Heavy quarks in the presence of higher derivative corrections from AdS/CFT
K. Bitaghsir Fadafan
Physics Department, Shahrood University of Technology,
P.O.Box 3619995161, Shahrood, Iran
E-mails<EMAIL_ADDRESS>

Abstract

We use the gauge-string duality to study heavy quarks in the presence of
higher derivative corrections. These corrections correspond to the finite
coupling corrections on the properties of heavy quarks in a hot plasma. In
particular, we study the effects of these corrections on the energy loss and
the dissociation length of a quark-antiquark pair. We show that the calculated
energy loss of heavy quarks through the plasma increases. We also find in
general that the dissociation length becomes shorter with the increase of
coupling parameters of higher curvature terms.

###### Contents

  1. 1 Introduction
  2. 2 Energy loss of heavy quark at finite coupling
    1. 2.1 Set up of calculations
    2. 2.2 Positive couplings
    3. 2.3 non-positive couplings
    4. 2.4 analytic solution
  3. 3 dissociation length of quark-antiquark pair at finite coupling
    1. 3.1 dissociation length from AdS/CFT
    2. 3.2 Numerical Solutions
  4. 4 Conclusion
  5. 5 Review of Quasi-topological gravity

##  1 Introduction

The experiments of Relativistic Heavy Ion Collisions (RHIC) have produced a
strongly-coupled quark$-$gluon plasma (QGP)[1]. There are no known
quantitative methods to study strong coupling phenomena in QCD which are not
visible in perturbation theory (except by lattice simulation). A new method
for studying different aspects of QGP is the $AdS/CFT$ correspondence [2, 3,
4, 5]. This method has yielded many important insights into the dynamics of
strongly-coupled gauge theories. It has been used to investigate
hydrodynamical transport quantities in various interesting strongly-coupled
gauge theories where perturbation theory is not applicable [6]. Methods based
on $AdS/CFT$ relate gravity in $AdS_{5}$ space to the conformal field theory
on the four-dimensional boundary. It was shown that an $AdS$ space with a
black brane is dual to a conformal field theory at finite temperature.

The universality of the ratio of shear viscosity $\eta$ to entropy density $s$
[7, 8, 9, 10] for all gauge theories with Einstein gravity dual raised the
tantalizing prospect of a connection between string theory and RHIC. The
results were obtained for a class of gauge theories whose holographic duals
are dictated by classical Einstein gravity. Recently, $\frac{\eta}{s}$ has
been studied for a class of CFTs in flat space with higher derivative
corrections [11, 13, 12, 15, 16, 14]. In these studies, the effects of $R^{2}$
corrections to the gravitational action in AdS space have been computed and it
was shown that the conjecture lower bound on the $\frac{\eta}{s}$ can be
violated. For example, in the Reissner$-$Nordström$-$AdS black brane solution
in Gauss$-$Bonnet gravity, the $\frac{\eta}{s}$ bound is violated and the
Maxwell charge slightly reduces the deviation [16]. Regarding this study and
motivated by the vastness of the string landscape [18], we explored the
modification of the jet quenching parameter and drag force on a moving heavy
quark in the strongly-coupled plasma in [19].

Recently, a new higher derivative theory of gravity in five-dimensional
spacetime which contains not only the Gauss-Bonnet term but also a curvature-
cubed interaction introduced [33, 34]. This theory is known as quasi-
topological gravity theory which is thought to be dual to the large $N$ limit
of some conformal field theory without supersymmetry. Unlike Lovelock gravity,
this cubic term is not purely topological. Therefor it would be useful to
consider curvature-cubed terms as the higher derivative corrections and
investigate behavior of the heavy quarks by means of the $AdS/CFT$
correspondence. Holographic investigation of Quasi-topological gravity have
been done in [35]. Also it was shown that the lower bound of the ratio of
shear viscosity to density entropy can be violated in this background [36].

In this paper we use the $AdS/CFT$ correspondence to study effect of higher
derivative corrections to the properties of the heavy quarks.111 In general,
we do not know about forms of higher derivative corrections in string theory,
but it is known that due to the string landscape one expects that generic
corrections can occur. One should notice that string theory contains higher
derivative corrections from stringy or quantum effects, and such corrections
correspond to $1/\lambda$ and $1/N$ corrections. In the case of
$\mathcal{N}=4$ super Yang$-$Mills theory, the dual corresponds to the type
$\amalg B$ string theory on $AdS_{5}\times S^{5}$ background. The leading
order corrections in $1/\lambda$ arise from stringy corrections to the low
energy effective action of type $\amalg B$ supergravity, $\alpha^{\prime
3}R^{4}$.

Employing numerical methods, we investigate energy loss and dissociation
length of heavy quarks in section 2 and 3, respectively. One finds that the
energy loss of heavy quarks increases by increasing higher derivative
corrections. Also the dissociation length becomes shorter with the increase of
coupling parameters of higher curvature terms. We summarize the effects of
these corrections in the last section. In the appendix, we give a brief review
of [33].

##  2 Energy loss of heavy quark at finite coupling

In this section, we investigate the finite-coupling corrections to the energy
loss of a moving heavy quark in the Super Yang-Mills plasma using the AdS/CFT.
These corrections are related to the curvature corrections to the AdS black
brane solution.

The effect of curvature-squared corrections to the drag force on a moving
heavy quark in the Super Yang-Mills plasma is investigated in [39]. It is
shown that the corrections to the drag force depend on the velocity of heavy
quark. This dependance is such that for $v>v_{c}$ including the corrections
increase the drag force. This means that at the critical velocity $v_{c}$ the
curvature-squared corrections have the minimum effect on the drag force. For
the particular case of Gauss-Bonnet gravity, we do not expect a critical
velocity [39]. Also in this background, the drag force is larger than the
$\mathcal{N}=4$ case if $\lambda$ (Gauss-Bonnet gravity constant) is positive
while it is smaller than the $\mathcal{N}=4$ case if $\lambda$ is negative.

Now we continue with considering curvature-cubic corrections. Our purpose is
finding a general rule for considering higher derivative terms. We use the
proposal of [33, 34] and study new higher derivative theory of gravity in
five-dimensional spacetime which contains not only the Gauss-Bonnet term but
also a curvature-cubed interaction.

We should emphasize that in the case of these corrections, one can not predict
a result for $\mathcal{N}=4$ SYM because the first higher derivative
correction in weakly curved type IIB backgrounds enters at order
${\cal{R}}^{4}$. These corrections on the drag force have been studied in [40]
and it was found that the drag force for a heavy quark moving through
$\mathcal{N}=4$ SYM plasma is generally enhanced by the leading correction due
to finite ’t Hooft coupling. We will compare our results with this observation
and interestingly find a general rule for curvature corrections.

###  2.1 Set up of calculations

In the framework of $AdS/CFT$, an external quark is represented as a string
dangling from the boundary of $AdS_{5}-$Schwarzschild and a dynamical quark is
represented as a string ending on flavor D7-brane and extending down to some
finite radius in $AdS$ black brane background. We consider the $AdS$ black
hole solution in quasi-topological gravity 222one finds a quick review of this
background in the appendix. as

$ds^{2}=r^{2}\left(-N^{2}\,f(r)dt^{2}+d\vec{x}^{2}\right)+\frac{dr^{2}}{r^{2}\,f(r)},$
(1)

notice that we work in units where the radius of $AdS$ is one. Here $r$
denotes the radial coordinate of the black brane geometry and $t,\vec{x}$
label the directions along the boundary at the spatial infinity. In these
coordinates the event horizon is located at $r_{h}$ and it is found by solving
$f(r_{h})=0$ equation. The boundary is located at infinity and the geometry
will be as asymptotically $AdS$. The constant $N^{2}$ specifies the speed of
light of the boundary gauge theory and one can choose it to be unity. We name
$f(r)$ at the boundary where $r\rightarrow\infty$, as $f_{\infty}$ and one
finds that

