vertex sum of a vertex is defined to be the sum of the
labels of the in-edges minus the that of the out-edges incident to the vertex.
It was conjectured by Hefetz, Mütze, and Schwartz [5] in 2010 that every
connected simple graph admits an antimagic orientation. In this note, we prove
that the Mycielski construction of any simple graph with at most one isolated
vertex admits an antimagic orientations, regardless of whether the original
graph admits an antimagic orientation.

Keywords: antimagic labeling, antimagic orientation, Euler circuit, Mycielski
construction.

##  1 Introduction

In this note, unless we particularly mention, all graphs are simple. Most of
the notation and terminology follow from [21]. For a graph $G$, an antimagic
labeling of $G$ is a bijection from the edge set $E(G)$ to the set
$\\{1,2,\ldots,|E(G)|\\}$ such that the vertex sums are pairwise distinct,
where the vertex sum of a vertex $v\in V(G)$ is defined as the sum of the
labels of all edges incident to $v$. It was conjectured by Hartsfied and
Ringel [8] in 90’s that every connected graph other than $K_{2}$ has an
antimagic labeling. The conjecture has been verified to be true for various
graphs [1, 4, 16, 3, 12, 10, 2], but is still widely open. For more details,
we refer the readers to the survey [6].

The study of antimagic labeling on directed graphs began much later. It was
initiated by Hefetz, Mütze, and Schwartz [5] in 2010. For a directed graph
$G$, the vertex sum of a vertex $v$ is defined as the sum of the labels of all
edges entering $v$ minus the sum of the labels of all edges leaving $v$. As
before, We say $G$ is antimagic if the vertex sums are pairwise distinct.
Hefetz et al. pointed out that the directed $C_{3}$ and $P_{3}$ are not
antimagic. On the other hand, it is not hard to show among all of the directed
graphs whose underlying graph is $P_{3}$ (or $C_{3}$), there exists a directed
graph that is antimagic. So they proposed two questions:

###### Question 1.1

Is every connected directed graph with at least 4 vertices antimagic?

###### Conjecture 1.2

Every connected graph admits an antimagic orientation.

While Question 1.1 seems to be very difficult, Conjecture 1.2 has been studied
intensively recently. We say a graph $G$ admits an antimagic orientation, if
there exists an orientation $D$ on $E(G)$ and a bijection $\tau$ from $E(G)$
to $\\{1,2\ldots,|E(G)|\\}$ such that the vertex sums $s_{(D,\tau)}(v)$,
defined earlier in this paragraph, of all vertices are pairwise distinct.

There are several families of graphs proved to satisfy Conjecture 1.2 by
different groups of researchers: $K_{n}$ with $n\geq 4$, $S_{n}$ with $n\geq
3$, $W_{n}$ with $n\geq 3$, the $(2d+1)$-regular graphs with $d\geq 0$ by
Hefetz et al [5]; the $2d$-regular graphs with $d\geq 2$ by Li, Song, Wang,
Yang, and Zhang [9], by Yang [22], and by Song and Hao [18]; the biregular
graphs by Shan and Yu [15]; the Halin graphs by Yu, Chang, and Zhou [24]; the
caterpillars by Lozano [11]; the lobsters by Gao and Shan [7]; and the
complete $k$-ary trees by Song and Hao [17]. In addition to the above special
graphs, Yang, Carson, Owens, Perry, Singgih, Song, Zhang, and Xhang [23],
proved that every connected graph with at least $n\geq 9$ vertices and maximum
degree at least $n-5$ admits an antimagic labeling, and Song, Yang, and Zhang
[19] proved that every graph $G$ with independent number at most 4 or least
$|V(G)|/2$ admits an antimagic orientation. Some of the above results are also
true for disconnected graphs.

Let $G$ be a graph with $V(G)=\\{v_{1},\ldots,v_{n}\\}$. The _Mycielski
construction_ of $G$, $M(G)$ is a new graph with
$V(M(G))=\\{v_{1},\ldots,v_{n}\\}\cup\\{u_{1},\ldots,u_{n}\\}\cup\\{w\\}$ and
$E(M(G))=E(G)\cup\\{v_{i}u_{j}\mid\mbox{ if }v_{i}v_{j}\in
E(G)\\}\cup\\{wu_{i}\mid 1\leq i\leq n\\}$. Moreover, we call
$\\{u_{1},\ldots,u_{n}\\}$ the image vertices of $V(G)$. An example is
$M(K_{2})=C_{5}$. This graph operation was introduced by Mycielski [13] to
construct the graphs having the large chromatic number but the small clique
number. In this note, we study the antimagic orientation problem for the
Mycielski construction of simple graphs. We manage to prove that the Mycielski
construction of any graph with at most one isolated vertex admits an antimagic
orientation. The proof of this result will be presented in the next section.

##  2 Main result

To prove our result, we need a technical lemma which is used frequently for
constructing the antimagic orientations in recent papers. A closed walk in a
multiple graph $G$ is an Euler circuit if it traverses every edge in $G$
exactly once. The following theorem is classical:

######  Theorem 2.1 (Euler, 1736)

A connected multiple graph has an Euler circuit if and only if the degree of
every vertex is even.

Using the Euler circuit, one can estimate the vertex sum of each vertex
induced by the circuit. Our first lemma below is almost the same as Lemma 2.1
in [9], Lemma 2.2 in [19], and Lemma 7 in [14]. For the sake of completeness,
we present the proof in the note.

###### Lemma 2.2

Let $p\geq 0$ be an integer, and $G$ be a graph with $|E(G)|=m\geq 1$. Then
there exist an orientation $D$ of $G$ and a bijection
$\tau:E(G)\rightarrow\\{p+1,p+2,\ldots,p+m\\}$ such that for any $v\in V(G)$,

$-(2m+p-2)\leq s_{(D,\tau)}\left(v\right)\leq p+m+(d_{G}(v)-1)/2.$

Proof. It suffices to prove the lemma for connected graphs. Let
$V(G)=\\{v_{1},v_{2},\ldots,v_{n}\\}$ and $V_{1}=\\{v\in V(G)\mid
d_{G}(v)\mbox{ is odd}\\}$. By the handshaking theorem, we see that $|V_{1}|$
is even. If $V_{1}\neq\varnothing$, we may assume, without loss of generality,
$V_{1}=\\{v_{1},v_{2},\ldots,v_{2t}\\}$. Define $G^{*}=G$ if $t=0$, else
$G^{*}$ is obtained by adding $t$ new edges
$e^{*}_{1},e^{*}_{2},\ldots,e^{*}_{t}$ to $G$, where $e^{*}_{i}$ is an edge
connecting $v_{i}$ and $v_{i+t}$. Thus, $d_{G^{*}}(v_{i})$ is even for every
vertex $v_{i}\in G^{*}$.

Pick an Euler circuit $C$ in $G^{*}$ as following:

$C:u_{1},e_{1},u_{2},e_{2},\ldots,u_{m+t},e_{m+t},u_{1},$

where the $u_{i}$’s are vertices in $V(G)$ may have repetition, and
$e_{1},e_{2},\ldots,e_{m+t}$ are exactly the edges in $E(G^{*})$. Let
$e_{i_{1}},e_{i_{2}},\ldots,e_{i_{m}}$ be the edges in $E(G)$. By orienting
each edge a direction the same as it is in $C$, we obtain an orientation $D$
of $G$. Observe that $|d^{+}_{D}(v_{i})-d^{-}_{D}(v_{i})|\leq 1$.

Define $\tau:E\rightarrow\\{p+1,p+2,\ldots,p+m\\}$ with
$\tau(e_{i_{j}})=p+m+1-j$ for all $1\leq j\leq m$. For any vertex $v_{i}$ with
$d^{+}_{D}(v_{i})=d^{-}_{D}(v_{i})=d_{G}(v_{i})/2$, either
$S_{(D,\tau)}(v_{i})=-m+d_{G}(v_{i})/2$ if $v_{i}$ is incident to both the
edges labeled with $p+m$ and $p+1$, or $S_{(D,\tau)}(v_{i})=d_{G}(v_{i})/2$
otherwise. For any vertex $v_{i}$ with $d^{+}_{D}(v_{i})-d^{-}_{D}(v_{i})=1$,
let $e$ be the edge leaving $v_{i}$, whose predecessor in $C$ is some
$e^{*}_{j}$. We have

$S_{(D,\tau)}(v_{i})=\left\\{\begin{array}[]{ll}-m+(d_{G}(v_{i})-1)/2-\tau(e),&v_{i}\mbox{
is incident to both the edges}\\\ &\mbox{labeled with $p+m$ and $p+1$ };\\\
(d_{G}(v_{i})-1)/2-\tau(e),&\mbox{otherwise}.\end{array}\right.$

Similarly, for any vertex $v_{i}$ with $d^{+}_{D}(v_{i})-d^{-}_{D}(v_{i})=-1$,
we have

$S_{(D,\tau)}(v_{i})=\left\\{\begin{array}[]{ll}-m+(d_{G}(v_{i})-1)/2+\tau(e),&v_{i}\mbox{
is incident to both the edges}\\\ &\mbox{labeled with $p+m$ and $p+1$ };\\\
(d_{G}(v_{i})-1)/2+\tau(e),&\mbox{otherwise},\end{array}\right.$

where $e$ is the edge entering $v_{i}$, whose successor in $C$ is some
$e^{*}_{j}$. Now the lower and upper bounds of the vertex sums are obtained
from the first and last cases of a vertex of odd degree. $\square$

Conventionally, we define the vertex sum of an isolated vertex to be zero.
Clearly, a graph with more than one isolated vertex does not admit an
antimagic orientation. The next is our main result in the note.

###### Theorem 2.3

Let $G$ be a graph with at most one isolate vertex. Then the Mycielski
construction of $G$, $M(G)$ admits an antimagic orientation.

Proof. First suppose that $G$ contains no isolated vertex. Let
$V(G)=\\{v_{1},v_{2},\ldots,v_{n}\\}$. Define
$V(M(G))=\\{v_{1},\ldots,v_{n}\\}\cup\\{u_{1},\ldots,u_{n}\\}\cup\\{w\\}$ and
$E(M(G))=E(G)\cup\\{v_{i}u_{j}\mid\mbox{ if }v_{i}v_{j}\in
E(G)\\}\cup\\{wu_{i}\mid 1\leq i\leq n\\}$. Let $|E(G)|=m$. Then
$|E(M(G))|=3m+n$. We partition the label set $\\{1,2,\ldots,3m+n\\}$ into
$\\{1,\ldots,m\\}$, $\\{m+1,\ldots,3m-n\\}$, $\\{3m-n+1,\ldots,3m\\}$, and
$\\{3m+1,\ldots,3m+n\\}$.

