# Blume-Emery-Griffiths dynamics in social networks

Yao-Hui Yang  Department of Mathematics and Physics, Chongqing University of
Science and Technology, Chongqing $401331$, China

###### Abstract

We introduce the Blume-Emery-Griffiths (BEG) model in a social networks to
describe the three-state dynamics of opinion formation. It shows that the
probability distribution function of the time series of opinion is a Gaussian-
like distribution. We also study the response of BEG model to the external
periodic perturbation. One can observe that both the interior thermo-noise and
the external field result in phase transition, which is a split phenomena of
the opinion distributions. It is opposite between the effect acted on the
opinion systems of the amplitude of the external field and of the thermo-
noise.

###### pacs:

02.50.-r, 87.23.Ge, 89.75.-k, 05.45.-a,

##  I INTRODUCTION

Over the last few years, the study of opinion formation in complex networks
has attracted a growing amount of works and becomes the major trend of
sociophysics Intro-1 . Many models have been proposed, like those of Deffuant
Intro-2 , Galam Intro-3 , Krause-Hegselmann (KH) Intro-4 , and Sznajd Intro-5
. But most models in the literature consider two-state opinion agents, in
favor ($+1$) or against ($-1$) about a certain topic. In the Galam’s majority
rule and the Sznajd’s updating rule, the interaction between the agents is
randomly changed during the evolution, and the time to reach consensus is
associated with the initial traction $p$ of $+1$ state. The consensus time $T$
reaches its maximal value at $p=0.5$. In the Sznajd model, a pair of nearest
neighbors convinces its neighbors to adopt the pair opinion if and only if
both members have the same opinion. Otherwise the pair and its neighbors do
not change opinion. In the KH consensus model, the opinions between $0$ and
$1$ and a confidence bound parameter is introduced. The agent $i$ would take
the average opinion of all neighboring agents that are within a confidence
bound during the evolution. In the Deffuant model, the opinion of two randomly
selected neighboring agents $i$ and $j$ would remain unchanged, if their
opinions $\sigma_{i}$ and $\sigma_{j}$ differ by more than a fixed threshold
parameter. Otherwise, each opinion moves into the direction of the other by an
amount $\mu\times\mid\sigma_{i}-\sigma_{j}\mid$.

Additionally, complex networks have received much attention in recent years.
Topologically, a network is consisted of nodes and links. The complex network
models, such as the lattice network, the random network Intr_6 ; Intr_7 ;
Intr_8 , the small-world network Intr_9 ; Intr_10 , and the scale-free network
Intr_11 , are studied in many branches of science. It is meaningful to mention
that opinion formation models are set up in complex networks.

In the present work, we investigate the implication of a social network in a
stochastic opinion formation model. We first introduce the Blume-Emery-
Griffiths (BEG) model Intr_12 ; Intr_13 ; Intr_14  to describe the dynamics of
opinion formation, and the model of complex networks we used is social network
which is more reality. Our simulation focuses on the average opinion for
different situation. And we also simulated the system under the influence of
external field.

In the rest of this paper we will give a description of this dynamic model and
how to generate the underlying networks. In Sec.III, we show the simulation
results without external filed. In Sec. IV we present the results with the
influence of external field. The final section presents further discussion and
conclusion.

##  II The model

Generally speaking, social networks include some essential characteristics,
such as short average path lengths, high clustering, assortative mixing
Model-1 ; Model-2 , the existence of community structure, and broad degree
distributions Model-3 ; Model-4 . As a result, we use Riitta Toivonen’s social
network model in our present work Model-5 . This network is structured by two
processes: $1)$ attachment to random vertices, and $2)$ attachment to the
neighborhood of the random vertices, giving rise to implicit preferential
attachment. These processes give rise to essential characteristics for social
networks. The second process gives rise to assortativity, high clustering and
community structure. The degree distribution is also determined by the number
of edges generated by the second process for each random attachment.

Figure 1: Degree distribution of networks with $N=10000$. Result is averages
over $20$ simulation runs. The number of initial contacts is distributed as
$p(n_{init}=1)=0.25$, $p(n_{init}=2)=0.75$, and the number of secondary
contacts from each initial contact $n_{2nd}\sim U[0,3]$.

In this paper, the network is grown from a chain with $10$ nodes. The number
of initial contacts is distributed as $p(n_{init}=1)=0.25$,
$p(n_{init}=2)=0.75$, and the number of secondary contacts from each initial
contact $n_{2nd}\sim U[0,3]$ (uniformly distributed between $0$ and $3$). The
total number of nodes in the social network structure is $N=10000$. The degree
distribution of simulated networks is displayed in Fig. 1. We note that the
degree distributon $P(k)$ is a power-law functional form and a peak around the
degree $k=5$, also that consistent with real world observations Intr_11 ;
Model-6 .

Now, we consider a system with $N$ agents, which is represented by nodes on a
social network. For each node, we consider three states which are represented
by $+1$, $0$, and $-1$. A practical example could be the decision to agree
$\sigma_{i}(t)=+1$, disagree $\sigma_{i}(t)=-1$, or neutral $\sigma_{i}(t)=0$.
The states are updated according to the stochastic parallel spin-flip dynamics
defined by the transition probabilities

$Prob\left(\sigma_{i,t+1}=s^{\prime}|\sigma_{N}(t)\right)=\frac{\exp\left\\{-\beta\epsilon_{i}\left[s^{\prime}|\sigma_{N}(t)\right]\right\\}}{\sum_{s}\exp\left\\{-\beta\epsilon_{i}[s|\sigma_{N}(t)]\right\\}}$
(1)

where $s,s^{\prime}\in\\{+1,0,-1\\}$, and $\beta=a/T$, $a$ represents the
active degree of system, defined as $a=\left<\sigma_{N}^{2}(t)\right>$. The
energy potential $\epsilon_{i}\left[s|\sigma_{N}(t)\right]$ is defined by

$\epsilon_{i}\left[s|\sigma_{N}(t)\right]=-sh_{i}\left(\sigma_{N}(t)\right)-s^{2}\theta_{i}\left(\sigma_{N}(t)\right),$
(2)

where the following local field in node $i$ carries all information

$\displaystyle h_{N,i}(t)$ $\displaystyle=$ $\displaystyle\sum_{j\neq
i}J_{ij}\sigma_{j}(t),$ $\displaystyle\theta_{N,i}(t)$ $\displaystyle=$
$\displaystyle\sum_{j\neq i}K_{ij}\sigma_{j}^{2}(t).$

Here, we define coupling $J_{ij}$ and $K_{ij}$ are positive numbers less than
or equal to $1$, and with Gaussian distribution. $h_{N,i}(t)$ represents the
time dependent interaction strengths between the node $i$ and his $n_{i}$
nearest neighboring nodes. $\theta_{N,i}(t)$ instead the strengths of feedback
and $T$ is interior thermo-noise. So the average opinion is defined by

$r(t)=\frac{1}{N}\sum_{j=1}^{N}\sigma_{j}(t).$ (3)

##  III Simulation results

Figure 2: (a) Time series of average opinion with the total time steps is
$t=10000$, (b) the distribution functions $P(R)$, and (c) the autocorrelation
function $c(\tau)$. The parameters used in the simulation are
$p(n_{init}=1)=0.95$, $N=10000$, $T=1.0$ and $L=10000$. The parameter $J_{ij}$
and $K_{ij}$ are positive numbers which are not larger than $1$ in whole
simulations. All the results in this paper are obtained over $20$ realizations
of the social networks.

At first we investigate the time series of average opinion, as illustrated in
Fig. 2(a). It shows there exists the fluctuation around the average opinion
$r=0$. In order to compare the fluctuation of different scales, the time
series have been normalized according to

$R(t)=\left(r(t)-\left<r(t)\right>_{\tau}\right)/\delta\left(r(t)\right),$

where $\left<r(t)\right>_{\tau}$ and $\delta(r(t))$ denote the average and the
standard deviation over the period considered, respectively. In Fig. 2(b), we
present the distribution functions $P(R)$ associated with the time series. It
is clear that this function $P(R)$ is a Gaussian form.

We calculate the autocorrelation function $c(\tau)$ of our model. For a time
series of $L$ samples, $r(t)$ for $t=1,2,\ldots,L$, $c(\tau)$ is defined by

$c(\tau)=\frac{\sum_{t=1}^{L-\tau}(r(t)-\bar{r})(r(t+\tau)-\bar{r})}{\sum_{t=1}^{L-\tau}(r(t)-\bar{r})^{2}},$
(4)

where $\tau$ is the time delay and $\bar{r}$ represents the average over the
period under consideration. Fig. 2(c) shows the result of autocorrelation
function of our model. It is found that $c(\tau)$ decreases rapidly in very
small rang of $\tau$. It means the system has short-time memory effects. As is
now well known, the stock market has nontrivial memory effects simulation-2 .
For example, the autocorrelation funciton of Dow Jones (DJ), also in the small
rang of $\tau$, decreases rapidly from $1$ to $0$. From this point, perhaps
our model is helpful to understand the financial markets.

##  IV The influence of external field

Figure 3: Time series of the average opinion with different values of
amplitude $A=0.08$, $0.12$, $0.16$, $0.22$, $0.28$, $0.32$. Parameters are
$T=1.0$, $\omega=\pi/3$, and $\varphi=0$.

In order to explore what phenomena maybe happen to system under the influence
of external field. We add a period external field to the energy potential
$\epsilon_{i}$,

$\epsilon_{i}\left[s^{\prime}|\sigma_{N}(t)\right]=-sh_{i}\left(\sigma_{N}(t)\right)-s^{2}\theta_{i}\left(\sigma_{N}(t)\right)-s\left[A\cos(\omega
t+\varphi)\right],$ (5)

where $A$ is the amplitude of period external field, $\omega$ is frequency and
$\varphi$ denotes the initial phase of external field.

We investigate the effect of amplitude $A$ by fixing other parameters. In Fig.
3 we plot the time series of the average opinion $r(t)$ under different values
of $A$. It is obvious that the distribution functions have a remarkable change
with increasing $A$. With increasing strength of external field, the average
opinion comes into several discrete parts. For small amplitude $A=0.02$,
$P(R)$ is still a Gaussian form. When $A=0.08$, it begins to appear two
fluctuation around nonzero symmetric values of average opinions. Then, four
nonzero average opinions appear at $A=0.16$. Note that the intervals among the
discrete average opinions increase with increase in the strength $A$ of
external fields. Fig. 3 gives the process from two wave crests to four
independent parts. And the average opinion of the whole system will jump from
one part to the other parts at all times.

Figure 4: (a) The distribution functions $P(R)$ of average opinion time series
under different amplitudes $A$. Parameters are $T=1.0$, $\omega=\pi/3$, and
$\varphi=0$. (b) $P(R)$ for different frequencies $\omega$. Parameters are
$A=0.06$, $\varphi=\pi/2$, and $T=1.0$.