$N^{2}=\frac{1}{f_{\infty}},$ (2)

The temperature of the hot plasma is given by the Hawking temperature of the
black hole

$T=\frac{N\,r_{h}}{\pi}.$ (3)

The relevant string dynamics is captured by the Nambu-Goto action

$S=-\frac{1}{2\pi\alpha^{\prime}}\int d\tau d\sigma\sqrt{-det\,g_{ab}},$ (4)

where the coordinates $(\sigma,\tau)$ parameterize the induced metric $g_{ab}$
on the string world-sheet and $X^{\mu}(\sigma,\tau)$ is a map from the string
world-sheet into the space-time. Defining $\dot{X}=\partial_{\tau}X$,
$X^{\prime}=\partial_{\sigma}X$, and $V\cdot W=V^{\mu}W^{\nu}G_{\mu\nu}$ where
$G_{\mu\nu}$ is the AdS black hole solution in Quasi-topological gravity (1),
then

$-det\,g_{ab}=(\dot{X}\cdot X^{\prime})^{2}-(X^{\prime})^{2}(\dot{X})^{2}.$
(5)

We follow [37, 38] and focus on the dual configuration of the external quark
moving in the $x$ direction on the plasma. The string in this case, trails
behind its boundary endpoint as it moves at constant speed $v$ in the $x$
direction

$x(r,t)=vt+\xi(r),\,\,\,\,\,y=0,\,\,\,z=0.$ (6)

One finds the lagrangian in the static gauge $(\sigma=r,\tau=t)$ as follows

$\mathcal{L}=\sqrt{-det\,g_{ab}}=N^{2}+r^{4}N^{2}f(r)x^{\prime
2}-\frac{\dot{x}^{2}}{f(r)},$ (7)

The equation of motion for $\xi$ implies that $\frac{\partial
L}{\partial\xi^{\prime}}$ is a constant. We name this constant as $\Pi_{\xi}$
and solve this relation for $\xi^{\prime}$, the result is

$\xi^{\prime
2}=\frac{\left(\frac{\Pi_{\xi}^{2}}{f(r)}\right)\left(-N^{2}f(r)+v^{2}\right)}{r^{4}N^{2}f(r)\left(-r^{4}N^{2}f(r)+\Pi_{\xi}^{2}\right)}.$
(8)

We are interested in a string that stretches from the boundary to the horizon.
In such a string, $\xi^{\prime 2}$ remains positive everywhere on the string.
Hence both numerator and denominator change sign at the same point and with
this condition, one finds the constant of motion $\Pi_{\xi}$ in terms of the
critical value of $r_{c}$ as follows

$\Pi_{\xi}=v\,r_{c}^{2},$ (9)

The drag force that is experienced by the heavy quark is calculated by the
current density for momentum along $x^{1}$ direction. After straightforward
calculations, the drag force is easily simplified in terms of $\Pi_{\xi}$

$F=-\frac{1}{2\pi\alpha^{\prime}}\Pi_{\xi}.$ (10)

As a result, to find the drag force one should find the constant of motion,
$\Pi_{\xi}$ from (9). Numerator and denominator in (8) change sign at $r_{c}$
and it can be found by solving this equation

$f(r_{c})-\frac{v^{2}}{N^{2}}=0.$ (11)

As it is clear in the appendix, Gauss-Bonnet coupling and curvature-cubed
interaction constant are $\lambda$ and $\mu$, respectively and the precise
form of $f(r)$ depends on $\lambda$ and $\mu$. It was found that there are
three different $AdS$ black hole solutions in quasi-topological gravity which
are determined by $f_{1}(r),f_{2}(r)$ and $f_{3}(r)$ in (32). Then for
different values of coupling constant $\lambda$ and $\mu$, one should choose
appropriate form of $f(r)$ from (32) and solve (11). However (11) is
complicated one can solve it numerically. Then, we assume different values for
$\mu$ and $\lambda$ and discuss behavior of the drag force in terms of these
coupling constants.

###  2.2 Positive couplings

We assume both coupling parameters $\mu$ and $\lambda$ are positive. As
pointed out in [33], for this case, only $f_{3}(r)$ in (32) leads to a stable
AdS black hole solution. The drag force versus the velocity of the heavy quark
has been plotted in Fig. 1. In the right and left plots of this figure, Gauss-
Bonnet coupling constant is $\lambda=0.01$ and $\lambda=0.20$, respectively.
Also different values of cubic-curvature coupling interaction are assumed.

As one finds from [40], by increasing $\lambda$ the value of the drag force
increases. This behavior of the drag force is clearly seen in these plots. One
finds that by increasing Gauss-Bonnet coupling constant from $\lambda=0.01$ to
$\lambda=0.20$, the drag force increases. In the plots of Fig. 1, one finds
that by increasing $\mu$ the value of the drag force also increases. Though at
the small velocities, the cubic-curvature interactions have minimum effect on
the drag force. As a result, the main effect of increasing cubic-curvature
coupling constant is increasing the drag force value. This is the same as the
case of $R^{2}$ and $R^{4}$ case [39, 40]. We should check this result in the
case of non-positive $\mu$ and $\lambda$.

,

Figure 1: The drag force versus the velocity of the heavy quark for _positive_
values of cubic-curvature coupling $\mu$ at fixed _positive_ Gauss-Bonnet
coupling constant.

###  2.3 non-positive couplings

Now we intend to study the effect of the non-positive coupling constants
$(\lambda,\mu)$ to the drag force. Three distinct $AdS$ black hole backgrounds
are discussed in (32). These solutions for different regimes of the parameter
space of $(\lambda,\mu)$ are discussed in the table 1 of [33]. As it is
explained in this table, to study the non-positive coupling constants, one
needs $f_{1}(r),f_{2}(r)$ and $f_{3}(r)$ from (32). In the case of positive
$\mu$ and negative $\lambda$, only $f_{3}(r)$ in (32) leads to a stable AdS
black hole solution. In Fig. 2, we assume $\lambda=-0.2$ and
$\mu=+0.01,+0.02,+0.25$ and plot the drag force versus the velocity of the
heavy quark. Also here, one finds that by increasing $\mu$ the value of drag
force increases. One should notice that at the small velocities, the cubic-
curvature corrections have the minimum effects. Therefor we confirm the
previous result. If one assumes negative $\mu$ and positive $\lambda$, also
finds that for larger $\mu$ the value of the drag force becomes larger.

###  2.4 analytic solution

Fortunately, we find an analytic result for the drag force in the special case
of $\mu=-\frac{\lambda^{2}}{3}$ which corresponds to $p=0$ in (34), as

$F_{R^{2}+R^{3}}=-\frac{1}{2\pi\alpha^{\prime}}\left(\frac{\sqrt{3}\,\pi^{2}\,T^{2}\,v}{N^{\frac{1}{2}}\sqrt{-v^{6}\,\lambda^{2}+3v^{4}\,\lambda\,N-3\,v^{2}\,N^{2}+3N^{3}}}\right),$
(12)

where $N$ is defined in (2).

It would be interesting to compare the drag force in the presence of higher
derivative corrections with the case of $\mathcal{N}=4$ strongly-coupled SYM
plasma $F_{\mathcal{N}=4}$. The authors of [38, 37] have obtained

$F_{\mathcal{N}=4}=-\left(\frac{\pi\,\sqrt{\tilde{\lambda}}\,T_{0}^{2}}{2}\right)\,\frac{v}{\sqrt{1-v^{2}}}.$
(13)

where $\tilde{\lambda}$ is ’t Hooft coupling333Notice that
$\alpha^{\prime-2}=\tilde{\lambda}$. and $T_{0}$ is the temperature of AdS
black hole solution without any corrections. Let us consider the case of
$\lambda\rightarrow 0$ in (12). In this limit, one does not consider any
correction in the action (24) and finds that the drag force is nothing but the
drag force in the case of $\mathcal{N}=4$ strongly-coupled SYM plasma
$F_{\mathcal{N}=4}$.