First apply Lemma 2.2 to obtain an orientation $D_{1}$ on $E(G)$ and $\tau$
from $E(G)$ to $\\{1,\ldots,m\\}$ so that $s_{(D_{1},\tau)}\left(v\right)\leq
m+(d_{G}(v_{i})-1)/2$ for each $v_{i}\in V(G)$. We extend the orientation
$D_{1}$ to $D$ on $E(M(G))$. Since every edge in $E(M(G))-E(G)$ is incident to
some image vertex $u_{i}$, we define the direction of it to be entering the
$u_{i}$’s. For each vertex $v_{i}$, we select an edge in $E(M(G))-E(G)$
incident to it as $e_{v_{i}}$. We arbitrary label the edges of the form
$v_{i}u_{j}$, except for the $e_{v_{i}}$’s, with labels in
$\\{m+1,\ldots,3m-n\\}$. Let $s^{\prime}_{(D,\tau)}(v_{i})$ be the partial
(lacking of the label of $e_{v_{i}}$) vertex sum of $v_{i}$. Without loss of
generality, we may assume

$s^{\prime}_{(D,\tau)}(v_{1})\leq s^{\prime}_{(D,\tau)}(v_{2})\leq\cdots\leq
s^{\prime}_{(D,\tau)}(v_{n}).$

Then we label $e_{v_{i}}$ with $3m+1-i$ for $1\leq i\leq n$. Since $e_{v_{i}}$
is leaving $v_{i}$, we have
$s_{(D,\tau)}(v_{i})=s^{\prime}_{(D,\tau)}(v_{i})-3m-1+i$ for $1\leq i\leq n$,
which implies $s_{(D,\tau)}(v_{i})\neq s_{(D,\tau)}(v_{j})$ for $i\neq j$.
Finally, for the image vertices $u_{1},u_{2},\ldots,u_{n}$, suppose we have
the partial (lacking of the label of $wu_{i}$) vertex sums satisfying the
following:

$s^{\prime}_{(D,\tau)}(u_{i_{1}})\leq
s^{\prime}_{(D,\tau)}(u_{i_{2}})\leq\cdots\leq
s^{\prime}_{(D,\tau)}(u_{i_{n}}).$

Then we label $wu_{i_{j}}$ with $3m+j$ for $1\leq j\leq n$. Since $wu_{i_{j}}$
is entering $u_{i_{j}}$, we have
$s_{(D,\tau)}(u_{i_{j}})=s^{\prime}_{(D,\tau)}(u_{i_{j}})+3m+j$ for $1\leq
j\leq n$, which implies $s_{(D,\tau)}(u_{i_{j}})\neq s_{(D,\tau)}(u_{i_{k}})$
for $j\neq k$.

Our orientation $D$ gives $s_{(D,\tau)}(u_{i})>0$ for $1\leq i\leq n$. Let us
see $s_{(D,\tau)}(v_{i})<0$ for $1\leq i\leq n$. By the fact that every edge
of the form $v_{i}u_{j}$ is leaving $v_{i}$ and by Lemma 2.2, we have

$\displaystyle s_{(D,\tau)}(v_{i})$ $\displaystyle\leq$ $\displaystyle
s_{(D_{1},\tau)}(v_{i})-d_{G}(v_{i})(m+1)$ $\displaystyle\leq$ $\displaystyle
m+(d_{G}(v_{i})-1)/2-d_{G}(v_{i})(m+1)$ $\displaystyle<$ $\displaystyle 0.$

It remains to show $s_{(D,\tau)}(v_{i})\neq s_{(D,\tau)}(w)$ for $1\leq i\leq
n$. For each $v_{i}$, there are $d_{G}(v_{i})$ edges leaving $v_{i}$ and
entering the image vertices, and one of them is $e_{v_{i}}$. The largest
possible sum of labels of these edge is
$(\sum_{j=1}^{d_{G}(v_{i})-1}3m-n+1-j)+3m$. By Lemma 2.2, the smallest
$s_{(D_{1},\tau)}(v_{i})$ is at least $-2m+2$. Thus,

$\displaystyle s_{(D,\tau)}(v_{i})$ $\displaystyle\geq$ $\displaystyle
s_{(D_{1},\tau)}(v_{i})-\left(\sum_{j=1}^{d_{G}(v_{i})-1}3m-n+1-j\right)-3m$
$\displaystyle\geq$
$\displaystyle-2m+2-\left(\sum_{j=1}^{n-2}3m-n+1-j\right)-3m$ $\displaystyle>$
$\displaystyle-\sum_{j=1}^{n}3m+j=s_{(D,\tau)}(w).$

Now if $G$ contains an isolated vertex, we first apply the previous arguments
to obtain an orientation $D$ and a bijection $\tau$ for $G-v$. It is clear
$V(M(G))=V(M(G-v))\cup\\{v,u\\}$, where $u$ is the image vertex of $v$, and
$E(M(G))=E(M(G-v)\cup\\{wu\\}$. Now let the direction of $wu$ be leaving $w$,
label $wu$ with the smallest $\tau(wu_{i})$ for all $u_{i}\in
V(M(G))-\\{u\\}$, and replace the label of $wu_{i}$ with $\tau(wu_{i})+1$ for
$u_{i}\in V(M(G))-\\{u\\}$. Clearly, all the vertex sums of the image vertices
are still distinct and positive, the vertex sums of every vertex in
$V(G)-\\{v\\}$ remains unchanged, and the vertex sum of $w$ decreases.
Therefore the proof is completed. $\square$

##  3 Conclusions

If $G$ contains exactly an isolated vertex, then one can show the independence
number of $M(G)$ is $(|V(M(G))|+1)/2$, and our result follows from the result
in [19]. However, the independence number of $M(G)$ is $(|V(M(G))|-1)/2$ in
general, and thus we cannot use the result in [19] to show Theorem 2.3.

In an early paper, Wang and Hsiao [20] studied the Cartesian product and the
lexicographic product of graphs, and proved that for some types of graphs the
above products of the graphs will produce graphs that are antimagic. In
addition to the Mycielski construction, it should be interesting to study
whether of not a graph operation can produce graphs that also admit an
antimagic orientation.

## References

  * [1] N. Alon, G. Kaplan, A. Lev, Y. Roditty, and R. Yuster, Dense graphs are antimagic, Journal of Graph Theory, 47 (2004), 297-309.
  * [2] F.-H. Chang, P. Chin, W.-T. Li. and Z. Pan, The strongly antimagic labelings of double spider, Indian Journal of Discrete Mathematics, 6 (2020), 43-68.
  * [3] F.-H. Chang, Y.-C. Liang, Z. Pan, and X. Zhu, Antimagic labeling of regular graphs, Journal of Graph Theory, 82 (2016), 339-349.
  * [4] D. W. Cranston, Y.-C. Liang, and X. Zhu, Regular graphs of odd degree are antimagic, Journal of Graph Theory, 80 (2015), 28-33.
  * [5] D. Hefetz, T. Mütze, and J. Schwartz, On antimagic directed graphs. Journal of Graph Theory, 64 (2010), 219–232.
  * [6] J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics (2019), #DS6
  * [7] Y. Gao and S. Shan, Antimagic orientation of lobsters, Discrete Applied Mathematics, 287 (2020), 21-26.
  * [8] N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, INC., Boston 1990.
  * [9] T. Li, Z.-X. Song, G. Wang , D. Yang, and C.-Q. Zhang, Antimagic orientations of even regular graphs. Journal of Graph Theory, 90 (2019), 46–53.
  * [10] Y.-C. Liang, T. Wong, and X. Zhu, Anti-magic labeling of trees, Discrete Mathematics, 331 (2014), 9-14.
  * [11] A. Lozano, Caterpillars have antimagic orientations, Analele Universitatii ”Ovidius” Constanta - Seria Matematica, 26(3) (2018), 171–180.
  * [12] A. Lozano, M. Mora, C. Seara, and J. Tey, Caterpillars are antimagic, Computer Science Appl. Math. Comput. (2017).
  * [13] J. Mycielski, Sur le coloriage des graphes, Colloquium Mathematicum 3 (1955), 161-162
  * [14] S. Shan, Antimagic orientation of graphs with minimum degree at least 33, arXiv:2005.01829.
  * [15] S. Shan and X. Yu, Antimagic orientation of biregular bipartite graphs, Electronic Journal of Combinatorics, 24(4) (2017), paper P4.31.
  * [16] J.-L. Shang, Spiders are antimagic, Ars Combinatoria, 118 (2015) 367-372.
  * [17] C. Song and R. X. Hao, Antimagic orientations for the complete $k$-ary trees, Journal Combinatorial Optimization, 38 (2019), 1077–1085.
  * [18] C. Song and R. X. Hao, Antimagic orientations of disconnected even regular graphs, Discrete Mathematics, 342(8) (2019), 2350-2355.
  * [19] Z.-X. Song, D. Yang, and F. Zhang, Antimagic orientations of graphs with given independence number, arXiv:1909.10928.
  * [20] T.-M. Wang and C.-C. Hsiao, On anti-magic labeling for graph products, Discrete Mathematics, 308 (2008), 3624-3633.
  * [21] D. B. West, Introduction to Graph Theory, 2nd edition. Pearson, 2001.
  * [22] D. Yang, A note on anti-magic orientation of even regular graphs, Discrete Applied Mathematics, 267 (2019), 224-228.
  * [23] D. Yang, J. Carlson, A. Owens, K. E. Perry, I. Singgih, Z.-X. Song, F. Zhang, and X. Xhang, Antimagic orientations of graph with large maximum degree, Discrete Mathematics, 343 (2020), 112-123.
  * [24] X. Yu, Y. Chang, and S. Zhou, Antimagic orientation of Halin graphs, Discrete Mathematics, 342 (2019), 3160-3165.


# GraMuFeN: Graph-based Multi-modal Fake News Detection in Social Media

Makan Kananian
Department of Computer Engineering
Sharif University of Technology,
International Campus
Kish Island, Iran
<EMAIL_ADDRESS>
& Faitma Badiei
Department of Computer Engineering
Sharif University of Technology,
International Campus
Kish Island, Iran
<EMAIL_ADDRESS>
& S. AmirAli Gh. Ghahramnai
Department of Computer Engineering
Sharif University of Technology
International Campus
Kish Island, Iran
<EMAIL_ADDRESS>

###### Abstract

The proliferation of social media platforms such as Twitter, Instagram, and
Weibo has significantly enhanced the dissemination of false information. This
phenomenon grants both individuals and governmental entities the ability to
shape public opinions, highlighting the need for deploying effective detection
methods. In this paper, we propose GraMuFeN, a model designed to detect fake
content by analyzing both the textual and image content of news. GraMuFeN
comprises two primary components: a text encoder and an image encoder. For
textual analysis, GraMuFeN treats each text as a graph and employs a Graph
Convolutional Neural Network (GCN) as the text encoder. Additionally, the pre-
trained ResNet-152, as a Convolutional Neural Network (CNN), has been utilized
as the image encoder. By integrating the outputs from these two encoders and
implementing a contrastive similarity loss function, GraMuFeN achieves
remarkable results. Extensive evaluations conducted on two publicly available
benchmark datasets for social media news indicate a 10 % increase in micro
F1-Score, signifying improvement over existing state-of-the-art models. These
findings underscore the effectiveness of combining GCN and CNN models for
detecting fake news in multi-modal data, all while minimizing the additional
computational burden imposed by model parameters.