In Fig. 4, we present the distribution function $P(R)$ of the average opinion.
Again, it is easy to verify that the average opinions oscillate among serval
separate symmetric nonzero values under the external periodic driving force
[see Fig. 4(a)]. A similar oscillation behavior is observed for simulation on
the influence of the frequency $\omega$ which is shown in Fig. 4(b). Noted
that $P(R)$ for the frequency $\omega=\pi/3$ is same to the case for
$\omega=2\pi/3$, and the same distribution is observed between $\omega=\pi/6$
and $\omega=5\pi/6$. But there are distinct difference for $\omega=0$ and
$\omega=\pi/2$. It indicates a possible period $\pi$ in the case of fixed
other parameters.

Figure 5: The distribution functions $P(R)$ of average opinion time series
under different initial phases $\varphi$. Parameters are $A=0.16$,
$\omega=\pi/3$, and $T=1.0$.

Fig. 5 shows the distribution functions $P(R)$ of average opinion time series
for different initial phases $\varphi$. For $\varphi=0$, the average opinion
vibrates among four symmetric nonzero values. When $\varphi$ increases to
$\pi/2$, clearly, the average opinion comes into a $3$-value oscillation.
Additionally, note that the distribution functions is almost same for
$\varphi=0$ and $\varphi=\pi$ (or $\varphi=\pi/2$ and $\varphi=3\pi/2$).
Again, one can conjecture $P(R)$ is a $\pi$-period behavior. We also observe
the system’s average opinion time series only have two types of distribution
functions in different values of initial phases $\varphi$.

Figure 6: The distribution functions $P(R)$ of average opinion time series
under different interior thermo-noises $T$. The parameters used in the
simulation are $A=0.16$, $\omega=\pi/3$, and $\varphi=0$.

Another important parameter for the systems is the interior thermo-noise $T$.
We explore its effects with (or without) external fields. It is found that
there is not remarkable influence on the system without external field.
Contrarily, in the case of external field, $P(R)$ shows a similar oscillation
with it in Fig. 4(a) (see Fig. 6). Note that their influences are opposite. In
Fig. 6, with increasing $T$ the forms of $P(R)$ transform from four-peak to
two-peak gradually, and merge into only one-peak at last. At the same time,
the average opinion $r$ is expanded from some separate regions to the whole
more expansive scale for larger $T$.

By comparing the Fig. 4(a) with the Fig. 6, it is clear that the amplitude $A$
and interior thermo-noise $T$ have opposite effects acting on the systems. It
looks like a couple of contradictory parameters, even though both lead to the
split phenomena of the distribution of average opinion $P(R)$ and the nonzero
average $R$.

It exists similar behaviors in the Ising ferromagnetic systems. In Ising
model, the order-disorder transition is a second order transition. It will be
a non-zero magnetization $\pm|M_{sp}|$ for a finite system. There is a nonzero
probability for ever that the system from near $+|M_{sp}|$ to near
$-|M_{sp}|$, and vice versa external-1 . In our model under the influence of
external field, it is also observed the phenomena of phase transition caused
by $T$ (or by $A$), which is similar to the Ising paramagnetic-
antiferromagnetic transition.

As discussed above, the energy potential increases with increasing $T$, and
the system’s entropy becomes larger (more disordered). But the external field
tends to restrict the disordered effects in the system and reduces the
disordered strength into several separate regions.

##  V Conclusion

In the present work we introduce Blume-Emery-Griffiths model on opinion
formation with three-state. Considering the characters of real social systems,
we construct a social network to link between agents. In this BEG model, each
person’s opinion is influenced not only by his specific local information from
his neighbors but also by the average opinion of the whole network.

Moreover, we focus on the behaviors of BEG systems under external
perturbation. The simulation results show that this system is sensitive to the
external field. As discussed in Sec. III, the parameters in the external
periodic perturbation, such as amplitude $A$, initial phase $\varphi$, and
frequency $\omega$, have obvious impacts on the opinion systems. Besides, the
effect of the amplitude $A$ or interior thermo-noise $T$ is similar to the
Ising paramagnetic-antiferromagnetic transition, and the influence acted on
systems from $A$ and $T$ is opposite.

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# On the action of the Steenrod algebra on the modular invariants of special
linear group

Nguyễn Sum  Department of Mathematics, Quy Nhơn University, 170 An Dương
Vương, Quy Nhơn, Bình Định, Viet Nam<EMAIL_ADDRESS>
###### Abstract.

We compute the action of the Steenrod algebra on generators of algebras of
invariants of special linear group ${SL_{n}=SL(n,\mathbb{Z}/p)}$ in the
polynomial algebra with $p$ an odd prime number.

###### Key words and phrases:

Invariant theory, Dickson-Mùi invariants, Steenrod-Milnor operations

###### 2010 Mathematics Subject Classification:

Primary 55S10; Secondary 55S05

##  1\. Introduction

For an an odd prime $p$, let $SL_{n}$ denote the special linear subgroup of
$GL(n,\mathbb{Z}/p)$, which acts naturally on the cohomology algebra
$H^{*}(B(\mathbb{Z}/p)^{n})$. Here and in what follows, the cohomology is
always taken with coefficients in the prime field $\mathbb{Z}/p$.

According to [3], $H^{*}(B(\mathbb{Z}/p)^{n})=E(x_{1},\ldots,x_{n})\otimes
P(y_{1},\ldots,y_{n})$ with $\dim x_{i}=1$, $y_{i}=\beta x_{i}$, where $\beta$
is the Bockstein homomorphism, $E(.,\ldots,.)$ and $P(.,\ldots,.)$ are the
exterior and polynomial algebras over $\mathbb{Z}/p$ generated by the
variables indicated. Let $(e_{k+1},\ldots,e_{n}),k\geqslant 0,$be a sequence
of non-negative integers. Following Mùi [2], we define

$[k;e_{k+1},\ldots,e_{n}]=[k;e_{k+1},\ldots,e_{n}](x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})$

by

$[k;e_{k+1},\ldots,e_{n}]=\frac{1}{k!}\begin{vmatrix}x_{1}&\cdots&x_{n}\\\
\vdots&\cdots&\vdots\\\ x_{1}&\cdots&x_{n}\\\
y_{1}^{p^{e_{k+1}}}&\cdots&y_{n}^{p^{e_{k+1}}}\\\ \vdots&\cdots&\vdots\\\
y_{1}^{p^{e_{n}}}&\cdots&y_{n}^{p^{e_{n}}}\end{vmatrix}.$

The precise meaning of the right hand side is given in [2]. For $k=0$, we
write

$[0;e_{1},\ldots,e_{n}]=[e_{1},\ldots,e_{n}]=\det\left(y_{i}^{p^{e_{j}}}\right).$

We set

$\displaystyle L_{n,s}$ $\displaystyle=[0,\ldots,\hat{s},\ldots,n],\
0\leqslant s\leqslant n,$ $\displaystyle L_{n}$
$\displaystyle=L_{n,n}=[0,\ldots,n-1].$

Each $[k;e_{k+1},\ldots,e_{n}]$ is an invariant of $SL_{n}$ and
$[e_{1},\ldots,e_{n}]$ is divisible by $L_{n}$. Then Dickson invariants
$Q_{n,s},\ 0\leqslant s\leqslant n,$ and Mùi invariants
$M_{n,s_{1},\ldots,s_{k}}$, $0\leq\ s_{1}<\ldots<s_{k}\leq n$ are defined by

$\displaystyle Q_{n,s}$ $\displaystyle=L_{n,s}/L_{n},$ $\displaystyle
M_{n,s_{1},\ldots,s_{k}}$
$\displaystyle=[k;0,\ldots,\hat{s}_{1},\ldots,\hat{s}_{k},\ldots,n-1].$

Note that $Q_{n,n}=1$, $Q_{n,0}=L_{n}^{p-1}$,
$M_{n,0,\ldots,n-1}=[n;\emptyset]=x_{1}\ldots x_{n}$.

Mùi proved in [2] that $H^{*}(B(\mathbb{Z}/p)^{n})^{SL_{n}}$ is the free
module over the Dickson algebra $P(L_{n},Q_{n,1},\ldots,Q_{n,n-1})$ generated
by 1 and $M_{n,s_{1},\ldots,s_{k}}$ with $0\leq\ s_{1}<\ldots<s_{k}\leq n$.

The Steenrod algebra $\mathcal{A}(p)$ acts on $H^{*}(B(\mathbb{Z}/p)^{n})$ by
well-known rules. Since this action commutes with the action of $SL_{n}$, it
induces an action of $\mathcal{A}(p)$ on
$H^{*}(B(\mathbb{Z}/p)^{n})^{SL_{n}}$.

Let $\tau_{s}$ and $\xi_{i}$ be the Milnor elements of dimensions $2p^{s}-1$
and $2p^{i}-2$, respectively, in the dual algebra $\mathcal{A}(p)^{*}$ of
$\mathcal{A}(p)$. Milnor showed in [5] that

$\mathcal{A}(p)^{*}=E(\tau_{0},\tau_{1},\ldots)\otimes
P(\xi_{1},\xi_{2},\ldots).$

So, $\mathcal{A}(p)^{*}$ has a basis consisting of all monomials
$\tau_{S}\xi^{R}=\tau_{s_{1}}\ldots\tau_{s_{t}}\xi_{1}^{r_{1}}\ldots\xi_{m}^{r_{m}}$
with $S=(s_{1},\ldots,s_{t})$, $0\leqslant s_{1}<\ldots<s_{t}$,
$R=(r_{1},\ldots,r_{m})$. Let $St^{S,R}\in\mathcal{A}(p)$ denote the dual of
$\tau_{S}\xi^{R}$ with respect to this basis of $\mathcal{A}(p)^{*}$. Then
$\mathcal{A}(p)$ has a new basis consisting of all operations $St^{S,R}$. In
particular, for $S=\emptyset$, $R=(k)$, $St^{S,R}$ is nothing but the Steenrod
operation $P^{k}$.

The action of $P^{k}$ on Dickson and Mùi invariants was explicitly computed by
Hưng and Minh [4]. The action of $St^{S,R}$ on the invariant
$[n,\emptyset]=x_{1}\ldots x_{n}$ was computed by Mùi [3].

In this paper, we compute the action of $St^{S,R}$ on
$[k;e_{k+1},\ldots,e_{n}]$ and prove a nice relation between the invariants
$[k;e_{k+1},\ldots,e_{n}+s]$, $0\leqslant s\leqslant n$, and the Dickson
invariants. Using these results, we explicitly compute the action of $P^{k}$
on Mùi invariants $M_{n,s_{1},\ldots,s_{k}}$ which was first computed in Hưng
and Minh [4] by another method.