The analytical result in (12), shows the effect of the higher derivative
corrections on the drag force. It is clearly seen that the corrections appear
in the denominator of (12) and as a result the drag force increases.

Figure 2: The drag force versus the velocity of the heavy quark for _positive_
values of cubic-curvature coupling $\mu$ at fixed _negative_ Gauss-Bonnet
coupling constant $\lambda$.

##  3 dissociation length of quark-antiquark pair at finite coupling

In this section we investigate the effect of the higher derivative terms to
the dissociation length of quark-antiquark pair.

In the usual fashion, the two endpoints of the classical open string at the
boundary are seen as a quark and antiquark pair which may be considered as a
meson [31]. Based on lattice results and experiments, it is found that the
meson shows interesting behavior as the temperature of the plasma increases.
It is known that heavy quark bound states can survive in a QGP to temperatures
higher than the confinement/deconfinement transition [32]. Thermal properties
of _static_ quark-antiquark systems have been studied in [20, 21] in an AdS-
Schwarzschild black hole setting using the AdS/CFT correspondence. In [29], a
rotating quark-antiquark in the presence of higher derivative corrections is
studied. In the case of Gauss-Bonnet corrections, it is shown that as the
Gauss-Bonnet coupling constant $\lambda$ increases the string endpoints become
less separated i.e. the radius of the rotating open string at the boundary
decreases but the tip of the U-shaped string does not change considerably.

The heavy quark potential in the presence of curvature-squared corrections is
calculated in [41]. It is shown that the potential can be calculated as a
power series in $LT<<1$, where $T$ is the temperature of the hot plasma. One
finds that at fixed temperature, as the Gauss-Bonnet coupling constant
$\lambda$ increases the interquark distance $L$ decreases. It would be
interesting to investigate this observation in the case of higher derivative
corrections. To do this, we consider Quasi-topological gravity and study
effect of curvature-cubed corrections to the dissociation length. Because of
the complicated feature in this background, we use numerical methods.

###  3.1 dissociation length from AdS/CFT

To find the dissociation length, one should study the heavy quark potential,
$V_{q\bar{q}}(L)$,444We call quark-antiquark potential in (15) as ”heavy quark
potential”. where $L$ is the distance between two quarks [28]. One finds that
the heavy quark potential can be _negative_ , _positive_ or _zero_. If
$V_{q\bar{q}}(L)<0$, the dominant string configuration becomes the one for the
U-shaped string which can be interpreted as a heavy meson. When
$V_{q\bar{q}}(L)>0$, the heavy meson dissociates to two free quarks and the
string configuration changes. This phenomena happens at special length $L=d$
which is obtained from $V_{q\bar{q}}(L=d)=0$. We call $d$ as a dissociation
length. Thus by studying the heavy quark potential in the quasi-topological
gravity, we will find the effect of higher derivative terms to this quantity.

The heavy quark potential is given by the expectation value of the following
static Wilson loop

$W(C)=\frac{1}{N}Tr\,P\,e^{i\,\int A_{\mu}dx^{\mu}},$ (14)

where $C$ denotes a closed loop in spacetime and the trace is over the
fundamental representation of $SU(N)$ group. We consider a rectangular loop
along the time coordinate $t$ and spatial extension $L$. The static heavy
quark potential is related to the expectation value of this rectangular Wilson
loop in the limit of $t\rightarrow\infty$,

$\langle W(C)\rangle\sim e^{-t\,V_{q\bar{q}}(L)},$ (15)

This expectation value can be calculated from $AdS/CFT$ correspondence [20,
21]. In this set up, one should consider an infinitely massive quark in the
fundamental representation of $SU(N)$ group in $\mathcal{N}=4$ Yang-Mills
gauge theory. This quark is dual to a classical string hanging down to the
horizon from a probe brane at the boundary. The classical string hanging in
the bulk space and connecting two endpoints has a characteristic U-shaped. We
name $r_{*}$ as the tip of the U-shaped string and we let it to define the
nearest point between the string and the horizon of the black hole; i.e.
$r_{*}>r_{h}$. Let us emphasize that for non-physical states we would have
$r_{*}<r_{h}$ [20].

The dynamics of the U-shaped string is given by the Euclidean version of the
Nambu-Goto action in (4). To calculate the heavy quark potential, one has to
subtract the infinite self-energy of two independent heavy quarks and from the
$AdS/CFT$ correspondence. These massive quarks are dual to two straight
strings that extend from the probe brane at the boundary to the horizon. The
regularized action is shown by $\bigtriangleup S$ and it is related to the
expectation value of Wilson loop in (15) by this equation

$\langle W(C)\rangle\sim e^{-\bigtriangleup S},$ (16)

As a result, the heavy quark potential is

$V_{q\bar{q}}(L)=\frac{\bigtriangleup S}{t}.$ (17)

The heavy quark potential in the vacuum and in the strongly coupled $N=4$ SYM
gauge theory was found in [20]

$V_{q\bar{q}(L)}=-\frac{4\pi^{2}\sqrt{\lambda}}{\Gamma(1/4)^{2}}\left(\frac{1}{L}\right).$
(18)

We consider $X^{\mu}=(t,x,0,0,r(x))$ for the coordinates of U-shaped string in
the static gauge $\sigma=x,\,\tau=t$. As a result, the Euclidean version of
Nambu-Goto action in (4) can be found as

$S=\frac{N\,t}{2\pi\alpha^{\prime}}\int\,dx\sqrt{r^{4}\,f(r)+r^{\prime 2}},$
(19)

Notice that $r$ depends on $x$. The Hamiltonian density of this action is
constant and it is

$H=-\frac{N\,t}{2\pi\alpha^{\prime}}\frac{r^{4}\,f(r)}{\sqrt{r^{4}\,f(r)+r^{\prime
2}}},$ (20)

This constant is found at special point $r(0)=r_{*}$, where
$r^{\prime}_{*}=0$, as

$H=-\frac{N\,t}{2\pi\alpha^{\prime}}\sqrt{r_{*}^{4}\,f(r_{*})}.$ (21)

Then it is possible to find $L$ as follows

$\frac{L}{2}=\int_{r_{*}}^{\infty}\,dr\left(\frac{1}{r^{4}\,f(r)\left(\frac{r^{4}\,f(r)}{r_{*}^{4}\,f(r_{*})}-1\right)}\right)^{1/2}.$
(22)

Finally, the heavy quark potential is given by

$V_{q\bar{q}}(L)=\frac{\,N}{\pi\alpha^{\prime}}\,\int_{r_{*}}^{\infty}\,dr\left(\left(\frac{\frac{r^{4}\,f(r)}{r_{*}^{4}f(r_{*})}}{\frac{r^{4}\,f(r)}{r_{*}^{4}f(r_{*})}-1}\right)^{\frac{1}{2}}-1\right)-\frac{\,N}{\pi\alpha^{\prime}}\,\int_{r_{h}}^{r_{*}}\,dr.$
(23)

We intend to study the effect of the higher derivative corrections to the
heavy quark potential in (23) and the interquark distance in (22). For
different values of coupling constants $(\lambda,\mu)$, one should consider
three distinct $AdS$ black hole backgrounds which are discussed in (32).
However, we can not solve (23) and (22) analytically and we have to resort to
numerical methods. Also the coefficient $\frac{N\,}{\pi\alpha^{\prime}}$ does
not play any role in our physical discussion.

###  3.2 Numerical Solutions

We illustrate behavior of $V_{q\bar{q}}(L)$ as a function of $L$ at fixed
temperature $(r_{h}=1)$ in Fig. 3. It is clearly seen that there is a maximal
interquark distance, $L_{max}$. It has been shown that for $L<L_{max}$ there
are two kinds of strings; long strings and short strings [42, 43, 44, 45].
These strings correspond to the upper and lower parts of $V_{q\bar{q}}(L)$ in
Fig. 3, respectively. The stability analysis has shown that short strings are
favorable [42, 43, 44, 45]. One concludes that only the lower part is
physical[21].