_Keywords_ Fake news detection $\cdot$ Social media $\cdot$ Graph
Convolutional Networks $\cdot$ CNN $\cdot$ Graph Neural Networks $\cdot$ GCN
$\cdot$ ResNet-152 $\cdot$ Multi-modal $\cdot$ Text classification $\cdot$
Image analysis $\cdot$ F1-score.

##  1 Introduction

In the current landscape of information dissemination, the proliferation of
social media platforms, including Twitter, Instagram, Weibo, and others, has
resulted in an era of interconnectedness and instantaneous communication. This
digital revolution has not only facilitated global connectivity but has also
inadvertently given rise to a formidable challenge - the rapid propagation of
fake news Wessel et al., (2016).

In sharp contrast to the controlled environments of traditional media, social
media’s dynamic nature accelerates the spread of fake information, rendering
the dissemination of fake news more pervasive than ever before. This
concerning phenomenon has led to the potential manipulation of public
sentiments by both malicious entities and governmental actors Woolley and
Howard, (2018) Kalsnes, (2018), thereby compromising the authenticity of the
information.

Given this urgent situation, identifying and addressing fake news has become
an important concern for researchers, experts, and decision-makers. While
traditional methods for spotting fake news mostly focused on analyzing text,
the diverse aspects of social media now require new approaches that can
comprehend the complex interplay between text, images, and various forms of
media.

In our research, we tackle the challenge of detecting fake news in a multi-
modal context, where news articles comprise both textual and image-based
content. To address this complex problem, we propose the Graph-based Multi-
modal Fake News Detection model (GraMuFeN). Our approach involves the
integration of text and image encoders, utilizing the power of three distinct
neural network architectures: Long Short Term Memory (LSTM), Graph
Convolutional Neural Network (GCN), and Convolutional Neural Network (CNN).

Within the Text Encoder component, each textual input is treated as both
sequential data and a graph-like structure, enabling the combination of LSTM
and GCN networks. This enables us to capture complex relationships within the
textual content effectively. Simultaneously, the Image Encoder utilizes the
pre-trained ResNet-152 He et al., (2016) as the CNN model, enabling the
extraction of features from the visual elements contained within each news.

Subsequently, the feature vectors extracted by the Text and Image Encoders
were combined to predict the label associated with each news by minimizing the
cross entropy loss between the predictions and the ground-truth labels. At the
same time and by employing a contrastive similarity loss, the GraMuFeN is
trained to produce similar feature vectors for each text and its corresponding
image. This dual training process enhances the model’s ability to align text
and image representations effectively while simultaneously improving its
capacity for accurate label prediction. As a result, we achieved a 10 percent
enhancement in the F1-score micro for multi-modal fake news detection on well-
established benchmark datasets like Twitter Boididou et al., (2015).

The rest of this paper is organized as follows. In Section 2, we present a
brief review of the methods for spotting fake news in social media. We
formally define the problem that is going to be solved in this research in
Section 3. Section 4 provides the details related to the Text Encoder, the
Image Encoder, the contrastive similarity loss, and the classifier used in our
method. We describe the characteristics of the datasets used in this paper in
Section 5. Our extensive evaluations and their outcomes are summarized in
Section 6. Finally, we conclude the paper in Section 7.

##  2 Literature Review

Following the guidance of key studies Ruchansky et al., (2017) Shu et al.,
(2017), we describe ’fake news’ as stories that are intentionally made up and
can be proven wrong with real facts. Identifying fake news is not easy because
it involves analyzing different aspects of the story including the message it
is trying to convey, its place in society, and the images used. Research into
detecting fake news is a field that is constantly growing, with many different
methods being developed to tackle it.

In the early stages of fake news detection, most research focused on using
single-mode data, mainly text. Wu et al., used a special kind of system called
a graph kernel-based hybrid support vector machine to understand how news
spreads and to spot fake news Wu et al., (2015). Some researchers looked at
the way language was used in tweets to figure out if the news was fake Shu et
al., (2017), while others used structural and cognitive features to detect
fake news on social networks Kwon et al., (2013). But these early methods had
problems. They were only good for specific topics and relied too much on
manually picking out features, which could lead to biased results Shu et al.,
(2017); Kwon et al., (2013).

Later on, researchers started using other kinds of data like images from
social media to find fake news Ping Tian et al., (2013); Gupta et al., (2012).
The use of deep learning models also started, improving the accuracy of fake
news detection Jin et al., (2017). Steinebach et al., delve into the problem
of fake images accompanying fake news Steinebach et al., (2019). They
highlight that these manipulated images, often photo montages spliced from
several images, are designed to make the fake news appear more authentic. To
combat this, they developed a concept based on feature detection, indexing
these features using a nearest-neighbor algorithm. This allows for the rapid
comparison of a large number of images, identifying montages even if they have
been manipulated in various ways.

In the paper written by Meng et al. Meng et al., (2022), the authors propose a
proactive strategy to counteract fake news using traceable and authenticable
image tagging. This strategy leverages a Decoupled Invertible Neural Network
(DINN) to embed dual tags into news images before publication. However, the
approach is not without its limitations. One significant concern is its
generalization to unseen manipulations; the current setup might struggle to
handle new types of manipulations that were not included in the training
dataset. This indicates a potential area for further development to enhance
the robustness of the system against a broader array of manipulations. Even
though now we are moving towards using multiple types of data, these early
single-type data methods helped build the base for today’s more advanced fake
news detection techniques.

As research progressed, it became clear that using different types of data
together could make fake news detection more accurate. For example, some
systems were designed to answer questions using deep learning networks Xi et
al., (2020). Others combined text and images to better detect fake news
Farajtabar et al., (2017).

A big step forward was the creation of EANN Wang et al., (2018), Unlike
traditional methods that often rely on event-specific features, EANN is
designed to identify fake news across various events, making it adaptable to
new and unexpected news topics. It achieves this through a combination of
multi-modal feature extraction and an event discriminator, ensuring both
textual and visual content are analyzed while filtering out event-specific
biases. This approach enhances its versatility in the ever-changing landscape
of social media news. Following this, other powerful models like MVAE Khattar
et al., (2019) emerged. MVAE is a special model designed to understand and
analyze both these types of content together. It works by creating a combined
representation of the text and images in news articles. This combined
understanding helps MVAE decide if an article is fake or real. We should also
mention SpotFake Singhal et al., (2019), which is designed to detect fake news
by looking at both the words and pictures in an article. It uses transformers
like BERT for understanding the text and convolutional neural networks like
VGG-19 for analyzing images.

New methods kept coming, with CARMN Song et al., (2021). The special way that
the CARMN model uses to look at both words and pictures in news stories is
called "Crossmodal Attention Residual." This technique allows the model to
focus on how words and pictures relate to each other. Additionally, this model
uses "Multichannel convolutional neural Networks" to process and understand
the information deeply. In simpler terms, "Crossmodal Attention Residual"
helps CARMN pay special attention to the connections between text and images,
while "Multichannel convolutional neural Networks" allow it to analyze the
content from different perspectives or channels. This combination makes CARMN
effective at detecting fake news.

A significant advancement was AMFB Kumari and Ekbal, (2021) which stands out
because it understands how image and text intertwine. The heart of AMFB lies
in its "Attention-based multimodal Factorized Bilinear Pooling" technique.
This means that AMFB meticulously focuses on the most crucial parts of the
words and pictures, discerning which elements are vital for determining the
authenticity of the news. It adeptly processes multiple types of information
simultaneously, such as text and visuals. Furthermore, instead of merely
merging the information from words and pictures, AMFB employs a sophisticated
method of blending them, capturing their complex relationship. This approach
empowers AMFB to discriminate between real and fake news stories with high
accuracy.

Another model called FNR Ghorbanpour et al., (2023) was developed which
evaluates how closely the content of a news story (both text and images)
matches with known genuine news. By gauging this similarity, it gets an
initial sense of the story’s authenticity, FNR uses transformers (Bert and ViT
(vision transformer) to deeply analyze the content, capturing intricate
patterns.

In the paper written by Zhou et al.  Zhou et al., (2020), the authors
introduce the Similarity-Aware Fake news detection (SAFE) method. The
technique focuses on the identification of mismatches between text and images
in news content, utilizing a multi-modal neural network to analyze the
relationship between textual and visual elements and identify inconsistencies
that are indicative of fake news.

###  2.1 Comparison with Existing Methods

In this study, we present the GraMuFeN, a model that leverages the power of
Graph Convolutional Networks for text and ResNet-152 for images.

Table 1 compares our proposed model with existing state-of-the-art models,
including EANN, MVAE, SpotFake, CARMN, AMFB, and FNR, in terms of the type of
text encoder, image encoder and the fusion used to combine textual and image
features.

Table 1: Comparison of GraMuFeN with state-of-the-art models Method | Text Encoder | Image Encoder | Fusion Type
---|---|---|---
EANN Wang et al., (2018) | Text-CNN | VGG19 | Concat
MVAE Khattar et al., (2019) | BiLSTM | VGG19 | auto-encoder
SpotFake Singhal et al., (2019) | BERT | VGG19 | Concat
CARMN Song et al., (2021) |  Word level sentence embeddings | VGG19 |  Concat + Crossmodal attention
AMFB Kumari and Ekbal, (2021) |  Attention-based BiLSTM |  Attention-based CNN-RNN |  Element wise multiplication
FNR Ghorbanpour et al., (2023) | BERT |  Visual transformer (ViT) |  Concat + Similarity
GraMuFeN (Our Model) | GCN-LSTM |  CNN (resnet-152) |  Concat + Similarity

##  3 Problem statement

Both of our datasets (Twitter and Weibo) contain some text that has been
posted by a user with one or several images corresponding to that text. Each
text $T$ has an image $T_{I}$ and a label $T_{L}$. For the same events with
different texts, the image could be the same, especially in the Twitter
dataset. Our model aims to use $T$ and $T_{I}$ to predict $T_{L}$, whether it
is Fake or Real. In the next section, we delve deeper into our model.