To state the main results, we introduce some notations. Let $J=(J_{0},J_{1},$
$\ldots,J_{m})$ with $J_{s}\subset\\{{k+1},\ldots,n\\}$, $0\leqslant
s\leqslant m$, and $\coprod_{s=0}^{m}J_{s}=\\{{k+1},\ldots,n\\}$ (disjoint
union). We define the sequence $R_{J}=(r_{J_{1}},\ldots,r_{J_{m}})$ and the
function $\Phi_{J}:\\{{k+1},\ldots,n\\}\to\\{{1},\ldots,m\\}$ by setting

$\displaystyle r_{J_{s}}$ $\displaystyle=\sum_{j\in J_{s}}p^{e_{j}},\
0\leqslant s\leqslant m,$ $\displaystyle\Phi_{J}(i)$ $\displaystyle=s\ \text{
if }\ i\in J_{s},\ k+1\leqslant i\leqslant n.$

The main result of this paper is

######  Theorem 1.1.

Suppose that $e_{i}\neq e_{j}$ for $i\neq j$, $S=(s_{1},\ldots,s_{t})$,
$s_{1}<\ldots<s_{t}<m$. Under the above notation we have

$St^{S,R}[k;e_{k+1},\ldots,e_{n}]\\\
=\begin{cases}(-1)^{t(k-t)}[k-t,s_{1},\ldots,s_{t},e_{k+1}+\Phi_{J}(k+1),\ldots,e_{n}+\Phi_{J}(n)],\\\
\hskip 156.49014ptR=R_{J},\ \text{ for some }J,\\\ 0,\hskip 142.26378pt\text{
otherwise. }\end{cases}$

We have also the following relation from which we can explicitly compute
$St^{S,R}[k;e_{k+1},\ldots,e_{n}]$ in terms of Dickson and Mùi invariants.

######  Proposition 1.2.

For $0\leqslant k\leqslant n$,

$[k;e_{k+1},\ldots,e_{n-1},e_{n}+n]=\sum_{s=0}^{n-1}(-1)^{n+s-1}[k;e_{k+1},\ldots,e_{n-1},e_{n}+s]Q_{n,s}^{p^{e_{n}}}.$

Using Theorem 1.1 and Proposition 1.2 we explicitly compute the action of
$St^{S,R}$ on Mùi invariant $M_{n,s_{1},\ldots,s_{k}}$ when $S,\ R$ are
special. Particularly, we prove

######  Theorem 1.3 (Hưng and Minh [4]).

For $s_{0}=-1<s_{1}<\ldots<s_{k}<s_{k+1}=n$,

$\displaystyle P^{t}$ $\displaystyle M_{n,s_{1},\ldots,s_{k}}=$
$\displaystyle\begin{cases}M_{n,t_{1},\ldots,t_{k}},&t=\underset{i=1}{\overset{k}{\sum}}\frac{p^{s_{i}}-p^{t_{i}}}{p-1},\text{
with }s_{i-1}<t_{i}\leqslant s_{i},\\\
\underset{i=1}{\overset{k+1}{\sum}}(-1)^{k+1-i}M_{n,t_{1},\ldots,\hat{t}_{i}\ldots,t_{k+1}}Q_{n,t_{i}},&t=\underset{i=1}{\overset{k+1}{\sum}}\frac{p^{s_{i}}-p^{t_{i}}}{p-1},\text{
with }s_{i-1}<t_{i}\leqslant s_{i},\\\ &1\leqslant i\leqslant k+1,\
t_{k+1}<s_{k+1}=n,\\\ 0,&\text{otherwise. }\end{cases}$

## Acknowledgment

I would like to thank Professor Huỳnh Mùi for his generous help and inspiring
guidance.

##  2\. Proof of Theorem 1.1

First we recall Mùi’s results on the homomorphism $d_{m}^{*}P_{m}$ and the
operations $St^{S,R}$.

Let $\mathcal{A}_{p^{m}}$ be the alternating group on $p^{m}$ letters. Suppose
that $X$ is a topological space, $W\mathcal{A}_{p^{m}}$ is a contractible
$\mathcal{A}_{p^{m}}$-free space. Then we have the Steenrod power map

$P_{m}:H^{q}(X)\longrightarrow
H^{p^{m}q}\big{(}W\mathcal{A}_{p^{m}}\underset{\mathcal{A}_{p^{m}}}{\times}X^{p^{m}}\big{)},$

which sends $u$ to $1\otimes u^{p^{m}}$ at the cochain level (see [6; Chap.
VII]).

The inclusion $(\mathbb{Z}/p)^{n}\subset\mathcal{A}_{p^{m}}$ together with the
diagonal map $X\to X^{p^{m}}$ and the Künneth formula induces the homomorphism

$d_{m}:H^{*}\big{(}W\mathcal{A}_{p^{m}}\underset{\mathcal{A}_{p^{m}}}{\times}X^{p^{m}}\big{)}\longrightarrow
H^{*}(B(\mathbb{Z}/p)^{n})\otimes H^{*}(X).$

Set $\tilde{M}_{m,s}=M_{m,s}L_{m}^{h-1},\ 0\leqslant s<m,\
\tilde{L}_{m}=L_{m}^{h},\ h=(p-1)/2$. We have

######  Theorem 2.1 (Mùi [3; 1.3]).

Let $u\in H^{q}(X),\ \mu(q)=(-1)^{hq(q-1)/2}(h!)^{q}$. Then

$d_{m}^{*}P_{m}(u)=\mu(q)^{m}\sum_{S,R}(-1)^{r(S,R)}\tilde{M}_{m,s_{1}}\ldots\tilde{M}_{m,s_{t}}\tilde{L}_{m}^{r_{0}}Q_{m,1}^{r_{1}}\ldots
Q_{m,m-1}^{r_{m-1}}\otimes St^{S,R}u.$

Here the summation runs over all $(S,R)$ with $S=(s_{1},\ldots,s_{t})$,
$0\leqslant s_{1}<\ldots<s_{t}<m$, $R=(r_{1},\ldots,r_{m})$,
$r_{0}=q-t-2(r_{1}+\ldots+r_{m})\geqslant 0$,
$r(S,R)=t+s_{1}+\ldots+s_{t}+r_{1}+2r_{2}+\ldots+mr_{m}$.

######  Proposition 2.2 (Mùi [2, 3]).

i) $d_{m}^{*}P_{m}$ is a natural homomorphism preserving cup product up to a
sign. Precisely,

$d_{m}^{*}P_{m}(uv)=(-1)^{mhqr}d_{m}^{*}P_{m}ud_{m}^{*}P_{m}v,$

with $q=\dim u,\ r=\dim v$.

ii)
$d_{m}^{*}P_{m}y_{i}=\underset{s=0}{\overset{m}{\sum}}(-1)^{m+s}Q_{m,s}\otimes
y_{i}^{p^{s}}$.

iii) $d_{m}^{*}P_{m}(x_{1}\ldots x_{n})=$

$\mu(n)^{m}\underset{0\leqslant
s_{1}<\ldots<s_{t}<m}{\sum}(-1)^{t(n-t)+r(S,0)}\tilde{M}_{m,s_{1}}\ldots\tilde{M}_{m,s_{t}}\tilde{L}_{m}^{n-t}\otimes[n-t,s_{1},\ldots,s_{t}].$

Here $x_{i}$ and $y_{i}$ are defined as in the introduction.

######  Lemma 2.3.

If $e_{i}\neq e_{j}$ for $i\neq j$, then

$d_{m}^{*}P_{m}[e_{1},\ldots,e_{n}]\\\
=\sum_{J=(J_{0},\ldots,J_{m})}(-1)^{mn+r(\emptyset,R_{J})}\tilde{L}_{m}^{2r_{J_{0}}}Q_{m,1}^{r_{j_{1}}}\ldots
Q_{m,m-1}^{r_{j_{m-1}}}\\\
\otimes[e_{1}+\Phi_{J}(1),\ldots,e_{n}+\Phi_{J}(n)],$

where $R_{J}$ and $\Phi_{J}$ are defined as in Theorem 1.1.

###### Proof.

Let $\Sigma_{n}$ be the symmetric group on $n$ letters. Then

$[e_{1},\ldots,e_{n}]=\sum_{\sigma\in\Sigma_{n}}\text{sign}\
\\!\sigma\prod_{i=1}^{n}y_{i}^{p^{e_{\sigma(i)}}}.$

From Proposition 1.2, we have

$\displaystyle
d_{m}^{*}P_{m}\Big{(}\prod_{i=1}^{n}y_{i}^{p^{e_{\sigma(i)}}}\Big{)}$
$\displaystyle=\prod_{i=1}^{n}\big{(}d_{m}^{*}P_{m}y_{i}\big{)}^{p^{e_{\sigma(i)}}}$
$\displaystyle=\prod_{i=1}^{n}\Big{(}\underset{s=0}{\overset{m}{\sum}}(-1)^{m+s}Q_{m,s}^{p^{e_{\sigma(i)}}}\otimes
y_{i}^{p^{e_{\sigma(i)}+s}}\Big{)}.$

Expanding this product and using the definitions of $\Phi_{J},R_{J}$ and the
assumption of the lemma, we get

$d_{m}^{*}P_{m}\Big{(}\prod_{i=1}^{n}y_{i}^{p^{e_{\sigma(i)}}}\Big{)}=\sum_{J}(-1)^{mn+r(\emptyset,R_{J})}Q_{m,0}^{r_{J_{0}}}\ldots
Q_{m,m-1}^{r_{j_{m-1}}}\otimes\prod_{i=1}^{n}y_{i}^{p^{e_{\sigma(i)}+\Phi_{J}(\sigma(i))}}.$

Hence, from the above equalities we obtain

$\displaystyle d_{m}^{*}P_{m}$ $\displaystyle[e_{1},\ldots,e_{n}]$
$\displaystyle=\sum_{J}(-1)^{mn+r(\emptyset,R_{J})}Q_{m,0}^{r_{J_{0}}}\ldots
Q_{m,m-1}^{r_{j_{m-1}}}\otimes\sum_{\sigma\in\Sigma_{n}}\text{sign}\
\\!\sigma\prod_{i=1}^{n}y_{i}^{p^{e_{\sigma(i)}+\Phi_{J}(\sigma(i))}}$
$\displaystyle=\sum_{J}(-1)^{mn+r(\emptyset,R_{J})}Q_{m,0}^{r_{J_{0}}}\ldots
Q_{m,m-1}^{r_{j_{m-1}}}\otimes[e_{1}+\Phi_{J}(1),\ldots,e_{n}+\Phi_{J}(n)].$