By analyzing Fig. 3., we investigate behavior of the dissociation length for
different values for $\mu$ and $\lambda$. In this figure, the heavy quark
potential (23) is plotted versus the interquark distance (22). We take that
different values of cubic-curvature coupling $\mu$ while the Gauss-Bonnet
coupling constant $\lambda$ is fixed in each frame. Notice that in this case
the corresponding black hole backgrounds are specified by $f_{3}$. In this
figure, from left to right the Gauss-Bonnet coupling constant $\lambda$ is
increasing, $\lambda=-0.2,0.01$ and $0.2$. By increasing Gauss-Bonnet coupling
constant, the dissociation length of meson decreases. This phenomena has been
found also in the case of a rotating meson [29].

Figure 3: The heavy quark potential versus the interquark distance for
different values of cubic-curvature coupling $\mu$ at fixed Gauss-Bonnet
coupling constant $\lambda$. Left:$\lambda=-0.2$. Middle:$\lambda=0.01$.
Right: $\lambda=0.2$.

What is the effect of increasing cubic-curvature coupling $\mu$ while Gauss-
Bonnet coupling $\lambda$ is fixed? In each plot of Fig. 3, $\lambda$ is fixed
and $\mu$ is increasing. For example in the left plot of this figure
$\lambda=-0.20$ and $\mu=0.01,0.20,0.25$ and $0.28$. One can see that the
interquark distance decreases by increasing $\mu$. This observation is clearly
seen in the middle and right plots of Fig. 3, too. Therefor by increasing
cubic-curvature coupling, the dissociation length of meson decreases.

As we pointed out, there are three distinct AdS black hole backgrounds which
correspond to $f_{1}(r),f_{2}(r)$ and $f_{3}(r)$ in (32). In the case of
$\lambda<0$ and $\mu<0$ one should consider $f_{1}(r)$. We show the heavy
quark potential versus the interquark distance in the left plot of Fig. 4. In
this plot, $\lambda=-0.9$ and from right to left curve $\mu$ is increasing
from $-0.2,-0.1$ to $-0.01$. As before, one finds that the dissociation length
decreases by increasing the cubic-curvature constant $\mu$. One should notice
that the rate of decreasing is not so large. In the case of $\lambda>0$ and
$\mu<0$, one should choose $f_{2}(r)$ to investigate behavior of the heavy
quark potential versus the interquark distance. We show the result in the
right plot of Fig. 4. In this plot $\lambda=0.20$ and $\mu$ is increasing from
$-0.01$ to $-0.0001$. It is clearly seen that by increasing $\mu$, the
dissociation length decreases. This observation is consistent with what we see
in Fig. 3.

Figure 4: The heavy quark potential versus the interquark distance.
Left:$\lambda=-0.9$ and from right to left $\mu=-0.2,\,-0.1,-0.01$. Right:
$\lambda=0.2$ and from right to left $\mu=-0.01,-0.0001$.

One infers that as Gauss-Bonnet coupling constant $\lambda$ increases the
interquark distance decreases. Also at fixed $\lambda$ by increasing cubic-
curvature constant $\mu$ the interquark distant decreases. As a result,
including the higher derivative corrections decrease the dissociation length.

##  4 Conclusion

The higher derivative corrections on the gravity side correspond to finite
coupling corrections on the gauge theory side. The main motivation to consider
these corrections comes from the fact that string theory contains higher
derivative corrections arising from stringy effects. On the gauge theory side,
computations are exactly valid when the ’t Hooft coupling constant goes to
infinity ($\tilde{\lambda}=g_{YM}^{2}N\rightarrow\infty$). An understanding of
how these computations are affected by finite $\lambda$ corrections may be
essential for more precise theoretical predictions.

Although AdS/CFT correspondence is not directly applicable to QCD, one expects
that results obtained from closely related non-abelian gauge theories should
shed qualitative (or even quantitative) insights into analogous questions in
QCD. This has motivated much work devoted to study various properties of
thermal SYM theories like the hydrodynamical transport quantities. In this
paper we have studied the energy loss and the interquark-antiquark distance in
the presence of higher derivative terms. We have considered the cubic-
curvature terms which is known as quasi-topological gravity.

We calculated the energy loss of heavy quark and from numerical analysis,
found that the drag force increases. Fortunately, we found an analytical
result in (12) which confirms our result.

We introduced the heavy quark potential in (15). As it is seen from Fig. 3,
the heavy quark potential is _negative_ , _positive_ or _zero_. By studying
the zero case, we investigated effect of higher derivative terms in quasi-
topological gravity on the dissociation length. We found that the interquark-
antiquark distance becomes shorter with the increase of coupling parameters of
higher curvature terms. This result is consistent with the case of rotating
quark-antiquark pair in [29]. We can therefore conclude that the higher
curvature corrections make the dissociation length shorter. Interestingly, the
subleading term of the strong coupling expansion of the heavy quark potential
in a $\mathcal{N}=4$ SYM plasma is studied in [46]. It is also found that this
correction reduces the magnitude of the heavy quark potential and leads to a
smaller screening radius.

## Acknowledgment

We would like to thank E. Azimfard for helpful discussions and specially thank
M. Ali-Akbari and M. Sohani for reading the manuscript and useful comments.

##  5 Review of Quasi-topological gravity

In this appendix we give a brief review of the quasi-topological gravity in
five-dimensional spacetime [33]. The bulk action is given by

$I=\frac{1}{16\pi G_{5}}\int dx^{5}\sqrt{-g}\left(R-\Lambda+\frac{\lambda
L^{2}}{2}\chi_{4}+\frac{7L^{4}\mu}{8}Z_{5}\right),$ (24)

where $\lambda$ and $\mu$ are Gauss-Bonnet coupling and curvature-cubed
interaction constant, respectively. The negative cosmological constant is
related to radius of AdS space by $\Lambda=-\frac{12}{L^{2}}$. The curvature-
squared interaction is given by $\chi_{4}$ as

$\chi_{4}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},$
(25)

and $Z_{5}$ is the new curvature-cubed interaction

$\displaystyle Z_{5}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}^{\,\,\,\,\,\,\rho\sigma}R_{\rho\sigma}^{\,\,\,\,\,\,\alpha\beta}R_{\alpha\beta}^{\,\,\,\,\,\,\mu\nu}+\frac{1}{14}\left(21R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,R-120\,R_{\mu\nu\rho\sigma}R^{\mu\nu\rho}_{\,\,\,\,\,\,\,\,\,\alpha}R^{\sigma\alpha}\right.$
(26)
$\displaystyle\left.144\,R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma}+128R_{\mu}^{\,\,\,\,\nu}R_{\nu}^{\,\,\,\,\rho}R_{\rho}^{\,\,\,\,\mu}-108R_{\mu}^{\,\,\,\,\nu}R_{\nu}^{\,\,\,\,\mu}R+11\,R^{3}\right).$

The planar AdS black hole solutions for different values of the coupling
constants were found in [33]. The solution, in units where the radius of $AdS$
is one, is

$ds^{2}=r^{2}\left(-N^{2}\,f(r)dt^{2}+d\vec{x}^{2}\right)+\frac{dr^{2}}{r^{2}\,f(r)},$
(27)

where $f(r)$ is determined by roots of the following equation

$1-f(r)+\lambda f(r)^{2}+\mu f(r)^{3}=\frac{r_{h}^{4}}{r^{4}}.$ (28)

Here $r$ denotes the radial coordinate of the black brane geometry and
$t,\vec{x}$ label the directions along the boundary at the spatial infinity.
In these coordinates the event horizon is located at $f(r_{h})=0$ where
$r_{h}$ is found by solving this equation. The boundary is located at infinity
and the geometry will be as asymptotically AdS . The constant $N^{2}$
specifies the speed of light of the boundary gauge theory and one can choose
it to be unity. We name $f(r)$ at the boundary where $r\rightarrow\infty$, as
$f_{\infty}$ and one finds that