##  4 Proposed Method

The architecture of GraMuFeN consists of three main components. The Text
Encoder, employs a sequence of multiple SageConv layers, generating embeddings
for the provided textual content. The second component, the Image Encoder,
employs the pre-trained ResNet-152 model to create embeddings corresponding to
the images accompanying each textual instance. Ultimately, the text and image
embeddings are combined and given to the third component, the classifier,
which undertakes the responsibility of predicting the label of the news item.
At the same time, the text and image embeddings are used to compute a
supervised contrastive similarity loss Khosla et al., (2020). In Figure 1, the
architecture of our model is presented. In the subsequent sections, we explain
each of the mentioned components in detail.

Figure 1: Architecture of GraMuFeN

###  4.1 The Text Encoder

The initial phase entails the transformation of each text (or sentence) into a
graph. For this purpose, each word within a sentence constitutes an individual
node, even instances of duplicate words. The establishment of connections
between nodes is achieved through the utilization of a fixed-size context
window. An example is provided in Figure 2. In this example, the context
window size is 2. This context window slides through the sentence, forging
links between nodes corresponding to words co-occurring within the window.
Thus, the underlying graph structure for each sentence is derived. Moreover,
to account for the requirements of GCN layers that will be applied later, we
add self-loops for each node of the resulting graph. This procedure is carried
out by the SentenceToGraph module, as shown in Figure 1. It is noteworthy,
however, that at this time the node’s features within the graph remain
unspecified.

Figure 2: Converting Text to Graph

Following the creation of a distinct graph corresponding to each text (or
sentence), the text is fed to an embedding layer. The embedding layer’s output
is then fed to an LSTM network creating hidden states for each word in the
sentence. At this time, these hidden states serve as the feature vectors for
the respective nodes in the corresponding graph. So far, we’ve successfully
transformed each sentence into a graph, where the words in the sentence become
nodes in the graph, and the LSTM-generated hidden states become the nodes’
features. Next, we input this graph into our GCN (Graph Convolutional Neural
Network) layers. Specifically, we utilize SageConv Hamilton et al., (2017) as
GCN layers in GraMuFeN. Our model consists of three GCN layers, each
incorporating a ReLU activation function to introduce non-linearity, enabling
us to capture greater complexity.

To aggregate information from the GCN layers, we employ Global Mean Pooling,
an aggregation technique that calculates the mean (average) of nodes’ feature
vectors for each graph, as follows:

$\text{Global Mean Pooling}(X)=\frac{1}{N}\sum_{i=1}^{N}X_{i}$ (1)

Here, $X$ represents the feature vectors of all nodes in a graph, $N$ is the
number of nodes in the graph, and $X_{i}$ represents the feature vector of the
$i$-th node. The result is a single feature vector representing the aggregated
information for each graph.

After obtaining the embeddings from the GCN, we employ a projection head
Ghorbanpour et al., (2023) to map these embeddings to a new representational
space:

$\begin{split}z_{text}=\omega_{2}\times(\textit{gelu}(\omega_{1}\times\textit{X}+b_{1}))+(\omega_{1}\times\textit{X}+b_{1})+b_{2}\end{split}$
(2)

Where $w_{1}$, $w_{2}$, $b_{1}$ and $b_{2}$ are weights and biases of linear
layers inside the text projector and X is the output of the Global Mean pool.
The structure of this projection head is illustrated in Figure 3. $z_{text}$
is the final embedding for texts with the shape of $(B,e_{text})$ where $B$ is
the batch size and $e_{text}$ is the size of the final embedding in the text
projection head.

Figure 3: Projection Head

###  4.2 Image Encoder

As the Image Encoder, we employ the pre-trained ResNet-152 network. This
network takes batches of images in the shape of (B, C, W, H) representing the
batch size, number of image channels, width, and height of input images.
ResNet-152 network projects the inputs into vectors of shape (B, 2048), where
B is the batch size and 2048 is the size of the last linear layer in
ResNet-152. Similar to the Text Encoder, we employ a projection head to map
the outputs of the Image Encoder into a new space:

$\begin{split}z_{img}=\omega_{4}\times(\textit{gelu}(\omega_{3}\times\textit{V}+b_{3}))+(\omega_{3}\times\textit{V}+b_{3})+b_{4}\end{split}$
(3)

Where $w_{3}$, $w_{4}$, $b_{3}$ and $b_{4}$ are weights and biases of linear
layers inside the image projector. The structure of the image projection head
is identical to that of the Text Encoder as illustrated in Figure 3. After
applying the image projector, we obtain $z_{img}$, which is the final
embedding for images with the shape of $(B,e_{img})$ where $B$ is the batch
size and $e_{img}$ is the size of the final embedding in the image projection
head.

###  4.3 Similarity Loss

To train the proposed model, we use a supervised contrastive similarity loss
Khosla et al., (2020). As explained previously, $z_{img}$ and $z_{text}$ are
the final embeddings for images and texts with the shape of $(B,e_{img})$ and
$(B,e_{text})$ for images and texts, where $e_{img}$ and $e_{text}$ are vector
size of image and text embedding respectively. To obtain the level of
similarity between these embeddings, their inner product is calculated:

$\begin{split}P=z_{text}z_{img}^{T}\end{split}$ (4)

Here, we consider $P$ as the prediction matrix with the shape of $(B,B)$.

The contrastive loss function tries to maximize the similarity of each text
and image to itself. To this aim and by means of inner product, we define the
expected matrix as the average similarity of text-to-text and image-to-image
based on the following formula Radford et al., (2021); Salama, (2021):

$\begin{split}E=\textit{softmax}(\frac{z_{img}z_{img}^{T}+z_{text}z_{text}^{T}}{2})\end{split}$
(5)

After calculating the $E$ matrix, we use cross-entropy to find the similarity
loss. The contrastive similarity loss is the average of the text similarity
loss $l_{text}$, and the image similarity loss $l_{img}$ Radford et al.,
(2021):

$\begin{split}l_{\text{text}}=-(E*\textit{log}(P)+(1-E)*\textit{log}(1-P))\end{split}$
(6)
$\begin{split}l_{img}=-(E^{T}*\textit{log}(P^{T})+(1-E^{T})*\textit{log}(1-P^{T}))\end{split}$
(7) $\begin{split}l_{s}=\frac{l_{text}+l_{img}}{2}\end{split}$ (8)

###  4.4 The Classifier

We concatenate the final text ($z_{text}$) and image ($z_{img})$ embeddings to
create the combined embedding vector as follows Ghorbanpour et al., (2023):

$\begin{split}z_{\text{combined}}=\text{Concat}(z_{\text{gcn}},z_{\text{img}})\end{split}$
(9)

This embedding vector is then passed through two linear layers for fake news
classification. After this linear mapping, we consider a vector of size (B, 2)
with two classes. According to Figure 4 and assuming $w_{6}$, $w_{5}$, $b_{4}$
and $b_{5}$ are weights and biases of linear layers inside the classifier, the
classifier output ($Z$) can be defined as follows:

$\begin{split}Z&=\text{softmax}\left(\omega_{6}\times\left(\text{gelu}\left(\omega_{5}\times
z_{\text{combined}}+b_{5}\right)+b_{6}\right)\right)\end{split}$ (10)

Figure 4: Classifier

After passing the vector through the softmax function, the classification loss
can be stated as follows:

$\begin{split}l_{c}=-(\textit{L}\times\textit{log}(Z)+(1-\textit{L})\times\textit{log}(1-Z))\end{split}$
(11)

The final total loss is obtained by combining the classification loss and the
similarity loss as follows:

$\begin{split}Loss=l_{c}+l_{s}\end{split}$ (12)

##  5 Datasets

In this section, we elaborate on datasets that we used to benchmark our model,
GraMuFeN. To this aim, we first begin with a description of the datasets used
in this study. Then we explain the pre-processing employed for handling text
and image data.

For the sake of evaluation, we choose two major datasets in this study:

  * •

Twitter dataset

  * •

Weibo dataset

###  5.1 Twiter Dataset

The Twitter dataset is from MediaEval Verifying Multimedia Use benchmark
Boididou et al., (2015), which is used for detecting fake content on Twitter.
This dataset has two parts: the train set and the test set. The tweets in this
dataset contain text content, attached images/videos, and additional social
context information. In this work, we focus on detecting fake news by
incorporating both text and image information. Thus, we remove the tweets
without any text or image. We translated the whole dataset into English
language using the Google Translate package For these two sets, there are no
overlapping events among them. We further split the training dataset into
training and validation sets, with the validation set comprising 20 percent of
our training dataset. The important characteristics of this dataset are
reported in Table 2

In previous studies, authors who worked with this Twitter dataset, like us,
performed several pre-processing steps, including the removal of links,
duplicates, and more. Upon examining the dataset, we discovered that there are
even more duplicates when tweets containing retweets (indicated by "RT" or
"rt" in their text) and mentions of other users (e.g., "@sample-user") are
removed. Previous works reported that the Twitter dataset comprises 12,237
tweets. However, after thorough pre-processing and the removal of all
duplicates, this number was reduced to 11,817 tweets.

Table 2: Twitter Dataset Information Dataset | Label | Count
---|---|---
Train | Fake | 6649
Real | 4599
Test | Fake | 545
Real | 444
All Data | 12237

We also analyzed the length of tweets in the Twitter dataset. The results are
illustrated in Figure 5. Notably, the length of tweets in the Twitter dataset
is shorter compared to the Weibo dataset with an average length of 10 words.

Figure 5: Tweet Length Distribution in Twitter Dataset

###  5.2 Weibo Dataset

This dataset Jin et al., (2017) was collected from Weibo social media from
2012 to 2016 and is written in the Chinese language. Each line of data in this
dataset also contains text, user information, and an image. The app’s
authentication system has tagged texts. This database is divided by Wang et
al., (2018) 111https://github.com/yaqingwang/EANN-KDD18 into three sets of
train, validation, and test data, as the news events of each set are
different. We utilized the same Datasets. We translated the whole dataset into
English language using the Google Translate package For these two sets. Only
the data with text and images are used. Moreover, in the Weibo dataset, each
text (tweet) was linked with a different number of images. In this study, we
considered only the first image linked to each text. Table 3 lists the number
of train/test data and fake/real data of the Weibo dataset.

Table 3: Weibo Dataset Information Dataset | Label | Count
---|---|---
Train | Fake | 3748
Real | 3758
Test | Fake | 999
Real | 995
All Data | 9500
Figure 6: Tweet length Distribution in Weibo Dataset

We also analyzed the length of tweets in the Weibo dataset. The results are
presented in Figure 6. As evident, the sequence length in the Weibo dataset is
significantly greater in comparison to the Twitter dataset.