Since $Q_{m,0}=\tilde{L}_{m}^{2}$, the lemma is proved. ∎

###### 2.4. Proof of Theorem 1.1.

Let $I$ be a subset of $\\{1,\ldots,n\\}$ and $I^{\prime}$ is its complement
in $\\{1,\ldots,n\\}$. Writing $I=(i_{1},\ldots,i_{k})$ and
$I^{\prime}=(i_{k+1},\ldots,i_{n})$ with $i_{1}<\ldots<i_{k}$ and
$i_{k+1}<\ldots<i_{n}$. We set $x_{I}=x_{i_{1}}\ldots x_{i_{k}}$,
$[e_{k+1},\ldots,e_{n}]_{I}=[e_{k+1},\ldots,e_{n}](y_{i_{k+1}},\ldots,y_{i_{n}})$
and $\sigma_{I}=\begin{pmatrix}1&\ldots&n\\\
i_{1}&\ldots&i_{n}\end{pmatrix}\in\Sigma_{n}$. In [2; I.4.2], Mùi showed that

$[k;e_{k+1},\ldots,e_{n-1},e_{n}]=\sum_{I}\text{sign}\
\\!\sigma_{I}x_{I}[e_{k+1},\ldots,e_{n}]_{I}.$

From Proposition 2.2 and Lemma 2.3 we have

$d_{m}^{*}P_{m}(x_{I})=\mu(k)^{m}\underset{0\leqslant
s_{1}<\ldots<s_{t}<m}{\sum}(-1)^{t(k-t)+r(S,0)}\tilde{M}_{m,s_{1}}\ldots\tilde{M}_{m,s_{t}}\tilde{L}_{m}^{k-t}\\\
\otimes[k-t,s_{1},\ldots,s_{t}]_{I},$

where
$[k-t,s_{1},\ldots,s_{t}]_{I}=[k-t,s_{1},\ldots,s_{t}](x_{i_{1}},\ldots,x_{i_{k}},y_{i_{1}},\ldots,y_{i_{k}})$,

$d_{m}^{*}P_{m}[e_{k+1},\ldots,e_{n}]_{I}\\\
=\sum_{J=(J_{0},\ldots,J_{m})}(-1)^{m(m-k)+r(\emptyset,R_{J})}\tilde{L}_{m}^{2r_{J_{0}}}Q_{m,1}^{r_{j_{1}}}\ldots
Q_{m,m-1}^{r_{j_{m-1}}}\otimes\\\
[e_{k+1}+\Phi_{J}(k+1),\ldots,e_{n}+\Phi_{J}(n)]_{I}.$

Set $q=\dim[k;e_{k+1},\ldots,e_{n}]=k+2(p^{e_{k+1}}+\ldots+p^{e_{n}}).$ An
easy computation shows that $\mu(q)=(-1)^{n-k}\mu(k)$ and
$r(S,0)+r(\emptyset,R)=r(S,R)$. Hence from Proposition 2.2 and the above
equalities we get

$d_{m}^{*}P_{m}[e_{k+1},\ldots,e_{n}]\\\
=\mu(q)^{m}\sum_{S,J}(-1)^{t(t-k)+r(S,R_{J})}M_{m,s_{1}}\ldots\tilde{M}_{m,s_{t}}\tilde{L}_{m}^{k-t+2r_{J_{0}}}Q_{m,1}^{r_{j_{1}}}\ldots
Q_{m,m-1}^{r_{j_{m-1}}}\otimes\\\ \sum_{I}\text{sign}\
\\!\sigma_{I}[k-t,s_{1},\ldots,s_{t}]_{I}[e_{k+1}+\Phi_{J}(k+1),\ldots,e_{n}+\Phi_{J}(n)]_{I}.$

Then, using the Laplace development we obtain

$d_{m}^{*}P_{m}[e_{k+1},\ldots,e_{n}]\\\
=\mu(q)^{m}\sum_{S,J}(-1)^{t(t-k)+r(S,R_{J})}M_{m,s_{1}}\ldots\tilde{M}_{m,s_{t}}\tilde{L}_{m}^{k-t+2r_{J_{0}}}Q_{m,1}^{r_{j_{1}}}\ldots
Q_{m,m-1}^{r_{j_{m-1}}}\otimes\\\
[k-t,s_{1},\ldots,s_{t},e_{k+1}+\Phi_{J}(k+1),\ldots,e_{n}+\Phi_{J}(n)].$

Theorem 1.1 now follows this equality and Theorem 2.1. ∎

##  3\. Proof of Proposition 1.2

First we prove the stated relation for $k=0$,

$\displaystyle[e_{1},\ldots,e_{n-1},e_{n}+n]=\sum_{s=0}^{n-1}(-1)^{n+s-1}[e_{1},\ldots,e_{n-1},e_{n}+s]Q_{n,s}^{p^{e_{n}}}.$
(3.1)

We will prove (3.1) and the following relation together by induction on $n$,

$\displaystyle[e_{1},\ldots,e_{n-1},e_{n}+n-1]=\sum_{s=0}^{n-2}(-1)^{n+s}[e_{1},\ldots,e_{n-1},e_{n}+s]Q_{n-1,s}^{p^{e_{n}}}$
$\displaystyle\hskip 170.71652pt+[e_{1},\ldots,e_{n-1}]V_{n}^{p^{e_{n}}}.$
(3.2)

Here, $V_{n}=L_{n}/L_{n-1}$.

We denote (3.1) and (3.2) when $n=m$ by 3.1$(m)$ and 3.2$(m)$, respectively.

When $n=2$ the proof is straightforward. Suppose that $n>2$ and that
3.1$(n-1)$ and 3.2$(n-1)$ are true.

By Laplace development and 3.2$(n-1)$ we have

$\displaystyle[e_{1},\ldots,e_{n-1},e_{n}+n-1]$
$\displaystyle=\sum_{t=1}^{n-1}(-1)^{n+t}[e_{1},\ldots,\hat{e}_{t},\ldots,e_{n-1},e_{n}+n-1]y_{n}^{p^{e_{t}}}+[e_{1},\ldots,e_{n-1}]y_{n}^{p^{e_{n}+n-1}}$
$\displaystyle=\sum_{t=1}^{n-1}(-1)^{n+t}\Big{(}\sum_{s=0}^{n-2}(-1)^{n+s}[e_{1},\ldots,\hat{e}_{t},\ldots,e_{n-1},e_{n}+s]Q_{n-1,s}^{p^{e_{n}}}\Big{)}y_{n}^{p^{e_{t}}}$
$\displaystyle\hskip 199.16928pt+[e_{1},\ldots,e_{n-1}]y_{n}^{p^{e_{n}+n-1}}$
$\displaystyle=\sum_{s=0}^{n-2}(-1)^{n+s}\Big{(}\sum_{t=1}^{n-1}(-1)^{n+t}[e_{1},\ldots,\hat{e}_{t},\ldots,e_{n-1},e_{n}+s]y_{n}^{p^{e_{t}}}\Big{)}Q_{n-1,s}^{p^{e_{n}}}$
$\displaystyle\hskip 199.16928pt+[e_{1},\ldots,e_{n-1}]y_{n}^{p^{e_{n}+n-1}}$
$\displaystyle=\sum_{s=0}^{n-2}(-1)^{n+s}[e_{1},\ldots,e_{n-1},e_{n}+s]Q_{n-1,s}^{p^{e_{n}}}$
$\displaystyle\hskip
113.81102pt+[e_{1},\ldots,e_{n-1}]\sum_{s=0}^{n-1}(-1)^{n+s-1}Q_{n-1,s}^{p^{e_{n}}}y_{n}^{p^{e_{n}+s}}.$

Since $V_{n}=\sum_{s=0}^{n-1}(-1)^{n+s-1}Q_{n-1,s}y_{n}^{p^{s}}$ (see [1],
[2]), 3.2$(n)$ is proved.

Now we prove 3.1$(n)$. From 3.2$(n)$ and the relation
$Q_{n,s}=Q_{n-1,s-1}^{p}+Q_{n-1,s}V_{n}^{p-1}$ (see [1], [2]) we obtain

$\displaystyle[e_{1},$ $\displaystyle\ldots,e_{n-1},e_{n}+n]$
$\displaystyle=\sum_{s=1}^{n-1}(-1)^{n+s-1}[e_{1},\ldots,e_{n-1},e_{n}+s]Q_{n-1,s-1}^{p^{e_{n}+1}}$
$\displaystyle\hskip 85.35826pt+[e_{1},\ldots,e_{n-1}]V_{n}^{p^{e_{n}+1}}$
$\displaystyle=\sum_{s=1}^{n-1}(-1)^{n+s-1}[e_{1},\ldots,e_{n-1},e_{n}+s]Q_{n,s}^{p^{e_{n}}}$
$\displaystyle\hskip
85.35826pt-[e_{1},\ldots,e_{n-1},e_{n}+n-1]V_{n}^{(p-1)p^{e_{n}}}$
$\displaystyle+\Big{(}\sum_{s=1}^{n-2}(-1)^{n+s}[e_{1},\ldots,e_{n-1},e_{n}+s]Q_{n-1,s}^{p^{e_{n}}}$
$\displaystyle\hskip
85.35826pt+[e_{1},\ldots,e_{n-1}]V_{n}^{p^{e_{n}}}\Big{)}V_{n}^{(p-1)p^{e_{n}}}.$

Combining this equality and 3.2$(n)$ we get

$\displaystyle[e_{1},e_{2},\ldots,e_{n-1},e_{n}+n]$
$\displaystyle=\sum_{s=1}^{n-1}(-1)^{n+s-1}[e_{1},\ldots,e_{n-1},e_{n}+s]Q_{n,s}^{p^{e_{n}}}$
$\displaystyle-(-1)^{n}[e_{1},\ldots,e_{n-1},e_{n}]Q_{n-1,0}^{p^{e_{n}}}V_{n}^{(p-1)p^{e_{n}}}.$

Since $Q_{n,0}=Q_{n-1,0}V_{n}^{p-1}$, the proof of 3.1$(n)$ is completed.

For $0<k<n$, Proposition 1.2 follows from (3.1) and [2; I.4.7] which asserts
that

$\displaystyle[k;e_{k+1},\ldots,e_{n}]=$ $\displaystyle\
(-1)^{k(k-1)/2}\sum_{0\leq
s_{1}<\ldots<s_{k}}(-1)^{s_{1}+\ldots+s_{k}}M_{n,s_{1},\ldots,s_{k}}[s_{1},\ldots,s_{k},e_{k+1},\ldots,e_{n}]/L_{n}.$

The proposition is completely proved.

##  4\. Some applications

In this section, using Theorem 1.1 and Proposition 1.2, we prove Theorem 1.3
and explicitly compute the action of $St^{S,R}$ on Mùi invariant
$M_{n,s_{1},\ldots,s_{k}}$ when $S,R$ are special. First we prove Theorem 1.3.