$N^{2}=\frac{1}{f_{\infty}},$ (29)

One also finds from (28) that $f_{\infty}$ satisfies

$1-f_{\infty}+\lambda f_{\infty}^{2}+\mu f_{\infty}^{3}=0.$ (30)

The temperature of the hot plasma is given by the Hawking temperature of the
black hole

$T=\frac{N\,r_{h}}{\pi}.$ (31)

Authors in [33], solved (28) and found $f(r)$ for different values of coupling
constants $\lambda$ and $\mu$. It is shown that there are three different
solutions of (28) in the $\mu-\lambda$ plane:

$\displaystyle f_{1}(r)$ $\displaystyle=$ $\displaystyle
u+v-\frac{\lambda}{3\mu},$ $\displaystyle f_{2}(r)$ $\displaystyle=$
$\displaystyle-\frac{u+v}{2}+i\,\frac{\sqrt{3}}{2}(u-v)-\frac{\lambda}{3\mu},$
$\displaystyle f_{3}(r)$ $\displaystyle=$
$\displaystyle-\frac{u+v}{2}-i\,\frac{\sqrt{3}}{2}(u-v)-\frac{\lambda}{3\mu},$
(32)

where

$u=(q+\sqrt{q^{2}-p^{3}})^{\frac{1}{3}},\,\,\,\,\,\,v=(q-\sqrt{q^{2}-p^{3}})^{\frac{1}{3}},$
(33)

and

$p=\frac{3\mu+\lambda^{2}}{9\mu^{2}},\,\,\,q=-\frac{2\lambda^{3}+9\mu\lambda+27\mu^{2}\left(1-\frac{r_{h}^{4}}{r^{4}}\right)}{54\mu^{3}}.$
(34)

There is a relation between Gauss-Bonnet coupling constant $\lambda$ and
cubic-curvature coupling constant $\mu$ as follows

$\mu=\frac{2}{27}-\frac{\lambda}{3}\pm\frac{2}{27}\left(1-3\lambda\right)^{\frac{3}{2}}.$
(35)

which shows the upper and lower bound on the cubic-curvature interaction
coupling. There is a special case $p=0$ in (34) which corresponds to
$\mu=-\frac{\lambda^{2}}{3}$. $f(r)$ is also found at this point.

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###### Abstract

Smile and Learn is an Ed-Tech company that runs a smart library with more that
100 applications, games and interactive stories, aimed at children aged two to
10 and their families. The platform gathers thousands of data points from the
interaction with the system to subsequently offer reports and recommendations.
Given the complexity of navigating all the content, the library implements a
recommender system. The purpose of this paper is to evaluate two aspects of
such system focused on children: the influence of the order of recommendations
on user exploratory behavior, and the impact of the choice of the
recommendation algorithm on engagement. The assessment, based on data
collected between 15 October 2018 and 1 December 2018, required the analysis
of the number of clicks performed on the recommendations depending on their
ordering, and an A/B/C testing where two standard recommendation algorithms
were compared with a random recommendation that served as baseline. The
results suggest a direct connection between the order of the recommendation
and the interest raised, and the superiority of recommendations based on
popularity against other alternatives.

###### keywords:

educational technologies; recommender systems; artificial intelligence

1910 8 19 Received: 12 February 2019; Accepted: 18 April 2019; Published: 22
April 2019 yes Data-Driven Interaction Review of an Ed-Tech Application
Alejandro Baldominos 1,2 and David Quintana 1,* Alejandro Baldominos and David
Quintana Correspondence<EMAIL_ADDRESS>Tel.: +34-91-624-9109

##  1 Introduction and Background

Smile and Learn’s smart library is an application in the educational
technology (Ed-Tech) space which is aimed at children. As of December 2018,
the platform features a total of 107 games, which are grouped according to
Gardner’s theory of multiple intelligences (Gardner, 1983).

Contents, which include games, stories and videos, are designed to be used in
different devices and rely on a common framework. The application registers
thousands of data points as a result of user interaction. Based of these, the
system can then generate personalized reports and recommendations relevant to
users, parents and educators.

The initial interaction with the application requires choosing among a large
set of alternatives that are organized according to broad categories. These
can be identified in Figure 1. The different games are grouped in so-called
“worlds”, with each world corresponding to an intelligence: science
(naturalistic), spatial (visual-spatial), multiplayer (group-interpersonal),
logic (logical-mathematical), literacy (verbal-linguistic), emotions
(emotional-intrapersonal), and arts (artistic). There is one additional world
named after the user, which consists on a virtual village where the child
interacts with characters to improve her wealth. Here, the dynamics require
abilities from all intelligences.

Once a world is chosen, apps are displayed grouped in categories, with one
category per line. The children can use the vertical scroll (swipe pattern) to
navigate through categories, and the horizontal scroll to navigate between
apps within a category. An example screenshot from the Science world is shown
in Figure 2.

Figure 1: Screenshot of the main menu of Smile and Learn, showing the
different worlds available in the application, each world corresponding to an
intelligence. Figure 2: Screenshot of a world menu (in this case, the science
world) in Smile and Learn, where apps belonging to the same world (or
intelligence) are listed grouped by category.

Due to the increase in the number of games, which is still growing month after
month, navigation through the application is becoming more cumbersome, meaning
that it can take more time and effort to search for a certain game within the
application. In order to alleviate this issue, we have introduced a navigation
ribbon at the bottom of the main menu which allows a fast access to some
chosen applications. As it can be seen at the bottom of Figure 1, this
navigation pattern consists of two tabs: “recommended” and “most played”. The
latter displays a list of the five top-played apps for the child using the
app, turning into a useful mechanism for providing a fast access to those
frequent apps.

Regarding the former tab, it displays a list of up to seven apps recommended
to children based on their usage behavior. The design motif behind this
recommender system is not only to enable fast access to those apps which might
be of interest for a child, but also to enhance exploration: letting children
discover games which they might not reach if they had to find them by
navigating through the app.

The range of strategies developed to assign users to items is broad. Some well
established alternatives would be the recommendation of the most popular items
(Jäschke et al., 2007); collaborative filtering (Schafer et al., 2007);
content-based approaches (Basilico and Hofmann, 2004) or usage context-based
similarity (Niemann and Wolpers, 2013). In addition to the basic
possibilities, there is room for hybrid strategies that combine the output of
several canonical strategies to generate their output.

Currently, two different strategies are implemented in Smile and Learn:

  * •

Popular: this approach recommends the most popular applications among other
users. The rationale behind this approach is that it is likely that some games
are specially enjoyable for most children, because of their quality, design or
playability, and therefore, they are safe to recommend, as there is a high
chance that other children will find them engaging.

  * •

Collaborative filtering: in this approach, the historical records of usage of
all children are used in order to generate recommendations. The idea behind
collaborative filtering is to find children similar to the one whose
recommendations are being generated, so that these recommendations consist of
games that have been played by these similar children, but not the child being
targeted.

Both these strategies are based on implicit feedback, since explicit feedback
given by children can turn out to be unreliable or difficult to interpret. Out
of the different implicit parameters that could be taken into account Rolando
Núñez-Valdez et al. (2015), we have used the number of games played by child
and app, as well as the duration of such games.

The specifics of the implementation of these algorithms are discussed in
relation to a prior version of the system by Ruiz–Iniesta et al. (2018).

In the navigation bar displayed at the bottom of the main menu,
recommendations from different recommenders could be combined. Also, it is
worth noting that while there is a maximum of seven recommended games, in some
screen sizes or formats it can occur that less than seven are displayed and
then the list becomes scrollable. Based on our experience, most screens are
able to show five recommendations, with the last two being hidden and
accessible only upon scroll (through a swipe pattern or by pressing the side
arrows, an example can be seen in Figure 1).