##  6 Results

We utilized PyTorch version 2.0.1 with CUDA 11.8 (cu118) support and Torch
Geometrics for the development and training of our proposed model. The
experiments were conducted on the Google Colab Pro platform, leveraging a T4
GPU with approximately 15GB of VRAM, 16GB of RAM, and 30GB of storage.

###  6.1 Results in Twitter Dataset

For the Twitter dataset, we first pre-train the text encoder and image encoder
separately on texts and images, respectively. Then we combine the pre-trained
encoders resulting in the combined model and we start to finetune it. In the
following sections, we elaborate on these training procedures.

####  6.1.1 Pre-training Text Encoder

We trained the Text Encoder separately on the text data in the Twitter
dataset. We employed a pre-trained embedding layer, specifically the Google
News Word2Vec embedding Mikolov et al., (2013). This embedding layer has been
trained on extensive text corpora and provides embeddings for 3,000,000 words,
each consisting of a 300-dimensional feature vector. It’s worth noting that we
have frozen all the pre-trained embeddings. Consequently, this layer generates
a 300-dimensional feature vector for each word. The necessary information for
the pretraining procedure is reported in Table 4 .

Table 4: The Text Encoder Pretraining for Twitter Dataset
---
Initial learning rate |  3e-3
batch size |  256
Epoch |  30
Optimizer |  AdamW

Furthermore, we incorporated a learning rate scheduler, ReduceLROnPlateau,
with a factor of 0.8 and a patience parameter of 3. Additionally, to enhance
training stability, we implemented gradient clipping, capping the gradients at
a maximum norm of 1, We utilized Optuna for hyperparameter tuning.

The configuration of the Text Encoder is presented in Table 5.

Table 5: Twitter Text Encoder Configuration Text Encoder for Twitter
---
Layers | Configs | Activation Function
Embedding |  | n vocabulary: 3,000,000
---
output dim: 300
LSTM |  | input dim: 300
---
n Layers: 3
Dropout = 0.3
hidden dim: 256
Sageconv |  | input dim: 256
---
n layers: 3
L2-normalized
hidden dim: 512
Relu
Global Mean pool |  | input dim:512
---
output dim:512
Dropout | Percentage: 50% |
Linear (Classifier) |  | input dim: 512
---
n Layers: 1
output dim: 2

The performance of the pre-trained Text Encoder over the Twitter dataset is
presented in Table 6. In this table, the proposed Text Encoder is named
GCNLSTM, and the performance of other models is presented based on what is
reported in Ghorbanpour et al., (2023).

Table 6: Pre-trained Text Encoder Performance on Twitter Dataset |  | Fake | Real
---|---|---|---
Accuracy | F1-micro | Precision | Recall | F1-score | Precision | Recall | F1-score
Text |  | Logistic
---
Regression
0.62 | 0.62 | 0.69 | 0.55 | 0.61 | 0.56 | 0.70 | 0.62
SVM | 061 | 0.61 | 0.68 | 0.55 | 0.61 | 0.55 | 0.68 | 0.62
BiLSTM | 0.61 | 0.60 | 0.62 | 0.73 | 0.67 | 0.58 | 0.45 | 0.51
Bert | 0.69 | 0.64 | 0.67 | 0.68 | 0.68 | 0.60 | 0.59 | 0.59
| GCN
---
LSTM
0.70 | 0.69 | 0.70 | 0.69 | 0.69 | 0.69 | 0.70 | 0.70

####  6.1.2 Pre-training Image Encoder

We also pre-trained our Image Encoder, Resnet-152, on Twitter images. The
Resnet-152 has been trained on the Imagnet Dataset and has a powerful
understanding of image embeddings.

Like our Text Encoder, we used a learning rate scheduler ReduceOnPlateau with
a factor of 0.7 and a patience parameter of 5. The necessary information for
the pretraining procedure of the image encoder is reported in Table 7.

Table 7: The Image Encoder Pretraining for Twitter Dataset
---
Initial learning rate |  1e-4
batch size |  128
Epoch |  10
Optimizer |  AdamW
Weight Decay |  0.07

The performance of the pre-trained Image Encoder over the Twitter Image
dataset is presented in Table 8. In this table, our Image Encoder is named
ResNet-152.

Table 8: Pre-trained Image Encoder Performance on Twitter Dataset |  | Fake | Real
---|---|---|---
Accuracy | F1-micro | Precision | Recall | F1-score | Precision | Recall | F1-score
Image | CNN | 0.62 | 0.62 | 0.69 | 0.55 | 0.61 | 0.56 | 0.55 | 0.61
VGG19 | 0.68 | 0.68 | 0.74 | 0.64 | 0.69 | 0.62 | 0.72 | 0.67
ViT | 0.72 | 0.76 | 0.74 | 0.86 | 0.80 | 0.79 | 0.63 | 0.7
| Resnet
---
152
0.74 | 0.74 | 0.74 | 0.91 | 0.82 | 0.73 | 0.44 | 0.55

####  6.1.3 Multi-modal Results

In this section, we report results related to our final model (GraMuFeN) that
combines the pre-trained Text and Image encoders as what was illustrated
previously in Figure 1. It is important to note that the embedding layer of
the Text Encoder has been frozen during the training procedure. Also, we
incorporated a learning rate scheduler, ReduceLROnPlateau, with a factor of
0.9 and a patience parameter of 2. Additionally, to enhance training
stability, we implemented gradient clipping, capping the gradients at a
maximum norm of 1, plus Torch AMP Auto cast for mixed precision training. The
necessary information related to the training procedure of this model is
reported in Table 9.

Table 9: The Model Training Setting for Twitter Dataset
---
batch size |  64
Epochs |  30
Optimizer |  AdamW
Weight Decay |  0.07
Learning Rate for Text Encoder |  1e-5
Learning Rate For Text Projection Head |  5e-3
Learning Rate for Image Encoder |  1e-7
Learning Rate for Image Projection Head |  5e-3
Learning Rate for Classifier |  5e-3

In Figure 7 the learning curve of training and validation is reported.
Obviously, the training procedure is smooth and stable.

Figure 7: Training and Validation loss in Twitter Dataset

We report the performance of our model (GraMuFeN) in Table 10. Results related
to other models are presented based on reports in Ghorbanpour et al., (2023).

Regarding the findings presented in this table, the combination of the Text
and Image Encoders has led to a noteworthy enhancement in accuracy, precision,
recall, and the F1 score. Concerning accuracy, there is an observed
advancement of 19 percent and 15 percent when compared to the Text Encoder
(see Table 4) and the Image Encoder (see Table 8), respectively. The joint
utilization of Text and Image Encoders yielded an F1-micro score of 0.89,
whereas the F1-micro scores for the Text Encoder and Image Encoder were 0.69
and 0.74, respectively. A similar pattern emerges for precision, recall, and
the F1-Score pertaining to both fake and real samples within the dataset.

Table 10: Performance of the GraMuFeN on Twitter Dataset |  | Fake | Real
---|---|---|---
Accuracy | F1-micro | Precision | Recall | F1-score | Precision | Recall | F1-score
Multi Model | EANN | 0.69 | 0.69 | 0.75 | 0.58 | 0.65 | 0.62 | 0.76 | 0.69
MVAE | 0.67 | 0.67 | 0.70 | 0.69 | 0.69 | 0.63 | 0.64 | 0.63
SpotFake | 0.77 | 0.76 | 0.72 | 0.92 | 0.81 | 0.85 | 0.56 | 0.68
CARMN | 0.73 | 0.73 | 0.70 | 0.88 | 0.78 | 0.78 | 0.54 | 0.64
AMFD | 0.75 | 0.75 | 0.76 | 0.79 | 0.78 | 0.73 | 0.70 | 0.71
FNR-S | 0.79 | 0.79 | 0.78 | 0.85 | 0.82 | 0.79 | 0.71 | 0.75
GraMuFeN | 0.89 | 0.89 | 0.92 | 0.85 | 0.89 | 0.86 | 0.93 | 0.89

When compared with other models in Table 10, GraMuFeN outperforms them across
various performance metrics. Specifically, in regard to accuracy, our proposed
model achieves an accuracy of 0.89, which is notably 10 percent higher than
the most advanced state-of-the-art model (referred to as FNR-S). This same
trend is evident for the F1-micro score as well.

For instances of fake samples, GraMuFeN attains a precision of 0.92,
surpassing the leading model by a substantial 14 percent. However, in terms of
recall, the SpotFake model takes the lead. Conversely, when it comes to the
F1-Score, GraMuFeN again emerges as the frontrunner, showcasing an improvement
of over 7 percent in comparison to the best existing model.

For the category of real samples, a parallel pattern unfolds. Our model
enhances performance metrics compared to the prevailing state-of-the-art
models. Notably, GraMuFeN achieves increases of 7 percent, 17 percent, and 14
percent in terms of precision, recall, and F1-score, respectively, when
matched against the existing state-of-the-art models.

###  6.2 Results in Weibo Dataset

In the case of the Weibo dataset, we began by pre-training the image encoder.
In contrast to the Twitter dataset, we did not employ pre-trained embeddings
for our Text Encoder. Instead, we allowed the embedding layer in the Text
Encoder to train concurrently with our projection heads and classifier during
our multi-modal training phase. We will thoroughly discuss the reasons for
choosing this approach for the Weibo dataset. In the following sections, we
will provide detailed explanations of these training procedures.

####  6.2.1 Text Encoder Results

In our approach for the Weibo dataset, we deliberately chose not to use pre-
trained embeddings in the Text Encoder. The reason behind this decision is
that, unlike the Twitter dataset, the Weibo dataset contains longer (as
illustrated in Figure 6) and more meaningful sentences with a greater variety
of words. As such, we decided to train our own embedding layer on Weibo
dataset from scratch.

With the aim of showcasing the performance of the Text Encoder, we started to
train it on the Weibo text dataset. The configuration of the Text Encoder is
presented in Table 12, We utilized Optuna for hyperparameter tuning. Necessary
information for using our Text Encoder during training is provided in Table 11

Table 11: The Image Encoder Pretraining for Weibo Dataset
---
Initial learning rate |  5e-3
batch size |  32
Epoch |  50
Optimizer |  AdamW

Furthermore, we incorporated a learning rate scheduler, ReduceLROnPlateau,
with a factor of 0.8 and a patience parameter of 3. Additionally, to enhance
training stability, we implemented gradient clipping, capping the gradients at
a maximum norm of 1.