###### 4.1. Proof of Theorem 1.3.

Recall that $P^{t}=St^{\emptyset,(t)}$. From Theorem 1.1 we have

$\displaystyle P^{t}$ $\displaystyle M_{n,s_{1},\ldots,s_{k}}$
$\displaystyle=\begin{cases}[k;0,\ldots,\hat{t}_{1},\ldots,\hat{t}_{k+1},\ldots,n],&t=\underset{i=1}{\overset{k+1}{\sum}}\frac{p^{s_{i}}-p^{t_{i}}}{p-1},\text{
with }\\\ &s_{i-1}<t_{i}\leqslant s_{i},1\leqslant i\leqslant k+1,\\\
0,&\text{otherwise. }\end{cases}$

If $t_{k+1}=s_{k+1}=n$, then
$[k;0,\ldots,\hat{t}_{1},\ldots,\hat{t}_{k+1},\ldots,n]=M_{n,t_{1},\ldots,t_{k}}$.
Suppose $t_{k+1}<n$. By Proposition 1.2 we have

$\displaystyle[k;0,\ldots,\hat{t}_{1},$
$\displaystyle\ldots,\hat{t}_{k+1},\ldots,n]$
$\displaystyle=\sum_{s=0}^{n-1}(-1)^{n+s-1}[k;0,\ldots,\hat{t}_{1},\ldots,\hat{t}_{k+1},\ldots,n-1,s]Q_{n,s}$
$\displaystyle=\sum_{s=0}^{n-1}(-1)^{k+1-i}M_{n,t_{1},\ldots,\hat{t}_{i},\ldots,t_{k+1}}Q_{n,t_{i}}$

Hence Theorem 1.3 follows. ∎

######  Notation 4.2.

Denote by $S^{\prime}:s_{k+1}<\dots<s_{n-1}$ the ordered complement of a
sequence $S:1\leqslant s_{1}<\ldots<s_{k}<n$ in $\\{1,\ldots,n-1\\}$. Set
$\Delta_{i}=(0,\ldots,1,\ldots,0)$ with 1 at the $i$-th place $(1\leqslant
i\leqslant n)$, $\Delta_{0}=(0,\ldots,0)$ and $R=(r_{1},\ldots,r_{n})$. Here,
the length of $\Delta_{i}$ is $n$.

The following was proved in Mùi [3; 5.3] for $R=\Delta_{0}$.

######  Proposition 4.3.

Set $s_{0}=0$. Under the above notations, we have

$St^{S^{\prime},R}M_{n,1,\ldots,n-1}=\begin{cases}(-1)^{(k-1)(n-1-k)+s_{t}-t}M_{s_{0},\ldots,\hat{s}_{t},\ldots,s_{k}},&R=\Delta_{s_{t}},\\\
\underset{t=0}{\overset{k}{\sum}}(-1)^{k(n-k)-t}M_{s_{0},\ldots,\hat{s}_{t},\ldots,s_{k}}Q_{n,s_{t}},&R=\Delta_{n},\\\
0,&\text{otherwise. }\end{cases}$

###### Proof.

Note that $M_{n,1,\ldots,n-1}=[n-1;0]$. From Theorem 1.1 we obtain

$St^{S^{\prime},R}M_{n,1,\ldots,n-1}=\begin{cases}(-1)^{k(n-1-k)}[k;1,\ldots,\hat{s}_{1},\ldots,\hat{s}_{k},\ldots,n-1,i],\\\
\hskip 28.45274ptR=\Delta_{i},\text{ with }i=s_{t},\ 0\leqslant t\leqslant
k,\text{ or }i=n,\\\ 0,\hskip 19.91684pt\text{otherwise. }\end{cases}$

It is easy to see that

$[k;1,\ldots,\hat{s}_{1},\ldots,\hat{s}_{k},\ldots,n-1,s_{t}]=(-1)^{n-1-k+s_{t}-t}M_{s_{0},\ldots,\hat{s}_{t},\ldots,s_{k}}.$

According to Proposition 1.2 we have

$\displaystyle[k;1,\ldots,$
$\displaystyle\hat{s}_{1},\ldots,\hat{s}_{k},\ldots,n-1,n]$
$\displaystyle=\sum_{s=0}^{n-1}(-1)^{n+s-1}[k;1,\ldots,\hat{s}_{1},\ldots,\hat{s}_{k},\ldots,n-1,s]Q_{n,s}$
$\displaystyle=\sum_{t=0}^{k}(-1)^{k-t}M_{s_{0},\ldots,\hat{s}_{t},\ldots,s_{k}}Q_{n,s_{t}}.$

From this the proposition follows. ∎

By the same argument as given in the proof of Theorem 1.3 and Proposition 4.3
we obtain the following results.

######  Proposition 4.4.

Let $\Delta_{i}$ be as in 4.2 and $s_{0}=0$. Then

$St^{\emptyset,\Delta_{i}}M_{n,s_{1},\ldots,s_{k}}=\begin{cases}(-1)^{s_{t}-t}M_{s_{0},\ldots,\hat{s}_{t},\ldots,s_{k}},&s_{1}>0,\
i=s_{t},\\\
\underset{t=0}{\overset{k}{\sum}}(-1)^{n-t-1}M_{s_{0},\ldots,\hat{s}_{t},\ldots,s_{k}}Q_{n,s_{t}},&s_{1}>0,\
i=n,\\\ 0,&\text{otherwise.}\end{cases}$

The following proposition was proved by Hưng and Minh [4] for $s=0$.

######  Proposition 4.5.

For $0\leqslant s\leqslant n$,

$St^{(s),(0)}M_{n,s_{1},\ldots,s_{k}}=\begin{cases}(-1)^{k+s_{t}-t}M_{s_{0},\ldots,\hat{s}_{t},\ldots,s_{k}},&s=s_{t},\\\
\underset{i=1}{\overset{k}{\sum}}(-1)^{n+k+t+1}M_{s_{1},\ldots,\hat{s}_{t},\ldots,s_{k}}Q_{n,s_{t}},&s=n,\\\
0,&\text{otherwise.}\end{cases}$

## References

  * [1] L. E. Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), 75-98 MR1500882
  * [2] H. Mùi, Modular invariant theory and the cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sec. IA Math. 22 (1975), 319-369 MR0422451
  * [3] H. Mùi, Cohomology operations derived from modular invariants, Math. Z. 193 (1986), 151-163 MR852916
  * [4] N.H.V. Hưng and P.A. Minh, The action of the mod $p$ Steenrod operations on the modular invariants of linear groups, Vietnam J. Math. 23 (1995), 39-56 MR1367491
  * [5] J. Milnor, Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150-171 MR0099653
  * [6] N.E. Steenrod and D.B.A. Epstein, Cohomology operations, Ann. of Math. No. 50, Princeton University Press, 1962 MR0145525


# Fake News Detection using Stance Classification: A Survey

Anders E. Lillie<EMAIL_ADDRESS>and Emil R. Middelboe<EMAIL_ADDRESS>
(December 11, 2018)

###### Abstract

This paper surveys and presents recent academic work carried out within the
field of stance classification and fake news detection. Echo chambers and the
model organism problem are examples that pose challenges to acquire data with
high quality, due to opinions being polarised in microblogs. Nevertheless it
is shown that several machine learning approaches achieve promising results in
classifying stance. Some use crowd stance for fake news detection, such as the
approach in [Dungs et al., 2018] using Hidden Markov Models. Furthermore
feature engineering have significant importance in several approaches, which
is shown in [Aker et al., 2017]. This paper additionally includes a proposal
of a system implementation based on the presented survey.

##  1 Introduction

Fake news detection currently relies on knowing the attitude that people
communicating on social media are expressing towards an idea. Figuring this
out is called stance classification, which is a Natural Language Processing
(NLP) task that seeks to classify the stance taken towards some claim. This
paper reviews different ideas and approaches towards accomplishing this goal.

NLP is a research area concerned with processing human language using language
models and computational approaches like machine learning (ML). With the
progress of ML, tools and techniques open up for various ways of designing the
algorithm for stance classification. It is interesting to investigate this
progress and gain insight into current state-of-the-art approaches.

The work presented in this paper is carried out in the ”Thesis Preparation”
course at the IT-University of Copenhagen on the third semester of the MSc
Software Development program. As such it is a project preparing for the thesis
in Spring, 2019. The following is the tentative research question for the
thesis project.

###  1.1 Research question

Stance classification and fake news detection is currently mostly concerned
with the English language. The thesis project will attempt to answer the
following questions: how do we build an automatic stance classification system
for Danish? Further, how do we apply this system to verify or refute rumours
and possibly detect fake news?

###  1.2 Overview

The objective of this paper will thus be to study the approaches used for
stance classification and fake news detection in the English language and what
methods might be applicable to build a system for the Danish language. In
particular section 2 will provide context and definition for the term stance
classification. Section 3 will discuss definitions of fake news detection,
refer to recent work and discuss a number of social and psychological aspects
in the area. Section 4 will cover data gathering, feature extraction and data
annotation, as well as give context for the structure of microblogs. Section 5
covers a number of different approaches taken to classify stance and detect
fake news. Section 6 will present proposals for the choice of approach, data
gathering and technology for the thesis project, in addition to a high-level
thesis plan. Finally section 7 will summarise the findings of this research
paper.

##  2 Stance classification

Literature on stance classification and stance detection systems is rather
new, as most of the papers are published within the last 10 years. One of the
first studies in the area is from [Qazvinian et al., 2011], in which they
gather data from Twitter containing more than 10,000 tweets over 5 different
topics. They propose a system for identifying misinformation in microblogs
using different Bayes classifiers, and extracting “content-based”, “network-
based”, and “Twitter specific memes” features. Different approaches and
objectives have since been set to tackle the computational task of classifying
stance given some data based on a number of claims.

Conversations in microblogs, such as Twitter, are typically used in
classifying the stance for each reply to the source post, which expresses some
claim. Many systems use the Support, Denying, Querying, and Commenting (SDQC)
labels for classifying these posts[Zubiaga et al., 2016]. Before stance
classification is further investigated, we discuss applications of stance
classification as well as related subjects.

###  2.1 Applications

Stance classification is an area with closely related subjects, including
veracity classification/detection and fake news detection. The reason for this
is that stance classification can be used in the task of veracity
classification, as well as fake news detection[Dungs et al., 2018, Shu et al.,
2017a]. In this paper the term stance classification refers to the task of
determining the opinion behind some text towards a specific target. As such,
stance detection is the task of using the classification system to
automatically discover stance, and this term is used interchangeably with
stance classification. The same goes for veracity classification which, on the
other hand, is the task of resolving some claim by analysing crowd
reactions[Derczynski et al., 2017].