In this paper, we want to design a sound experiment in order to determine
which recommender strategy is working best, i.e., which one is translating
into a higher engagement of children with the recommended games. We are also
interested in getting insights about the interaction patterns taken by
children when dealing with the recommendations. With this information, we
could improve the system and enhance user experience.

This work contributes valuable new evidence on the reaction of children to two
well-established methods of generating recommendations, and assessing the
importance of the order of presentation. Information like this, specifically
focused on children in the educational space, is both scarce and very hard to
obtain. Given that the related literature on children is very limited, we
consider that the results might be very relevant to developers of K-12 Ed-Tech
applications.

The rest of the document is structured as follows: in Section 2 we present
related works on recommender systems and interaction patterns within the
context of an Ed-Tech application. That will be followed by Section 3, where
we describe the experimental methodology. The results are then reported in
Section 4. Finally, conclusive remarks are presented in Section 5 along with
some suggested future lines of work.

##  2 Related Work

Applications of technology to educational processes (a field known under the
term “Ed-Tech”) are not particularly new. However, it has only been in the
last decade that the availability and ubiquity of technology devices (such as
smart-phones or tablets) has reached a point in which these applications can
be deployed in large-scale settings.

Additionally, the implementation of data science or machine learning
techniques are allowing to extend Ed-Tech far beyond the classical definition
of using technology to deliver contents to an audience (e.g., slides,
interactive videos, etc). Bhattacharya and Nath (2016) discuss how novel
development allow tracking the progress of individual users, detecting
strengths and weaknesses, and providing a customized experience to enhance the
learning process, among other possibilities.

Many examples of such novel applications can be found in the literature in
recent years. For example, Charleer et al. Charleer et al. (2017) presented a
learning analytics dashboard aimed at improving the communication in advising
sessions, helping to increase students’ motivation. Lange et al. de Lange et
al. (2018) also made related contributions in the space of virtual training
centers and Käser et al. (2017) have proposed an approach to student modelling
to represent and predict students’ knowledge and skills. Some authors have
been working on the prediction of academic performance (Hamoud et al., 2018)
and others, like Ueno and Miyazawa (2017) introduced a system designed to
scaffold learning in specific domains such as programming.

An important subset of Ed-Tech applications are those aiming at providing a
customized experience by suggesting users a certain topic or learning process.
This problem can generally be regarded as a recommendation problem.
Recommender systems, the instruments used to tackle the problem, have a three-
decade long history and have been the subject of a sizable amount of research.
These efforts on recommender systems in general have been surveyed by
Bobadilla et al. (2013) and, more recently, by Lu et al. (2015). If we focus
specifically in education, the literature reviews authored by Drachsler et al.
(2015) and Erdt et al. (2015) are specially insightful and comprehensive.

The design and implementation of recommender systems in the education space
poses a number of difficulties, as discussed by Tarus and Niu (2017), and so
does their evaluation (Lombardi and Marani, 2015). Among these challenges, we
can mention the selection of the right algorithm (or set of algorithms) and
the design of the interface. Even though the first aspect is not settled yet,
there have been valuable efforts to contribute evidence in this regard like a
recent work by Kopeinik et al. (2016).

These authors cite among the best-established and computationally inexpensive
tag and resource recommendation strategies for technology-enhanced learning:
base level learning equation with associative component (Kowald et al., 2015);
collaborative filtering (Schafer et al., 2007); content-based (Basilico and
Hofmann, 2004); most popular (Jäschke et al., 2007); SUSTAIN (Love et al.,
2004) or usage context-based similarity (Niemann and Wolpers, 2013).

In this regard, studies like Portugal et al. (2018) offer systematic reviews
of different algorithmic alternatives, while Drachsler et al. (2008);
Manouselis et al. (2011) provide specific lists of advantages and
disadvantages of the main recommendation techniques for technology-enhanced
learning. These are briefly summarized in Table 1, which adapts a very similar
one included in the last two mentioned studies, adding popularity-based
recommendations to the set of techniques.

Besides the alternatives displayed in the table, an additional category of
recommender systems are hybrid approaches. In this case, two or more of these
techniques are fused in order to alleviate some of the disadvantages of a
particular recommender. This possibility was discussed in detail in a survey
on Next Generation of Recommender Systems by Adomavicius and Tuzhilin (2005)
and, later by Aamir and Bhusry (2015).

Table 1: Characteristics of main recommendation techniques suitable for technology-enhanced learning. CF: collaborative filtering. Adapted from Drachsler et al. (2008) and Manouselis et al. (2011). Technique |  Description |  Advantages |  Disadvantages
---|---|---|---
Popularity |  Recommends items that are popular among all users. |  No new user problem.No content analysis. Domain-independent. |  Only popular taste.Insensitive to changes of preferences.
User-based CF |  Users who rated the same item similarly probably have the same taste. Based on this assumption, this technique recommends the unseen items already rated by similar users.
|  No content analysis.
Domain-independent.
Quality improves.
Bottom-up approach.
Serendipity. |  New user problem.
New item problem.
Popular taste.
Scalability.
Sparsity.
Cold start problem.
Item-based CF |  Focus on items, assuming that the items rated similarly are probably similar. It recommends items with the highest correlation (based on ratings for the items).
|  No content analysis.
Domain-independent.
Quality improves.
Bottom-up approach.
Serendipity. |  New item problem.
Popular taste.
Sparsity.
Cold start problem.
Stereotypes or demographics CF |  Users with similar attributes are matched, then it recommends items that are preferred by similar users (based on user data instead of ratings).
|  No cold start problem.
Domain-independent.
Serendipity. |  Obtaining information.
Insufficient information.
Only popular taste.
Obtaining metadata information.
Maintenance ontology.
Case-based reasoning |  Assumes that if a user likes a certain item, she or he will probably also like similar items. Recommends new but similar items.
|  No content analysis.
Domain-independent.
Quality improves |  New user problem.
Overspecialisation.
Sparsity.
Cold start problem.
Attribute-based techniques |  Recommends items based on the matching of their attributes to the user profile. Attributes could be weighted for their importance to the user. |  No cold start problem.
No new user/new item problem.
Sensitive to changes of preferences.
Can include non-item-related features.
Can map from user needs to items. |  Does not learn.
Only works with categories.
Ontology modelling and maintenance is required.
Overspecialisation.

Among the works that could illustrate this possibility, we could mention one
by Rodriguez et al. Rodríguez et al. (2015)

, who built a hybrid recommender combining content-based, collaborative
filtering and knowledge-based approaches to develop a recommender system that
suggests learning objects extracted from a repository in an educational
setting. Another example can be found in the work by Salehi and Kmalabadi
(2012), which combines content-based and collaborative filtering recommenders,
in this case to suggest appropriate learning materials. Bourkoukou and El
Bachari (2018) introduce a recommender system for web-based education that
adapts its recommendations of learning objects to the learning styles of
users. In this case, the hybrid recommender system is based on collaborative
filtering and association rule mining. Finally, Nafea et al. (2018) recently
introduced the ULEARN system, that also selects and sequences learning objects
that match the learning styles of the users. To this end, they rely on a
hybrid recommendation approach that includes collaborative filtering and
content filtering.

The second aspect is specially complicated when it involves children, as their
cognitive development makes their interests and capabilities change
dramatically over a period of relatively few years (Sears and Jacko, 2009),
and the fact that intuitions by adult designers might not be correct
(Bjorklund and Causey, 2017). Even though there have been advances in this
aspect, like the contribution of Wu et al. (2014) on interface design for
children, there is still a significant amount of work to be done.

If we focus on recommendation for children, even though we could mention some
relevant works like that by Pera and Ng (2014) the volume is still very
scarce. To illustrate this, it is worth mentioning that the first specialized
workshop, KidRec, took place in 2017 (Pera et al., 2018). As Deldjoo et al.
(2017) explain, recommender systems have been traditionally focused on adults
and, when it comes to children, the field is still in its infancy.