Table 12: Weibo Text Encoder Configuration Text Model Weibo
---
Layers | Configs | Activation Function
Embedding |  | n vocabulary: 8,991
---
output dim: 16
Dropout | Percentage: 50% |
LSTM |  | input dim: 16
---
n Layers: 2
hidden dim: 32
Sageconv |  | input dim: 32
---
n layers: 3
hidden dim: 32
Relu
Global Mean pool |  | input dim:32
---
output dim:32
Dropout | Percentage: 50% |
Linear (Classifier) |  | input dim: 32
---
n Layers: 1
output dim: 2

The performance of our Text Encoder over the Weibo dataset is presented in
Table 13. In this table, the proposed Text Encoder is named GCNLSTM.

Table 13: Text Encoder Performance on Weibo Dataset |  | Fake | Real
---|---|---|---
Accuracy | F1-micro | Precision | Recall | F1-score | Precision | Recall | F1-score
Text |  | Logistic
---
Regression
0.71 | 0.71 | 0.71 | 0.80 | 0.75 | 0.71 | 0.59 | 0.65
SVM | 0.70 | 0.70 | 0.72 | 0.74 | 0.73 | 0.67 | 0.65 | 0.66
BiLSTM | 0.66 | 0.66 | 0.62 | 0.78 | 0.69 | 0.73 | 0.55 | 0.63
Bert | 0.81 | 0.81 | 0.81 | 0.81 | 0.81 | 0.69 | 0.82 | 0.81
| GCN
---
LSTM
0.73 | 0.73 | 0.73 | 0.76 | 0.75 | 0.74 | 0.70 | 0.72

####  6.2.2 Pre-training Image Encoder

We did pre-train our Image Encoder, Resnet-152, on Weibo images. Like our Text
Encoder, we used a learning rate scheduler ReduceOnPlateau with a factor of
0.8 and a patience parameter of 5. Additionally, to enhance training
stability, we implemented gradient clipping, capping the gradients at a
maximum norm of 1. The necessary information for the pretraining procedure of
the image encoder is reported in Table 14.

Table 14: The Image Encoder Pretraining for Weibo Dataset
---
Initial learning rate |  1e-4
batch size |  64
Epoch |  30
Optimizer |  AdamW

The performance of the pre-trained Image Encoder over the Weibo Image dataset
is presented in Table 15. In this table, our Image Encoder is named
ResNet-152.

Table 15: Pre-trained Image Encoder Performance on Weibo Dataset |  | Fake | Real
---|---|---|---
Accuracy | F1-micro | Precision | Recall | F1-score | Precision | Recall | F1-score
Image | CNN | 0.52 | 0.50 | 0.79 | 0.24 | 0.38 | 0.58 | 0.55 | 0.63
VGG19 | 0.60 | 0.60 | 0.60 | 0.61 | 0.60 | 0.60 | 0.72 | 0.59
ViT | 0.68 | 0.68 | 0.67 | 0.69 | 0.68 | 0.79 | 0.68 | 0.67
| Resnet
---
152
0.71 | 0.71 | 0.72 | 0.72 | 0.72 | 0.70 | 0.70 | 0.70

####  6.2.3 Multi-modal Results

In this section, we report results related to our final model (GraMuFeN) that
combines the Text and pre-trained Image encoder as what was illustrated
previously in Figure 1. It is important to note that the text encoder is not
pre-trained and it will be trained from the ground up during this training
procedure. Also, we incorporated a learning rate scheduler, ReduceLROnPlateau,
with a factor of 0.9 and a patience parameter of 2 plus, Torch AMP Auto cast
for mixed precision training. Additionally, to enhance training stability, we
implemented gradient clipping, capping the gradients at a maximum norm of 1.
The necessary information related to the training procedure of this model is
reported in Table 16.

Table 16: The Model Training Setting for Weibo Dataset Batch size | 30
---|---
Epochs | 20
Optimizer | AdamW
Weight Decay for Image Encoder | 0.07
Weight Decay for Image Projection Head | 0.07
Learning Rate for Text Encoder | 5e-3
Learning Rate For Text Projection Head | 5e-3
Learning Rate for Image Encoder | 1e-7
Learning Rate for Image Projection Head | 5e-3
Learning Rate for Classifier | 5e-3

In Figure 8 the learning curve of training and validation is reported.
Obviously, the training procedure is smooth and stable. We report the
performance of our proposed model in Table 17.

Figure 8: Training and Validation loss in Weibo Dataset Table 17: Performance of the GraMuFeN on Weibo Dataset |  | Fake | Real
---|---|---|---
Accuracy | F1-micro | Precision | Recall | F1-score | Precision | Recall | F1-score
Multi Model | EANN | 0.81 | 0.81 | 0.89 | 0.66 | 0.76 | 0.77 | 0.93 | 0.85
MVAE | 0.79 | 0.79 | 0.89 | 0.65 | 0.75 | 0.74 | 0.93 | 0.82
SpotFake | 0.86 | 0.86 | 0.87 | 0.92 | 0.90 | 0.81 | 0.70 | 0.75
CARMN | 0.84 | 0.85 | 0.86 | 0.93 | 0.89 | 0.81 | 0.66 | 0.73
AMFD | 0.83 | 0.83 | 0.86 | 0.90 | 0.88 | 0.75 | 0.68 | 0.71
FNR-S | 0.87 | 0.87 | 0.87 | 0.89 | 0.88 | 0.88 | 0.87 | 0.88
GraMuFeN | 0.87 | 0.87 | 0.86 | 0.89 | 0.87 | 0.88 | 0.84 | 0.86

When compared to other models presented in Table 17, our proposed approach
outperforms all of them and has the same results as the FNR-S model, across
various performance metrics.

We attribute the primary challenge encountered during the training of our
model on the Weibo dataset to its length. Our proposed model can certainly
outperform most state-of-the-art models except FNR-S, where we achieved the
same results. There are several contributing factors. Notably, we did not
impose a specific word limit on the text, whereas in Ghorbanpour et al.,
(2023), texts were limited to 32 words on Twitter and 200 characters on Weibo.
Overall results are the same. The slight score difference where we fell short
could also be attributed to errors. The Weibo dataset is in the Chinese
language, and the translation method, as well as data pre-processing before
and after translation, can impact results significantly.

##  7 Conclusion and Future Works

In this paper, we introduced GraMuFeN, a model that leverages the capabilities
of Graph Convolutional Neural Networks (GCN) and Long Short-Term Memory (LSTM)
networks as the core of its text encoder, and Resnet-152 as its image encoder.
Graph neural networks have demonstrated exemplary performance across Natural
Language Processing (NLP) tasks. Our Vision was to illustrate the potential of
Graph Neural Networks in the domain of Fake News Detection. As evidenced by
our results, we achieved commendable outcomes with a model that is
considerably smaller compared to BERT and ViT. Our model was benchmarked using
two datasets, namely Twitter and Weibo, which have been utilized in previous
analogous works. As delineated in our results section 6, there is a noticeable
enhancement in the F1 score and the overall performance of the model. We
believe that there exists further scope for enhancement. For example,
incorporating more contextual features to the model, such as the current
event, the circle of followers and followings, the verification status of the
channel or user and clustering the events based on time and trends, may
further refine and improve the performance of the model.

## References

  * Boididou et al., [2015] Boididou, C., Andreadou, K., Papadopoulos, S., Dang-Nguyen, D.-T., Boato, G., Riegler, M., and Kompatsiaris, Y. (2015).  Verifying multimedia use at mediaeval 2015.  In MediaEval 2015 Multimedia Benchmark Workshop, volume 1436.
  * Farajtabar et al., [2017] Farajtabar, M., Yang, J., Ye, X., Xu, H., Trivedi, R., Khalil, E., Li, S., Song, L., and Zha, H. (2017).  Fake news mitigation via point process based intervention.  In International conference on machine learning, pages 1097–1106. PMLR.
  * Ghorbanpour et al., [2023] Ghorbanpour, F., Ramezani, M., Fazli, M. A., and Rabiee, H. R. (2023).  Fnr: a similarity and transformer-based approach to detect multi-modal fake news in social media.  Social Network Analysis and Mining, 13(1):56.
  * Gupta et al., [2012] Gupta, M., Zhao, P., and Han, J. (2012).  Evaluating event credibility on twitter.  In Proceedings of the 2012 SIAM international conference on data mining, pages 153–164. SIAM.
  * Hamilton et al., [2017] Hamilton, W., Ying, Z., and Leskovec, J. (2017).  Inductive representation learning on large graphs.  Advances in neural information processing systems, 30.
  * He et al., [2016] He, K., Zhang, X., Ren, S., and Sun, J. (2016).  Deep residual learning for image recognition.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778.
  * Jin et al., [2017] Jin, Z., Cao, J., Guo, H., Zhang, Y., and Luo, J. (2017).  Multimodal fusion with recurrent neural networks for rumor detection on microblogs.  In Proceedings of the 25th ACM international conference on Multimedia, pages 795–816.
  * Kalsnes, [2018] Kalsnes, B. (2018).  Fake news.  In Oxford Research Encyclopedia of Communication.
  * Khattar et al., [2019] Khattar, D., Goud, J. S., Gupta, M., and Varma, V. (2019).  Mvae: Multimodal variational autoencoder for fake news detection.  In The world wide web conference, pages 2915–2921.
  * Khosla et al., [2020] Khosla, P., Teterwak, P., Wang, C., Sarna, A., Tian, Y., Isola, P., Maschinot, A., Liu, C., and Krishnan, D. (2020).  Supervised contrastive learning.  Advances in neural information processing systems, 33:18661–18673.
  * Kumari and Ekbal, [2021] Kumari, R. and Ekbal, A. (2021).  Amfb: Attention based multimodal factorized bilinear pooling for multimodal fake news detection.  Expert Systems with Applications, 184:115412.
  * Kwon et al., [2013] Kwon, S., Cha, M., Jung, K., Chen, W., and Wang, Y. (2013).  Prominent features of rumor propagation in online social media.  In 2013 IEEE 13th international conference on data mining, pages 1103–1108. IEEE.
  * Meng et al., [2022] Meng, R., Zhou, Z., Cui, Q., Lam, K.-Y., and Kot, A. (2022).  Traceable and authenticable image tagging for fake news detection.
  * Mikolov et al., [2013] Mikolov, T., Chen, K., Corrado, G., and Dean, J. (2013).  Efficient estimation of word representations in vector space.  arXiv preprint arXiv:1301.3781.
  * Ping Tian et al., [2013] Ping Tian, D. et al. (2013).  A review on image feature extraction and representation techniques.  International Journal of Multimedia and Ubiquitous Engineering, 8(4):385–396.
  * Radford et al., [2021] Radford, A., Kim, J. W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., Krueger, G., and Sutskever, I. (2021).  Learning transferable visual models from natural language supervision.
  * Ruchansky et al., [2017] Ruchansky, N., Seo, S., and Liu, Y. (2017).  Csi: A hybrid deep model for fake news detection.  In Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, pages 797–806.
  * Salama, [2021] Salama, K. (2021).  Keras documentation: Natural language image search with a dual encoder.
  * Shu et al., [2017] Shu, K., Sliva, A., Wang, S., Tang, J., and Liu, H. (2017).  Fake news detection on social media: A data mining perspective.  ACM SIGKDD explorations newsletter, 19(1):22–36.
  * Singhal et al., [2019] Singhal, S., Shah, R. R., Chakraborty, T., Kumaraguru, P., and Satoh, S. (2019).  Spotfake: A multi-modal framework for fake news detection.  In 2019 IEEE fifth international conference on multimedia big data (BigMM), pages 39–47. IEEE.
  * Song et al., [2021] Song, C., Ning, N., Zhang, Y., and Wu, B. (2021).  A multimodal fake news detection model based on crossmodal attention residual and multichannel convolutional neural networks.  Information Processing & Management, 58(1):102437.
  * Steinebach et al., [2019] Steinebach, M., Gotkowski, K., and Liu, H. (2019).  Fake news detection by image montage recognition.  In Proceedings of the 14th International Conference on Availability, Reliability and Security, pages 1–9.
  * Wang et al., [2018] Wang, Y., Ma, F., Jin, Z., Yuan, Y., Xun, G., Jha, K., Su, L., and Gao, J. (2018).  Eann: Event adversarial neural networks for multi-modal fake news detection.  In Proceedings of the 24th acm sigkdd international conference on knowledge discovery & data mining, pages 849–857.
  * Wessel et al., [2016] Wessel, M., Thies, F., and Benlian, A. (2016).  The emergence and effects of fake social information: Evidence from crowdfunding.  Decision Support Systems, 90:75–85.
  * Woolley and Howard, [2018] Woolley, S. C. and Howard, P. N. (2018).  Computational propaganda: Political parties, politicians, and political manipulation on social media.  Oxford University Press.
  * Wu et al., [2015] Wu, K., Yang, S., and Zhu, K. Q. (2015).  False rumors detection on sina weibo by propagation structures.  In 2015 IEEE 31st international conference on data engineering, pages 651–662. IEEE.
  * Xi et al., [2020] Xi, Y., Zhang, Y., Ding, S., and Wan, S. (2020).  Visual question answering model based on visual relationship detection.  Signal Processing: Image Communication, 80:115648.
  * Zhou et al., [2020] Zhou, X., Wu, J., and Zafarani, R. (2020).  Safe: Similarity-aware multi-modal fake news detection.