The task of stance classification often comes in two variants: open and
target-specific[Aker et al., 2017]. Open stance classification is applied in
contexts, where no target/topic is known in advance, which makes it suitable
for rumour resolution. Since the attitudes(stances) from a crowd towards some
claim can be indicative of its truthfulness, it is as such applicable in
veracity detection[Dungs et al., 2018]. In target-specific stance
classification, on the other hand, cues about a target that is known in
advance are provided in the training data. This can make classification of
stance from unseen data, but with the same target, easier[Mohammad et al.,
2016].

Furthermore the above described variants of stance classification can be
either supervised or unsupervised. In the former case classification has prior
knowledge based on a ground truth, i.e. data is annotated, and in the latter
case classification must be inferred from data, since there is no prior
knowledge111https://towardsdatascience.com/supervised-vs-unsupervised-
learning-14f68e32ea8d. Visited 03-12-2018.

In the next section we introduce fake news detection and explore how stance
classification is used for rumour resolution.

##  3 Fake news detection

One definition of fake news is that “fake news is news articles that are
intentionally and verifiably false”[Shu et al., 2017a]. The key features of
this statement is (1) authenticity: fake news include false information that
can be verified, and (2) intent: fake news is created with dishonest intention
to mislead consumers. A related area is that of rumour classification, in
which the veracity of circulating information is yet to be verified at the
time of spreading[Shu et al., 2017a]. Thus the distinction is that fake news
is intentionally misleading and is something which can be proven to be fake.
The problem to solve for detecting rumours and fake news is however much the
same. In the context of Twitter for example, given a source tweet containing a
claim and a number of responses, the task is to determine whether the claim is
true or false.

PHEME is a project dealing with the fake news detection problem described
above, focusing on veracity of data in social media and on the web[Derczynski
and Bontcheva, 2014]. In particular four kinds of false claims are sought to
be identified in real time: rumours, disinformation, misinformation, and
speculation. Out of these four categories disinformation most precisely
describes the definition of fake news given above, i.e. information that is
spread deliberately to deceive, in contrast to misinformation, which is
unintentional. Since the start of PHEME in 2014, several studies and papers
have been published dealing with the task mentioned here, including [Kochkina
et al., 2017, Derczynski et al., 2017, Zubiaga et al., 2016].

The task of identifying false claims is also undertaken in the Fake News
Challenge[Pomerleau and Rao, 2017]. The goal in this challenge is to explore
how ML and NLP can be used to combat the “fake news problem”. Specifically the
task is broken down into stages, with the first stage being stance detection,
classifying whether a body text agrees, disagrees, discusses or is unrelated
to a headline. Note that this is quite different from the analysis of
microblog data, where the posts in a sense are dynamic due to its temporal
feature. However, related to the task of the Fake News Challenge is the work
of [Augenstein et al., 2016], in which they build a classification system to
interpret tweet stance towards previously unseen targets and where the target
is not always mentioned in the text. Specifically they build a model to
classify tweets regarding Donald Trump, where the training and development
data is based on the targets Climate Change is a Real Concern, Feminist
Movement, Atheism, Legalization of Abortion, and Hillary Clinton.

###  3.1 Social and psychological aspects

Since fake news revolve around people it is interesting to investigate which
social and psychological factors that have relevance and implications for fake
news detection.

Some concepts that may have effects for the data used in fake news detection
are confirmation bias and the echo chamber effect[Shu et al., 2017a].
Confirmation bias describes consumers who prefer to receive information that
confirms their existing views, while the echo chamber effect describes users
on social media that tend to form groups containing like-minded people with
polarised opinions. These phenomena are discussed in [Quattrociocchi et al.,
2016], which carries out research on a large Facebook dataset. The research
shows that users tend to polarise their interactions with users and pages of
the same kind. Furthermore it is shown that the degree of polarisation
correlates with the degree of sentiment extremity in the users’ comments.

Another concept describing sharing of information between users is filter
bubbles and is covered in [Bechmann and Nielbo, 2018]. Filter bubbles describe
isolated users receiving news and information which does not overlap with
information other users get. As such filter bubbles are much alike echo
chambers, however [Bechmann and Nielbo, 2018] has a focus on filter bubbles in
relation to the Facebook news feed. The paper concludes that respectively 10.0
and 27.8 percentage of users in the used data set were in a filter bubble,
depending on the approach. Furthermore it is noted that there is no clear
connection between age, education, living location or gender and being in a
filter bubble. However the users in filter bubbles had fewer friends, group
likes and page likes than users who were not.

While [Bechmann and Nielbo, 2018] and [Quattrociocchi et al., 2016] both
examine spread and isolation of information, it is important to note a key
difference between them. [Bechmann and Nielbo, 2018] covers information spread
of news content specifically on the Facebook news feed in relation to the
algorithm Edge Rank222http://edgerank.net/ visited 09-12-2018, while
[Quattrociocchi et al., 2016] examine the spread of information in regards to
shared posts, page likes and so forth.

The above findings show that it is important to keep these social and
psychological aspects in mind, while considering the data used from social
media platforms. Otherwise polarised or skewed data could have implications
for the results and later usefulness of research in other contexts. This leads
to the next section, where data and factors which influence its quality is
discussed.

##  4 Data

Gathering data for stance classification is a task in itself, as different
factors, such as bias and class distribution, can have significant
consequences for the resulting system. Social and psychological aspects in
this regard are discussed above in section 3.1. Furthermore classifiers
performs better with datasets with balanced class labels after annotation has
been performed. Otherwise you might end up with misleading/imprecise
classification systems: In [Kochkina et al., 2017] they build the best-
performing system for SemEval 2017 Task 8, subtask A, but due to unbalanced
data, the model is primarily able to just classify “commenting” instances,
with only few correct predictions of “denying” and “supporting” instances,
which are the more interesting classes.

###  4.1 Data gathering

This section will provide an overview of approaches to gather relevant data
for the stance classification task.

In [Castillo et al., 2011] a system is generated to gather data from Twitter
and filter newsworthy topics. First they monitor Twitter posts in a period of
2 months using a monitoring system333“Twitter Monitor”(currently unavailable)
from:
http://www.twittermonitor.net/, which detects bursts in the frequency of sets
of keywords found in messages. Then they query the system with specific
keywords and collect tweets that match them during the burst peaks. They
gather Twitter data in this way on over 2500 topics, and filter newsworthy
ones from pure conversations with the MTurk API444https://www.mturk.com/. The
paper also describes how the labels given from MTurk is used to train a J48
decision tree classifier to filter the topics automatically.

Similarly a dataset is generated from Twitter using regular expression queries
in [Qazvinian et al., 2011]. They utilise the Twitter API by searching for
data with queries that each represent a popular rumour that is deemed either
“false” or “partly true” by About.com555http://urbanlegends.about.com. Then
two annotators manually go over all the tweets collected and annotate whether
they are about a set of particular rumours.

More recent datasets include those in the SemEval tasks, such as SemEval 2016,
task 6[Mohammad et al., 2016] and SemEval 2017, Task 8[Derczynski et al.,
2017]. Alternative datasets are discussed in [Shu et al., 2017a], including
BuzzFeedNews, LIAR, BS Detector, and CREDBANK. They point out, however, that
these datasets have limitations that make them challenging for fake news
detection. As a result they are currently in the process of developing their
own dataset, which include news content and social context features[Shu et
al., 2018], which are the feature categories they find important for the fake
news detection task.

###  4.2 Feature extraction

Once data has been extracted for analysis, one must extract features relevant
for the task at hand. The subject of feature engineering could comprise a
whole paper in itself. As such this section will not try to compare features,
but will provide an overview of the most common features used for stance
classification and fake news detection[Castillo et al., 2011, Shu et al.,
2017a, Qazvinian et al., 2011, Aker et al., 2017, Kochkina et al., 2017,
Enayet and El-Beltagy, 2017]. Table 1 is a compact list of groups of similar
features with accompanying short descriptions. Additionally a popular approach
is to also include word embeddings using the word2vec algorithm[Google, 2013],
representing words by dense vectors, as done in [Kochkina et al., 2017].

Feature |  Description
---|---
Lexical |  Count of words and characters, ratio of capital letters, names, as well as presence of period, question mark, exclamation mark, and special words (e.g. negation words)
Attachments |  URLs, images, and/or hashtags content
Syntax |  Sentence-level features, e.g. n-grams, BOW, POS tags
User |  No. of posts written, user creation date, no. of followers, demographics
Post |  Source or reply, relation to other posts, sentiment (positive/negative polarity), temporal information
Table 1: An overview of the most common features used in stance classification
and fake news detection

With the progress of ML, tools and techniques open up for various ways of
tackling the task of stance classification. Several studies however show that
the most crucial part of stance classification is to extract and use the most
optimal features[Aker et al., 2017, Dungs et al., 2018]. In early work, it was
explored how features could be categorised into four classes, message-,
topic-, user-, and propagation-based [Castillo et al., 2011]. Although they
are Twitter-specific, they are claimed to be generic. The message-based
features deals with characteristics of messages, such as tweet-length. The
user-based features, on the other hand, deals with characteristics of the
user, which posts the message, such as registration age. Topic-based features
are then an aggregation computed from the message- and user-based, such as the
fraction of tweets containing URLs. Finally the propagation-based features
consider the conversation tree, e.g. depth of the tree.

Another study shows that more or less abandoning the idea of having many
features can provide significant results [Dungs et al., 2018]. Their
contribution shows how stance and tweet times alone achieve state-of-the-art
results in the task of veracity detection as opposed to approaches using
content and user based features as those introduced above. Along these lines
[Aker et al., 2017] shows how, by adding just six tweet confidence “problem-
specific” features to existing well-performing features, they achieve better
results than previous systems on the same data. They prove this by using a
decision tree stance classifier, which allegedly is simpler in its approach in
comparison to competing systems’.

###  4.3 Data structure in microblogs

This paper investigates stance classification over social media data and in
particular from microblog platforms, as the structure makes it applicable for
this task[Tolmie et al., 2018]. As an example of a microblog conversation,
figure 1 from [Kochkina et al., 2017] illustrates a Twitter conversation,
where a source post makes a claim and nested replies respond to it either
directly or indirectly. Note that the tweets are also annotated, which is
discussed in the next section.

Figure 1: A conversation thread with three branches. Source: [Kochkina et al.,
2017]

In [Procter et al., 2013] they analyse how rumours propagate in Twitter, which
we hypothesize also applies for similar microblogs such as
Reddit666https://www.reddit.com/. In short it comprises the following events,
which we have reformulated to be general for microblogs:

  1. 1.

A rumour starts with someone posting about the occurrence of an alleged
incident.

  2. 2.

The rumour spreads as the post is shared and some form of evidence may be
added in the process.

  3. 3.

Others begin to challenge its credibility.

  4. 4.

A consensus begins to emerge

This can be compared to figure 1, where user 0 starts a rumour and several
other users replies, some challenging its credibility by either querying the
rumour(user 2) or denying it(user 4). Had the example been bigger we might
also have seen other people actually re-posting the initial post, some
supporting it with URLs to the incident, and after some time a general
consensus could possible be inferred from the messages.