##  3 Materials and Methods

In this study we intend to compare the performance of two popular recommender
strategies to suggest potential games of interest to children. As it was
described earlier, these are providing recommendations based on apps commonly
played by most children, and collaborative filtering, whose recommendations
are based on games played by similar users, i.e., those who have a similar
record of played games.

In order to get a better understanding of which strategy was working best, we
tested the performance of both recommender strategies and, additionally, we
compared them against a baseline recommender which made random
recommendations. Given that the aim of the work in this regard was
descriptive, not prescriptive, an A/B/C testing, a well-established evaluation
approach in this context, was performed. This methodology was recently used by
Kalloori et al. (2018) to evaluate their recommender algorithm in Taobao
display advertising platform in production, and so was used by Hidasi and
Karatzoglou (2018) to evaluate theirs on a large scale online test on an
online video portal.

During the whole period in which the experiment was running, each child was
assigned one group (either A, B or C) randomly following a uniform
distribution. When running the recommender, children received recommendations
coming from a different strategy based on their group:

  1. A.

Popular strategy was used.

  2. B.

The collaborative filtering strategy was used.

  3. C.

The random strategy was used.

Instead of generating all the recommendations (a maximum of seven) from the
resulting strategy, only the first three recommendations were computed using
the recommender corresponding to the child’s group, and the remaining
recommendations were chosen randomly.

In all cases, some filters were applied in order to opt out of some
recommendations which might not be suitable:

  * •

Some games were blacklisted, meaning that they might not have enough quality
as to be recommended (e.g., they were in a beta stage).

  * •

Games were filtered out if they were not available in the version of the app
owned by the user (e.g., a game was introduced in version 4 and the child is
using version 3).

  * •

Games were filtered out if they were not designed for the range of age of the
target child (e.g., a game is aimed at children 4–6 years old, but the child
is 3 years old).

It is worth mentioning that these filters were applied in all cases, even in
the random recommendation strategy.

###  3.1 Algorithms

As stated earlier, three different recommendation strategies were followed.
One of these strategies was entirely random, and was used as a baseline. Its
implementation was trivial, as apps to be recommended were randomly sampled
from among the whole set.

####  3.1.1 Popular Recommender

The second strategy was the popular one. For the popular recommender, the
algorithm computed a metric of normalized interest, $\overline{I}(a)$, for
each app $a$, which was immediately derived from the number of games
($\\#Games(a)$) and the amount of time in days that the app has been available
in the library ($Age(a)$), as shown in Equation (1):

$\overline{I}(a)=\frac{\\#Games(a)}{Age(a)}.$ (1)

Once the normalized interest was computed for each app, then all apps were
ranked according to this value and the top-$k$ apps were returned by the
recommender.

Two design decisions about the popular recommendation system are worth
mentioning. First, only the amount of games played for an app was taken into
account, while the duration of these games was ignored. This was done on
purpose, as each app has its own particularities and some might naturally lead
to longer games than others, and we did not want to set these apps higher in
the ranking. However, it must be noted that games with a duration of less than
five seconds have been ignored. Second, the interest was normalized by the
amount of time that the app has been published in the library. While another
alternative could have been chosen (for example, considering only the data
from the last $d$ days), we have intentionally performed this normalization so
that new apps had a higher chance to be chosen by the recommender.

####  3.1.2 Collaborative Filtering Recommender

The third recommendation strategy was based on collaborative filtering. This
strategy relies on implicit rather than explicit feedback. This decision was
motivated by the fact that small children have their own interacting patterns,
and might be unable to properly give credit or score to a game after playing
it (Deldjoo et al., 2017). For this reason, a system based on explicit
feedback could be unreliable. Instead, we are basing recommendations on
implicit patterns of interaction, and in particular in how much and how long a
child played with a game.

The idea behind collaborative filtering was to recommend apps to a child which
can be of high interest based on the preferences of other children which are
deemed similar. Therefore, our implementation of collaborative filtering was
done in two stages. In the first stage, the neighborhood was formed by
choosing children similar to the one to whom recommendations are being
provided. In the second phase, a rating was predicted for each candidate app
based on the neighborhood and the top-$k$ apps were selected for the
recommendation.

In the neighborhood formation stage, for each target child to whom
recommendations were provided, we needed to select similar children on which
the recommendations relied. First of all, only children with at least one game
of the same application in common were considered for the neighborhood (or
equivalently, children whose set of played apps was disjoint with that of the
target child are systematically excluded for the neighborhood). Each child’s
profile included some demographic data including the age and the gender. In
this case, the age was used to form the neighborhood; however, the gender was
intentionally omitted to prevent unintended biases.

To establish the neighborhood, a similarity metric between two children
$c_{1}$ and $c_{2}$ was defined as shown in Equation (2):

$Sim(c_{1},c_{2})=0.4\cdot
Sim_{age}(c_{1},c_{2})+0.6\cdot\frac{\left|Apps(c_{1})\cap
Apps(c_{2})\right|}{\left|Apps(c_{1})\cup Apps(c_{2})\right|}.$ (2)

As it can be seen, this similarity measure relied on a weighted average of two
different criteria: the similarity in the age and the similarity of the gaming
history of the two children, and returned a value in the range $[0,1]$. In the
case of the former criterion, it was computed following Equation (3):

$Sim_{age}(c_{1},c_{2})=\begin{cases}1&\text{if }Age(c_{1})=Age(c_{2}),\\\
0.5&\text{if }|Age(c_{1})-Age(c_{2})|=1,\\\ 0&\text{otherwise}.\end{cases}$
(3)

In the previous equation, $Age(c)$ refers to the age of child $c$ in years.
The previous function has been designed to fit an educational reality: a
difference of more than one year in such an early stage of the educational
process (two to 10 years old) was very large, and two children with such an
age gap were likely be very different.

The latter part of Equation (2) refers to the similarity in terms of the used
apps, which was the main criterion for the implicit feedback collaborative
filtering. This was computed as the intersection-over-union of the sets of
apps played intensively by the two children. By “intensively” we refer to apps
that have a cumulative of at least 10 games and 60 s of game. This filtering
has been done to prevent considering apps which might been rarely played by a
children while examining or exploring Smile and Learn library.

Neighborhood formation was carried out by choosing the top-100 children with
higher similarity. A minimum similarity of 0.5 was enforced in order for a
child to be included in the neighborhood.

Finally, for each candidate app $a$ to be recommended we computed the average
of the ratings $r(n,a)$ for each children $n$ in the neighborhood $N$. Since
we wanted to deal with implicit feedback, this rating corresponds to the
number of games of children $c$ in app $a$. The average of ratings was
weighted by the similarity of the target children and each neighbor. In
summary, the interest of children $c$ in app $a$, $I(c,a)$ was computed as
shown in Equation (4):

$I(c,a)=\frac{\sum_{n\in N}\left(Sim(c,n)\cdot r(n,a)\right)}{|N|}.$ (4)

Finally, apps were sorted by their interest and the top-$k$ apps were provided
as recommendations.

###  3.2 Dataset

During the period of the experiment, from 15 October 2018 to 1 December 2018,
we recorded the following information, which we used to later evaluate the
system and report the results:

  * •

The recommender (popular, collaborative filtering, or random) assigned to each
child.

  * •

The recommendations generated, and also which recommender provided each of
these recommendations.

  * •

The date and time at which each recommendation is generated.

  * •

The games usage per child, including the times at which they play and the
duration for each game.

It is worth noting that regarding the second aspect, recommendations might not
always be generated using the desired recommender. For example, a child might
be assigned the collaborative filtering strategy, but this recommender might
not have enough information about usage as to generate useful recommendations,
or that all of these recommendations are filtered out. In that case, random
recommendations are provided instead.

With this data, we can explore the patterns of engagement of children with the
different games depending on whether those games were recommended or not.

###  3.3 Performance Metrics

The impact of the position in the ribbon of recommendations was be analyzed
using click-through frequencies. That way, we will determined whether the
first five visible items got more clicks than the last two, and whether the
ordering within the visible and invisible ones mattered.