[1]Jerzy Pryga

# Analysis of capability of detection of extensive air showers by simple
scintillator detectors

Weronika Stanek

###### Abstract

One of the main objectives of the CREDO project is to search for so-called
Cosmic-Ray Ensembles (CRE) [1]. To confirm the existence of such phenomena a
massive scale observation of even relatively low energy Extensive Air Showers
(EAS) and an analysis of their correlations in time must be performed. To make
such observations possible, an infrastructure of widely spread detectors
connected in a global network should be developed using low-cost devices
capable of collecting data for a long period of time. For each of these
detectors or small detector systems the probability of detection of an EAS has
to be determined. Such information is crucial in the analysis and
interpretation of collected data. In the case of large number of systems with
different properties the standard approach based on detailed and extensive
simulations is not possible, thus a faster method is developed. Knowing the
characteristics of EAS from more general simulations any required probability
is calculated using appropriate parameterization taking into account EAS
spectrum, energy dependence of particle density and zenith angle dependence.
This allows to estimate expected number of EAS events measured by a set of
small detectors [2] and compare results of calculations with these
measurements.

##  1 Introduction

The main motivation for this work is to provide helpful information about one
of methods of detection of Extensive Air Showers (EAS) proposed for Cosmic
Rays Extremely Distributed Observatory (CREDO) [1]. As the goal of this
project, which is a search of cosmic rays ensembles, require monitoring of
cosmic rays and EAS on a global scale, demand for a simple, cheap and reliable
detector rises. One of proposed ideas is analysed in this work: several
scintillator detectors connected in a coincidence system. When several devices
give signal at almost the same time it indicates an occurrence of a cosmic ray
cascade. Performed analysis aims to give answer if such very simple systems
can be reliable and efficient enough to use in a bigger network of devices and
collect useful data.

##  2 Analysis

###  2.1 Simulations

First step in the analysis was to perform shower simulations using CORSIKA
software [3]. These data is later analysed to characterise density of
particles form the shower on the ground as a function of cascade parameters.
Currently simulations of protons as a primary particle are used, with 18
different energies starting from 1 TeV up to 4000 TeV. For now, no heavier
nuclei were included but it is planned. For each energy 1000 to 20 000
cascades were simulated. To examine angular distribution of produced
particles, simulations of cascades at several selected zenith angles with
energy of 100 TeV were analysed.

###  2.2 Analysed system

Systems which are analysed in this work consists of several flat scintillator
detectors connected in a coincidence. It is assumed that particles can be
detected from all directions and different types of them can not be
distinguished. Time interval in which two or more signals from different
devices are considered a coincidence is called here coincidence time $\delta
T$. Other properties of considered detectors are their area of the surface
$A$, efficiency $\eta$ and frequency of non cosmic background signals
$f_{bg}$. All devices in the system are identical and are placed very close to
each other. With those properties it is possible to analyse such systems in
easy and relatively realistic way.

###  2.3 Background

Starting point in the analysis is to evaluate flux of background particles,
and probability of fake signals i.e. not caused by EAS. In this work, term
background means uncorrelated, single cosmic-rays which comes from all
direction with constant flux indicated as $I_{bg}$.

For short measurement time $\delta T$ the probability of a fake signal to
happen is approximately:

$P_{bg}=1-\exp\left(-\delta T\left(\eta\cdot A\cdot
I_{bg}+f_{bg}\right)\right)$ (1)

After that, chance of $k$ coincidence signals is computed for system made of
$n$ detectors. With probability of detection for each device equal $P_{bg}$ it
is denoted as $Q\left(n,k,P_{bg}\right)$ and given by binomial distribution.
Expected number of such events in the time of measurement $T$ is computed
using following formula:

$\langle
N_{bg}\left(k\right)\rangle=Q\left(n,k,P_{bg}\right)\cdot\frac{T}{\delta T}$
(2)

###  2.4 Signals from the cascades

Before starting any fitting or calculations some assumptions about EAS has to
be taken. All showers are treated as circularly symmetrical and produced by
protons. Because protons constitute about 78 % of primary cosmic rays, the
spectrum for all cosmic-ray particles was used. Frequency of those protons is
constant in time and indicated here as $j\left(E\right)$ [4]. This function is
obviously a model confirmed by various experiments [5].

The next step is to extract information from simulations. The easiest way to
characterise EAS is by is particles density $\rho$. In this work it is a
function of parameters such as energy of primary particle $E$, distance from
its centre $r$, zenith angle of primary particle approach direction $\theta$
and total number of produced particles $N_{part}$. Each of these quantities
were analysed more or less exhaustively and pre-defined functions were fitted
to data.

Final $\rho$ function is a product of all this factors given by following
formula:

$\rho\left(r,\theta,E,N_{part}\right)=\rho_{norm}\left(r\right)\cdot
F_{\theta}\left(\theta\right)\cdot F_{E}\left(E,r\right)\cdot
F_{N}\left(N_{part},r\right)$ (3)

  * •

$\rho_{norm}\left(r\right)$ \- The most important factor which represents
relation between particle density and distance from the centre of the shower.
This function is fitted for certain vertical cascade of chosen energy to which
other factors are normalised. Figure 1 presents shape of this function.

(a)

(b)

Figure 1: $\rho\left(r\right)$ function fitted to data for 1(a) muons and 1(b)
photons.

  * •

$F_{E}\left(E,r\right)$ \- This factor scales the density with different
energy and modifies its distance distribution. It is presented on figure 2.

(a)

(b)

Figure 2: Graph of $F_{E}\left(E,r\right)$ for 2(a) muons and 2(b) photons.

  * •

$F_{N}\left(N_{part},r\right)$ \- Scales the density and its distance
distribution as total number of produced particles changes. Their amount is
strongly connected with the altitude at which the cascade started to form, but
$N_{part}$ is a more convenient parameter to work with. It must be mentioned
that their form vary for different particle types and currently for
electromagnetic component of the showers $F_{N}$ factor is just a linear
proportion of $\frac{N}{N_{avr}}$ without dependence on distance $r$. It is
presented in Figure 3.

(a)

(b)

Figure 3: Histograms of $F_{N}\left(N_{part},r\right)$ for 3(a) muons and 3(b)
photons.

  * •

$F_{\theta}\left(\theta\right)$ \- This factor scales the density as zenith
angle of primary particle approach direction changes. Here a simplification
was made. Namely, the shape of shower footprint is not circularly symmetrical
as this angle increase. However, EAS comes from all directions with equal
probability thus this effect can be neglected. Figure 4 presents its shape.

(a)

(b)

Figure 4: $F_{\theta}\left(\theta\right)$ function fitted to data for 4(a)
muons and 4(b) photons.

Next step is to calculate probability of a signal to happen when there is a
cascade near the detector. Formula for a single signal probability is as
follows:

$P=1-\exp\left(-\eta\cdot A\cdot\rho\left(r,\theta,E,N_{part}\right)\right)$
(4)

There is also another effect in the cascades that could have impact on the
probability of coincidence signals. It is currently under investigation, but
early results suggest that when one particle that originates from the shower
is detected, then probability of finding another one in close region of such
event should increase be higher than it results from particles density
function. This "clustering" effect could be most likely explained by fact,
that particles in the shower are produced in at least pairs. Thus, when
originating not very high in the atmosphere they should hit the ground close
to each other as they may not have time to travel further. In this work this
results are included in calculations. The function which scale the probability
of multiple signals is a function of particles density and it impacts signals
of 2 and more coincidences.

To evaluate expected number of events caused by EAS an integration over all
energies of primary particles, area around the devices and spherical angle of
the sky must be performed. Obviously, assumption that cascades are circularly
symmetrical reduces integral over horizontal angle to the factor of $2\pi r$.
Formula below shows how it is computed:

$\langle
N\left(k\right)\rangle=\int_{0}^{r_{max}}\int_{E_{min}}^{E_{max}}\int_{0}^{\frac{\pi}{2}}Q\left(n,k,P\right)2\pi
r\ j\left(E\right)T\,d\Omega\,dE\,dr$ (5)

During numerical integration the number of produced particles in cascades was
chosen randomly according to distributions found earlier. Examples of those
are presented in figure 5.