###  4.4 Annotation

When data is gathered and features are extracted, the question is then which
kind of labels one should use for annotation. One annotation scheme that is
popular is SDQC which labels a text as either supporting, denying, querying or
commenting in regards to some rumour. This is discussed in section 4.4.1,
followed up by a comparison to topic classification in section 4.4.2, which
tries to label a text to be within some predefined category.

Manually annotating data does however come with some challenges. It is time
consuming to have experts and individuals annotate the data manually and the
annotations could be influenced by the individuals’ personal bias. Different
annotators might have different views of which labels are appropriate for some
microblog post. One example is [Stranisci et al., 2016], where 8 annotators
manually annotate over 8000 tweets. Each tweet is annotated twice by different
annotators, and there are disagreements on more than 2000 of the tweets. To
mitigate the disagreements from personal bias, a crowd sourcing platform is
utilised to give another set of
annotations777https://en.wikipedia.org/wiki/Figure_Eight_Inc. Visited
09-12-2018.

Not only is it important which labels are used, but also what data is being
annotated. Twitter is a popular platform for gathering data. It facilitates an
easy way to gather large amounts of text data which can circumvent
controversial debates or events. Using public datasets in research should help
enabling others to verify and improve on prior research.

While Twitter is a great platform for gathering data, it is not the only
source of data out there and this must be kept in mind. If data from Twitter
is primarily used, one could think that it might skew models and systems to be
optimised for text written in the context of that particular social media
platform and might be less useful elsewhere. This is further discussed in
section 4.5.

####  4.4.1 Labels and SDQC

The idea of using the SDQC labels stems from an article, which experimentally
analyses tweets sent during the August 2011 riots in England with a
“computationally assisted methodology”[Procter et al., 2013]. They develop a
code frame for annotating rumour tweets with 6 labels: claim with and without
evidence, counterclaim with and without evidence, appeal for more information,
and comment. This framework is extended and used in a more recent study, in
which they develop a methodology that enables to collect, identify and
annotate rumour data from Twitter[Zubiaga et al., 2016]. They assume two types
of tweets, namely the source tweet that initiates the rumour and the response
tweets that respond to it(See also figure 1). They categorise their labels in
three main dimensions, which express a mode of interaction; support/response
type, certainty, and evidentiality.

Support/response type is dependent on the tweet type, where a source tweet can
be labelled as supporting, denying or under-specified in regards to the
content of the statement. If it is a response tweet, it can be labelled as
agreed, disagreed, appeal for more information, and comment. These labels
corresponds to the codes using in the formerly mentioned paper, [Procter et
al., 2013]. In addition to their work, however, [Zubiaga et al., 2016] also
consider response types for nested responses, i.e. tweets not directly
responding to the source tweet.

The certainty measures the degree of confidence expressed by the author of a
tweet when posting a statement in the context of a rumour. The values for the
dimension include; certain, somewhat certain, and uncertain. Finally
evidentiality determines the type of evidence, if any, provided directly in
relation to the rumour being discussed. This includes seven labels, where
attachments and quotation are examples.

The methodology described above is the general approach for the articles
investigated in this paper as most of them work with data following the format
described in section 4.3. An important take-away is the observation that
nested posts/replies play a big role for the propagation of rumours.

####  4.4.2 A related annotation scheme: Topic classification

SDQC seems to be a fair annotation scheme, as the labels divide the classes
into very general opinion categories, supposedly making it very suitable for
stance classification. In comparison we could look at a different approach and
investigate another annotation scheme. One such example could be topic
classification.

Topic classification is somewhat similar to stance classification, but differs
in its objective, and thus its annotation scheme. Where the latter deals with
classifying opinions(stance) from text, the former deals with classifying
specific topics from the content of text. This approach is used in [Giménez et
al., 2017] to analyse tweets in regards to the Spanish election in 2015. They
introduce five categories for topic labelling: (1) political issues, (2)
policy issues, (3) personal issues, (4) campaign issues, and (5) other issues.

In regards to SDQC, where the labels are rather general and can be used in any
stance classification task, this annotation scheme is rather context specific.
They conclude that the task was complicated, in particular when the topics
were similar. One can indeed imagine that the tweet data would contain text
with generally more than one of the topics included, making it difficult to
annotate it with only one category. With SDQC, you would typically not see a
person both support and deny some claim. Thus stance classification is more
forgiving in comparison to topic classification when it comes to the
annotation scheme.

###  4.5 Twitter conversations as a social phenomenon

The methodology behind the SDQC annotation scheme is analysed in [Tolmie et
al., 2018], where they compare the sociological aspects of conversations and
compare them to that of microblogging, and in particular Twitter. They
conclude that microblogging cannot be treated as a face-to-face conversation
due to various factors, including the asynchronous nature of the technology
and limits in messages. They investigate microblogging as a turn-taking
system, in which one person initiates a message(and potentially a rumour),
from which users take turn in responding to. One interesting observation in
this regard is that the flow in face-to-face conversations allows for
naturally “self-selecting” next speakers, whereas there are no turn order in
microblogging because of the temporal gaps. They find out that rumours unfold
across multiple turns and that one needs to examine the organisational
characteristics of how specific phenomena unfold in social interaction in
order to understand how they work as social phenomena. This means that
focusing on one tweet in isolation is very limiting in regards to the
information that can be extracted in a social context. The annotation schema
is then based on the following observations in relation to Twitter, where 4
and 5 are specifically related to the production of rumours:

  1. 1.

Tweets are sequentially ordered

  2. 2.

Exchanges involve topic management

  3. 3.

Important accountability mechanisms are in play

  4. 4.

Agreement and disagreement

  5. 5.

How tweets are rendered trustworthy through production or evidence

More specifically 2 and 3 relate to the task of labelling source tweets as
either aligning or refuting a news event. 1, 2, 3, and 4 relate to the task of
labelling whether replies agree or disagree with the source tweet, and finally
certainty and evidentiality relate to 5.

####  4.5.1 Big data on social media

Related to the subject of Twitter conversations in a social context, [Tufekci,
2014] is a research paper on big data on social media, in which methodological
and conceptual challenges for this field are studied. Validity and
representativeness of social media big data is in focus. Issues in this regard
are introduced, in which some of them are of particular interest in the
context of stance classification and fake news detection.

One is the “model organism problem”, in which the use of a few specific social
media platforms are frequently used to generate data with no consideration of
potential bias. It is argued that online platforms such as Twitter raises
important questions of representation and visibility because of the difference
in behaviour depending on demography and social groups. The point is that we
might miss out on important data by making use of the same platforms over and
over again.

Another interesting issue is that big data analysis typically relies only on a
single social media platform, whereas it is rarely the case that such
information is only confined to one source. It is argued that such analysis
must take into account that there may be effects which are not visible because
relevant information is missing. Thus a request for more research on more than
one platform is given in order to understand broader patterns of connectivity.

Finally a point on human interaction on social media platforms argue that
human self-awareness needs to be taken into account in big data analysis as
humans behave differently when they know they are being observed. Along these
lines it is argued that one should take into account that people often are
aware of the mechanisms involved in social media communication, and as such
can exploit it for their own benefit. This is also related to the concept of
confirmation bias which is discussed in section 3.1.

To summarise, even if a social media platform such as Twitter provides easily
available data on news event, one should consider the actual data content. It
is important to investigate whether the data is representative, whether other
platforms can contribute and who the users communicating are and how they
behave.

##  5 Classification approaches

Once data has been gathered and annotated and features extracted, one must
decide which approach to use for the actual classification system. This
section will provide an overview of different approaches, as well as their
results(section 5.10), both for sole stance classification but also applied to
fake news detection.

###  5.1 Recurrent Neural Network and Long-Short Term Memory

Recurrent neural network(RNN) systems allow representing arbitrarily sized
structured inputs in a fixed-size vector, while paying attention to the
structure properties of the input. This makes it quite appropriate for the
type of data for stance classification. [Kochkina et al., 2017] implements
such a system in SemEval-2017 Task 8 subtask A, in which they implement a Long
Short-Term Memory(LSTM) version, feeding in whole conversation branches as
input, as opposed to the typical case of using words alone as input. They end
up with very nice results, even coming out as the best performing system for
the task.

[Augenstein et al., 2016], as earlier mentioned, utilises the LSTM model
differently. They use the LSTM to facilitate an encoding of target text and
source text, using a “Bidirectional conditional LSTM” approach. That is, input
is fed into the network in both directions, and one layer is dependent on
another layer’s state, thus making it conditional.

[Zarrella and Marsh, 2016] also implements an RNN, but takes an interesting
approach by pre-training their model on existing related twitter data
utilising the most frequent hash-tag as a label. The lack of domain-labelled
data was a challenge in their project and the pre-training of their model was
an attempt to tackle this, and yielded good results in comparison to a
randomly initialised model.

###  5.2 Support Vector Machine

Another SemEval-2017 Task 8 paper, which deals with both subtask A and B is
[Enayet and El-Beltagy, 2017]. They have focus on the latter(veracity
prediction), in which they scored best. They use a linear Support Vector
Machine (SVM) approach with BOW features and other manually selected features.
The SVM maps data to points in space, and then assign classes to the data
depending on the positions of output, as opposed to the probabilistic
classification approach typically used in neural networks.

###  5.3 Convolutional Neural Network

Another approach based on neural networks is that of convolutional neural
networks, which works particularly well with spatial data structures.

[Chen et al., 2017] implements a CNN model with leave-one-out(LOO) testing to
classify with SDQC. They use varying window sizes, while also training 5
models independently and using majority voting to find results. A similar
approach is implemented in [Kim, 2014], which deals with seven different NLP
tasks, experimenting with different CNN architectures. It is shown that a CNN
with “static”(non-trained in backpropagation) word embeddings and “non-
static”(trained in backpropagation) word embeddings performs well, whereas a
combination of the two, denoted as “CNN-multichannel” overall performs best.
As such it is concluded that unsupervised pre-training of word vectors is an
important factor for NLP deep learning, and that a one-layered CNN can perform
remarkably well.

###  5.4 Crowd stance

One approach for fake news detection, is to analyse the replies on the source
post of the news. Applying stance classification on microblog conversation
data enables analysis of the general stance/opinion of the “crowd”[Pomerleau
and Rao, 2017, Derczynski et al., 2017]. If there for example is a lot of
negative response to a post, it might show that the crowd is sceptical about
the claim/rumour in the source and vice versa.