(a)

(b)

Figure 5: Normalised distribution of produced particles for 5(a) muons with
fitted Gaussian distribution function and 5(b) photons with fitted Landau
distribution function. Both plots depicts vertical showers with primary energy
$E=100\ TeV$.

Of course the integration must be performed in certain limits, thus the
primary cosmic-rays energy range and maximal distance from the shower must be
chosen. Integration up to $\frac{\pi}{2}$ over zenith angle is obvious as it
was assumed that particles from all directions can be detected. Criteria of
choice of the maximal distance may be different and in this work three were
considered. First one was a radius in which $95\%$ percent of particles
produced by the cascade are included $R_{prc}$. Another one is a radius in
which shower particles density is greater than the average background
$R_{rho}$. Last one was just a fixed value $R_{max}$, chosen in a way that
cuts off the region in which particles density is relatively low. First two of
those quantities were analysed and characterised as a function of energy of
primary particle.

##  3 Comparison with simpler method

To check if the performed analysis was done right another approach to the
problem was tested. As muons have been studied for many years, an approximate
function for its density on the ground level can be found in the literature
[5]. It depends on total number of produced muons, which is a function of the
energy of primary particle, and distance from the centre of the shower. This
was also used for electromagnetic component of the cascades, however this
extrapolation is very rough as electrons and photons in EAS behave very
differently from muons. Thus, used formula is as follows:

$\rho\left(r,\theta,N_{part}\right)=F_{\theta}\left(\theta\right)\cdot\frac{1.25N_{part}(E)}{2\pi\Gamma\left(1.25\right)}\left(\frac{1}{320}\right)^{1.25}r^{-0.75}\left(1+\frac{r}{320}\right)^{-2.5}$
(6)

Where $F_{\theta}$ and further steps in the analysis are the same as described
in 2.4. This comparison should help to judge if analysis described earlier,
which is more general and gives more information about the behaviour of the
system, is reasonable and does not consist any unrealistic assumptions.

##  4 Results

Finally, in this sections results of above analysis calculated for exemplary
system are presented. Assumed properties of considered system are as follows.
It is a system built of 4 devices, namely Cosmic Watches [6] which surface
area is $25\ cm^{2}$ for each one. Coincidence time is set to $200\ ns$ and
efficiency of every device is $95\%$. Particles that gives signal are muons,
electrons, positrons and photons which energy is greater than $0.3\ GeV$. The
choice of above parameters are not arbitrary. They are meant to be the same as
in the system tested by prof. Tadeusz Wibig in his work [2], thus the time of
measurement has to be taken equal to this in the performed experiment which is
$7$ days. However, there are some uncertainties like which particles really
gives signal as well as efficiency and $f_{bg}$, here chosen arbitrary.

Assumptions described earlier imply that the flux of background particles
should be around $163\ particles\ m^{-2}\ s^{-1}$. Frequency of non cosmic
background noise was chosen to be $0.1\ Hz$.

To perform final results also integration limits had to be chosen. Reasonable
choice of energy limits was range between $1\ TeV$ and $10^{7}\ TeV$, as less
energetic particles almost does not produce cascades and more energetic are
very unlikely to occur in such measurement time. Maximal distance from the
system in which showers were considered was $R_{prc}$ described in 2.4.

Because chronically first assumptions was that only muons should give signal
also this case is presented here to show the difference in final results that
yields from such changes in assumptions. After carrying all steps of the
analysis the results are as follows:

| Background | Analysis | Simpler method | Measurement
---|---|---|---|---
| Only $\mu$ |  $\mu$, $e^{\pm}$, $\gamma$ | Only $\mu$ |  $\mu$, $e^{\pm}$, $\gamma$ | Only $\mu$ |  $\mu$, $e^{\pm}$, $\gamma$ |
$k$ | $\langle N(k)\rangle_{bg}$ | $\langle N(k)\rangle_{bg}$ | $\langle N(k)\rangle$ | $\langle N(k)\rangle$ | $\langle N(k)\rangle$ | $\langle N(k)\rangle$ | $N_{data}(k)$
1 | $860000$ | $1.17\cdot 10^{6}$ | $41000$ | $64000$ | $59000$ | $178000$ | -
2 | $0.092$ | $0.17$ | $0.179$ | $426$ | $0.213$ | $63$ | 94
3 | $\sim 10^{-9}$ | $\sim 10^{-8}$ | $0.0182$ | $31$ | $0.003$ | $0.406$ | 2
4 | $\sim 10^{-17}$ | $\sim 10^{-16}$ | $0.0068$ | $21$ | $0.0006$ | $0.143$ | 1
Table 1: Results of the calculations for cascades and background signals
compared with measurements [2].

As presented in table 1, measured number of events lay somewhere between
results of calculations which considered only muons and those that included
also electromagnetic component. Additional information about the measurement
is that it was held on the windowsill, which when assuming that
electromagnetic component of EAS can pass only through the window could be
taken into account in calculations. Considering that only $30\%$ of cascades
for which $\theta\geq 15^{o}$ contribute $e^{\pm}$ and $\gamma$ to density,
the results are as follows:

$k$ | $\langle N(k)\rangle_{\mu}$ | $\langle N(k)\rangle_{\mu,e,\gamma}$ | $\langle N(k)\rangle_{window}$ | $N_{data}(k)$
---|---|---|---|---
2 | $0.179$ | $426$ | $99.7$ | $94$
3 | $0.0182$ | $31$ | $5.9$ | $2$
4 | $0.0068$ | $21$ | $3.9$ | $1$

Table 2: Results of the calculations for cascades and background signals
compared with measurements [2]

.

##  5 Summary

There are several important conclusions that can be made from presented
results. As expected the average number of coincidence signals caused by EAS
is significantly higher than for the background. Thus, the level of confidence
that certain event indicates occurrence of the cascade in a close surrounding
of the system should be high. Assuming that only muons give signal is clearly
wrong, however not all electrons and photons should be able do it as well. It
must be clarified at which energy particles of each type give signal. Possibly
the electromagnetic component of the showers is blocked in the structure of
the building but more measurements in different conditions have to be
performed. There are still many significant assumptions and simplifications
like treating all showers as originating from protons while heavier nuclei
should contribute differently to the results. Background and effects like
particles production in the upper layers of the building are also poorly
understood and require more study. Therefore, one should treat presented
results with caution.

According to this analysis, this method of detection could even provide some
clues about primary energy of detected shower, which is easy to see when the
expected number of events for different number of coincidence signals is
plotted over energy spectrum:

(a)

(b)

Figure 6: Expected number of coincidence signals for different energies for
6(a) simpler method and 6(b) analysis.

Future development of this work will include more groups of heavier primary
nuclei and different energy thresholds of individual components of the showers
hopefully compared with more measurements.

## References

  * [1] P. Homola, D. Beznosko, G. Bhatta, Ł. Bibrzycki, M. Borczyńska, Ł. Bratek et al., _Cosmic-ray extremely distributed observatory_ , _Symmetry_ 12 (2020) 1835.
  * [2] M. Karbowiak, T. Wibig, D. Alvarez-Castillo, D. Beznosko, A.R. Duffy, D. Góra et al., _The first CREDO registration of extensive air shower_ , _Physics Education_ 55 (2020) 055021.
  * [3] D. Heck, G. Schatz, J. Knapp, T. Thouw and J. Capdevielle, _Corsika: A monte carlo code to simulate extensive air showers_ , Tech. Rep. (1998).
  * [4] M. Aartsen, R. Abbasi, Y. Abdou, M. Ackermann, J. Adams, J. Aguilar et al., _Measurement of the cosmic ray energy spectrum with icetop-73_ , _Physical Review D_ 88 (2013) 042004.
  * [5] P.D. Group et al., _Review of particle physics_ , _The European Physical Journal C-Particles and Fields_ 3 (1998) 1.
  * [6] S.N. Axani, K. Frankiewicz and J.M. Conrad, _The cosmicwatch desktop muon detector: a self-contained, pocket sized particle detector_ , _Journal of Instrumentation_ 13 (2018) P03019.

## Full Authors List: CREDO Collaboration

Jerzy Pryga1, Weronika Stanek2, Piotr Homola3, Tadeusz Wibig19, Oleksandr
Sushchov3, Peter Kovacs8, Vahab Nazari17,3, Jilberto Zamora-Saa20, Dmitriy
Beznosko3, Mikhail V. Medvedev10,11, Jarosław Stasielak3, Bartosz Łozowski9,
Nikolai Budnev4, Alok C. Gupta5, Arman Tursunov18, Karel Smolek16, Bohdan
Hnatyk6, David E. Alvarez Castillo3,17, Katarzyna Smelcerz12, Alona Mozgova6,
Michał Niedźwiecki12, Marcin Kasztelan7, Matías Rosas15, Krzysztof Rzecki2,
Sławomir Stuglik3.

1Jagiellonian University, Gołębia 24, 31-007 Kraków.
2AGH University of Science and Technology, Mickiewicz Ave., 30-059 Kraków,
Poland.
3Institute of Nuclear Physics Polish Academy of Sciences, Radzikowskiego 152,
31-342 Kraków, Poland.
4Irkutsk State University, Russia.
5Aryabhatta Research Institue of Observational Sciences (ARIES), Manora Peak,
Nainital 263001, India.
6Astronomical Observatory of Taras Shevchenko National University of Kyiv,
04053 Kyiv, Ukraine.
7National Centre for Nuclear Research, Andrzeja Soltana 7, 05-400 Otwock-
Świerk, Poland.
8Institute for Particle and Nuclear Physics, Wigner Research Centre for
Physics, 1121 Budapest, Konkoly-Thege Miklós út 29-33, Hungary.
9Faculty of Natural Sciences, University of Silesia in Katowice, Bankowa 9,
40-007 Katowice, Poland.
10Department of Physics and Astronomy, University of Kansas, Lawrence, KS
66045, USA.
11Laboratory for Nuclear Science, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA.
12Department of Computer Science, Faculty of Computer Science and
Telecommunications, Cracow University of Technology, Warszawska 24, 31-155
Kraków, Poland.
13ACC Cyfronet AGH-UST, 30-950 Kraków, Poland.
14Clayton State University, Morrow, Georgia, USA.
15Liceo 6 Francisco Bauzá, Montevideo, Uruguay.
16Institute of Experimental and Applied Physics, Czech Technical University in
Prague.
17Joint Institute for Nuclear Research, Dubna, 141980 Russia.
18Research Centre for Theoretical Physics and Astrophysics, Institute of
Physics, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava,
Czech Republic.
19University of Łódź, Faculty of Physics and Applied Informatics, 90-236 Łódź,
Pomorska 149/153, Poland.
20 Universidad Andres Bello, Departamento de Ciencias Fisicas, Facultad de
Ciencias Exactas, Avenida Republica 498, Santiago, Chile.