###  5.5 Hidden Markov Model

One example of a crowd stance implementation is the use of a Hidden Markov
Model(HMM) in [Dungs et al., 2018], which uses stance and tweets’ times alone
for automatic rumour veracity classification, achieving state-of-the-art
results. HMM is well known for temporal pattern recognition, which they
utilise as follows: Regard individual stances over a rumour’s lifetime as an
ordered sequence of observations, and then compare sequence occurrence
probabilities for true/false. Their results are obtained using gold stance
labels, but they also test it with automatically generated stance labels[Aker
et al., 2017], and observe only a marginal decrease in performance. This shows
that their veracity classification system has viable practical applications.

Furthermore they apply their system in a setting for early detection of
rumours. That is, they limit the number of tweets to the first 5 and 10 tweets
respectively. Surprisingly, even with only 5 tweets, their model still
outperforms the baselines, which use all of the tweets.

###  5.6 Tree classification

In [Aker et al., 2017] one of the approaches used is a J48 decision tree
classifier, which is build over a number of features from earlier work,
extended with some “problem-specific” features. The approach reaches state-of-
the-art performance for stance classification, and shows that a simple
classifier can work really well. A lot of work went into defining the features
for a tweet and shows that the features used to define a tweet are key to
having good results.

####  5.6.1 Ensemble with CNN

Another interesting approach is that of the winning team for the Fake News
Challenge[Pomerleau and Rao, 2017], which uses an ensemble of decision trees
and a CNN[Baird et al., 2017]. Specifically their model is based on a 50/50
weighted average between gradient-boosted decision trees and a deep CNN.
Neither of the models have impressive accuracy, but the 50/50 weighting in the
final classification step improves their results.

###  5.7 Multi-Layered Perceptron

The second best scoring team in the Fake News Challenge[Pomerleau and Rao,
2017], utilises a Multi-Layered Perceptron(MLP) [Hanselowski et al., 2017]
approach with several ReLU layers. The final system ensemble 5 instances of
the MLP model and decides the output label by majority voting between the
instances. The team scoring third best in the Fake News Challenge also
implements an MLP, employing lexical and similarity features with one hidden
ReLU layer[Riedel et al., 2017]. As noted in the paper, the results are quite
disappointing since the model is inaccurate at predicting the most interesting
classes, “agree” and “disagree”.

###  5.8 Matrix factorization

In [Shu et al., 2017b] an approach trying to exploit a tri-relationship
between publishers, news contents and social engagements is used to do fake
news detection. The approach(denoted “TriFN”) utilises non-negative matrix
factorisation to generate latent feature vectors for users and news. A
comprehensive mathematical model describing the problem as a minimisation
problem is described and formally optimised. The results of the TriFN
framework outperforms a number of baselines and yields positive results.

###  5.9 Bayes classification

A more mathematical approach than those described so far is classification
with Bayes classifiers, as implemented in [Qazvinian et al., 2011], being a
pioneering paper in the area of stance classification and fake news detection.
The approach is based on learning a linear function of different Bayes
classifiers as high level features to predict whether a user believes a
specific rumour or not. Each of the classifiers calculates the likelihood
ratio for a given tweet to be under either a positive model or negative model
with respect to the given feature(i.e. Bayes classifier).

###  5.10 Performance overview

In appendix A a brief overview of the used method, dataset and results for the
papers presented in this section is shown in table 2. Note that this is not
meant to be a direct comparison of the approaches taken in the papers, as some
of the papers use different metrics, degree of classification and datasets and
a comparison would as such not give much value.

The results for the Fake News Challenge systems are using a custom metric,
making it difficult to compare with other systems. However, [Hanselowski et
al., 2018] have reproduced the results and reports their $F_{1}$ score, which
are used here as well. Interestingly the teams’ rankings change when comparing
$F_{1}$ scores. Note also that the lower part of table 2(divided by double-
lines) reports results for binary classification for the fake news detection
task, whereas the upper part is for stance classification.

##  6 Proposal of system implementation approach

In this section we propose a concrete system implementation for the thesis
project as a result of the research in this paper. The objective is to build a
stance classification system for Danish over social media data, and apply it
to rumour resolution and possible to detect fake news. As such we need to
gather data, annotate it, build a classification system and deploy it888By
deploying it we mean to make it publicly available to use “out-of-the-box” on
GitHub. Furthermore we include a tentative thesis plan as a feasibility check
and guideline for the project.

###  6.1 Data gathering and annotation

For the data gathering and annotation phase we propose to use the social media
platform, Reddit, and in particular the official Danish sub-reddit on
https://www.reddit.com/r/Denmark/. Reddit has an open-source API allowing to
use their data for non-commercial use999https://www.reddit.com/wiki/api
Visited 04-12-2018. Clearly Twitter would be an obvious candidate to choose as
is clear from the research presented in this paper, but we have spent some
time on exploring Danish news on this platform, which is not really present.
Facebook would be another good candidate, but its data is not publicly
available.

For annotation we propose to use the SDQC approach, that is, four labels
indicating: support, denying, querying, and commenting.

Alternatively, if we do not succeed in finding a proper microblog platform, we
can use the same approach as the task in the Fake News Challenge[Pomerleau and
Rao, 2017], i.e. performing stance classification based on a headline and a
body text for a news event.

###  6.2 Classification and detection system

We propose to build a stance classifier model with a decision tree approach,
which is covered in section 5.6, as it has proved to be a simple yet effective
approach for stance classification.

Further, the approach described in section 5.5 is quite interesting, achieving
very nice results, implementing a HMM for veracity detection using few and
simple features, including stance and tweets’ times. We propose to use the
same approach for fake news detection based on historical events.

In the case where we do not have data in the form of a microblog, but as news
articles(see section 6.1 above), we propose to use an ensemble approach as the
one introduced in section 5.6.1, combing tree classification with deep
learning methods.

###  6.3 Technology

We also propose a framework and programming environment to work with for the
project. Python is a popular programming language for data analysis, including
ML and NLP, because of its plethora of libraries. A HMM model could be
implemented using hmmlearn101010https://github.com/hmmlearn/hmmlearn, a neural
network could be implemented using PyTorch111111https://pytorch.org/, and a
decision tree with the scikit-learn library121212https://scikit-learn.org/.
Python also features a rich library for data pre-processing,
NLKT131313https://www.nltk.org/, which among other things does automatic word
tokenization. Apart from the useful libraries, Python is a high-level language
allowing for a code-first approach, i.e. fast prototyping.

###  6.4 Thesis plan

The thesis project is carried out February through June 2019, with deadline
for hand-in June 3rd at 14.00 o’clock. Thus we have 17 weeks to implement the
system and write the thesis paper. A high-level thesis plan is sketched in
table 3, appendix B.

The plan comprises of work items for the implementation of the system and
thesis sections divided into the weeks we intend to carry them out. The first
month the focus will be on data gathering and annotation. Simultaneously and
the following month we will build a prototype for an early evaluation of the
data. The third month we will tune the system, run experiments and change the
parameters accordingly, allowing us to test and evaluate the system. Then we
will deploy it, making it publicly available.

The plan also consists of a week by week overview of the sections we will
focus on in the thesis paper. First off the general structure of the thesis as
well as data gathering will be in focus. In the following weeks the
annotation, baseline and dataset will be covered. After this the choice of
technology, data analysis and the prototype systems will be discussed. Then
the parameter space, optimal parameters and results will be reported after
running experiments. Finally the results will be analysed and discussed before
concluding the thesis.

##  7 Conclusion

The objective for this research paper has been to survey stance classification
and fake news detection. The task of classifying the opinion of a crowd
towards a rumour has been explored by many approaches within the last 10
years, resulting in very useful findings. One particularly interesting use of
stance classification is to use it for assessing whether some news event is
true or false. One challenge in this regard is the social and psychological
aspects occurring in microblogs, where polarised opinions take effect because
of filter bubbles and the echo chamber effect. Another challenge for analysing
data from microblogs, such as Twitter, is the model organism problem, which is
the prevalent issue of representation and visibility when continuously using
the same platforms in stance classification.

We have further investigated the process of data gathering and annotation, and
how imbalanced data can have significant impact of the results obtained in
stance classification. In particular feature engineering seem to be of great
importance, choosing representative information to extract that will do great
on the test data at hand while still being general-purpose oriented.

Different methods for stance classification and fake news detection have been
explored, but because of different data and metrics it has been difficult to
directly compare their results. However, one particular approach is very
relevant and interesting for the thesis project, which is the use of a HMM in
analysing rumours in microblog data, achieving very promising results.
Furthermore the use of a decision tree model for stance classification appear
to be a good choice. These are also the approaches we propose to use in the
thesis project, where we intend to gather a dataset in the Danish language,
annotate it, build the classifier/detection system and deploy it. In
conclusion the findings in this research paper will most likely prove very
useful as background knowledge in the coming thesis project.

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##  Appendix A Results overview

Approach | Acc | $F_{1}$ | Dataset
---|---|---|---
Transfer Learning RNN | - | 67.8 | [Mohammad et al., 2016]
Bidirectional LSTM | - | 58.3 | [Mohammad et al., 2016]
Branch LSTM | - | 43.4 | [Derczynski et al., 2017]
CNN | - | 53.6 | [Derczynski et al., 2017]
SVM | 53.0 | - | [Derczynski et al., 2017]
J48 | 79.02 | - | [Derczynski et al., 2017]
MLP | - | 60.4 | [Pomerleau and Rao, 2017]
MLP | - | 58.3 | [Pomerleau and Rao, 2017]
CNN and Tree ensemble | - | 58.2 | [Pomerleau and Rao, 2017]
Bayes | 94.1 | 92.5 | [Qazvinian et al., 2011]
Hidden Markov Models | - | 80.4 | [Zubiaga et al., 2016]
TriFN | - | 87.0 | [Shu et al., 2017b, BuzzFeed]
TriFN | - | 88.0 | [Shu et al., 2017b, Politifact]
Table 2: Overview of performance results for the different approaches for
stance classification(top) and fake news detection(bottom)

##  Appendix B Thesis plan

Week |  Work item |  Thesis |  Milestones
---|---|---|---
6 |  Data gathering |  General structure and data gathering |
7 |  Data annotation and
prototype |  Annotation |
8 |  Data annotation and
prototype |  Baseline |
9 |  Data annotation and
prototype |  Dataset |
10 |  Prototype testing |  Technology |  Working prototype and gathered dataset
11 |  Data evaluation |  Data analysis and
statistics |
12 |  Finalize prototype |  Describe prototype
system |
13 |  Finalize prototype |  Describe prototype
system |
14 |  Tune system
parameters |  Parameter space |  Intermediate results and finished prototype
15 |  Tune system
parameters |  Optimal parameters |
16 |  Test |  Experiment results |
17 |  Test |  Draft for final version |
18 |  Evaluation of results |  Result and error
analysis |  Draft for thesis and results gathered
19 |  Evaluation of results |  Result and error
analysis |
20 |  System revision |  Discussion |
21 |  Conclude |  Conclude, abstract |
22 |  Deploy system |  Proof read |
Table 3: Thesis plan