The core assessment of the recommendation algorithms will be made according to
two engagement metrics. One based on the number of games and another one on
game time.

The first one, the average number of games per user (ANG), is formally defined
in Equation (5).

$ANG=\frac{\sum_{i=1}^{NumUsers_{R}}Games_{Ri}}{NumUsers_{R}},$ (5)

where $Games_{Ri}$ is the number of games played by user $i$ on apps
recommended by algorithm $R$ (either collaborative filtering, popular or
random, and $NumUsers_{R}$ is the total number of users who were recommended
the apps by algorithm $R$ and acted on it.

The second aspect of engagement to be measured was the average game time
($AGT$) by users who acted on the recommendations. The expression used to
compute the indicator is described in Equation (6).

$AGT=\frac{\sum_{i=1}^{NumUsers_{R}}GameTime_{Ri}}{NumUsers_{R}},$ (6)

where $GameTime_{Ri}$ is the total time spent by user $i$ playing apps
recommended by algorithm $R$ (either collaborative filtering, popular or
random, and $NumUsers_{R}$ is the total number of users who were recommended
the apps by algorithm $R$ and used them.

We should note that, for the computation of these metrics, we define “game” as
the event where the user interacts with the application for 10 s or more.
Given the presence of some outliers, we also filtered out games of more than
3000 s and the instances where a game was played more than 60 times by the
same user. These accounted for less than 0.5% of the sample.

In order to provide a more complete picture, in addition to these engagement
metrics, we reported four standard performance indicators: accuracy,
precision, recall and F1 score.

To compute these metrics we have proceeded as follows: we have divided the
pilots period, comprising a total of 45 days, into two different time spans.
The first one comprised one month (from 15 October 2018 to 14 November 2018)
and the second comprised 15 days (from 15 November 2018 to 1 December 2018).
The former was used as the training set while the latter was used for
validation purposes.

We have considered the apps suggested by the recommender during the training
phase to each child and have then matched this information with the actual
apps played by children during the testing period. With this information, we
have been able to build a confusion matrix as follows: apps recommended to a
child during training and then played during testing constituted a true
positive ($TP$), apps recommended but not played counted as false positives
($FP$), apps played but not previously recommended were considered false
negatives ($FN$) and apps not recommended and not played constituted the true
negatives ($TN$). All confusion matrices for each child were summed up for
each recommendation strategy, and then accuracy, precision, recall and F1
score were computed following their standard definitions, which are shown in
Equations (7)–(10).

$Accuracy=\frac{TP+TN}{TP+TN+FP+FN},$ (7) $Precision=\frac{TP}{TP+FP},$ (8)
$Recall=\frac{TP}{TP+FN},$ (9) $F1\>Score=2\cdot\frac{Precision\cdot
Recall}{Precision+Recall}.$ (10)

##  4 Results and Discussion

As it was mentioned before, the first part of the study has to do with the
analysis of the impact of the position in the recommendation ribbon on the
click-through. During the pilots period we have measured a total of 30,516
clicks received by 89 different apps, which have been involved in a total of
472,498 games. The experimental results on this aspect are reported in Table
2. There, we can see the total number of clicks over the relevant period by
position for the apps that were recommended in all the possible slots.

Table 2: Accumulated click-through by position in the recommendation ribbon and mean for the visible and invisible positions for the period from 15 October 2018 to 1 December 2018. Position | Visible |  | Hidden
---|---|---|---
1 | 2 | 3 | 4 | 5 |  | 6 | 7
Clicks | 6329 | 5834 | 4516 | 2639 | 3960 |  | 3150 | 2937
Mean | - | - | 4830 | - | - |  | 3044 | -
Rank | 1 | 2 | 3 | 7 | 4 |  | 5 | 6

It is apparent that the first three positions grabbed much more interest than
the rest. The difference between the first two and the other five were
specially sizable. While it was true that the fourth received fewer clicks
than the ones that follow, other than that, the results were consistent with
the existence of a direct relationship between the order and the number of
times that users acted on the recommendations.

If we consider visibility, the ribbon only showed five recommendations at a
time. We expected that to be a relevant factor, as getting to the last two
requires a supplementary effort from the user. Interestingly, even though the
average number of clicks on the visible slots was 4830, higher than the 3044
average clicks on the hidden ones, the role of friction as an element that
drags click-through down could be questioned. If we consider the effect of
position, where apps on the left hand side gather more interest that those on
the right, and we infer the trend line, the recommendations on the sixth and
seventh positions get more clicks than expected.

Regarding engagement, a total of 12,229 games were completed in apps that were
being recommended at the moment by a total of 1387 children, summing up a
total of 598 gaming hours. From the study of engagement, we find that the
recommender algorithm based on popularity was superior across metrics. As we
can see in Table 3, it outperformed the other two both in terms of number of
games and accumulated use time by user. Unexpectedly, the proposed
collaborative filtering algorithm resulted in a slightly lower mean number
games vs. the random alternative. The sign of this difference, however, was
the opposite for game time.

Table 3: Engagement metrics for game apps by recommender for the period from 15 October 2018 to 1 December 2018. Engagement Metric | Mean | Median | Variance
---|---|---|---
Games |  |  |  |
| $NG_{Random}$ | 4.79 | 3 | 34.22
| $NG_{Popular}$ | 6.72 | 4 | 53.89
| $NG_{CF}$ | 4.22 | 2 | 24.75
Time |  |  |  |
| $GT_{Random}$ | 740.98 | 405.26 | 1,084,308.47
| $GT_{Popular}$ | 1202.76 | 639.79 | 3,286,849.96
| $GT_{CF}$ | 776.01 | 295.37 | 5,020,886.62

The statistical significance of the differences reported in Table 3 was
assessed according to the protocol that follows. First, we started testing the
normality of the distribution of the engagement metrics with
Kolmogorov–Smirnov test with Lilliefors correction. In case normality was
rejected, we applied Wilcoxon’s test. Otherwise, we tested for the presence
homoskedasticity using Levene test and, based on the result, we relied either
on Welch test, or the traditional t-test. The results were analogous for the
metrics. The superiority of the Popular algorithm over the other two was
significant at 1%. Regarding the comparison of the baseline vs the
implementation of collaborative filtering, the null hypothesis of equality
could not be rejected at the 5% conventional level for neither the number of
games nor the total game time.

In regards to the standard performance metrics, we report the main ones
(accuracy, precision, recall and F1 Score) in Table 4. As we can see,
recommendations based on popularity offered the largest precision and F1
score, while collaborative filtering and random recommendations provided the
highest values in terms of accuracy and recall, respectively.

Table 4: Performance metrics for game apps by recommender algorithm. Train: 15 October 2018 to 14 November 2018. Test: 15 November 2018 to 1 December 2018. Algorithm | Accuracy (%) | Precision (%) | Recall (%) | F1 Score (%)
---|---|---|---|---
Col. Filtering | 94.92 | 7.40 | 1.28 | 2.18
Popular | 94.73 | 12.93 | 3.30 | 5.25
Random | 93.51 | 6.64 | 3.57 | 4.64

These results are influenced by the nature of the algorithms and the variety
or recommendations that they generate. At the ends of the spectrum we would
have random recommendation and recommendations based on popularity. The former
approach is less constrained than the other two and, as a result, it foments
exploration to a higher extent. Among the remainder, even though both are more
stable, collaborative filtering provides a richer range of alternatives, as
unguided exploration by peers is likely to result in new app recommendations.

A second aspect to be considered regarding the performance indicators is that,
unlike the engagement ones, they only provide information on whether the user
felt compelled to try new apps, not whether the user liked them. We are more
interested in discovering this latter extent, which we find more important in
the scope of this study. For this reason, we find that the average number of
games and average time spent playing the recommended apps are the key
indicators to consider, whereas standard performance metrics can be useful
supplemental information to gain a better understanding of the situation.

To summarize the results, the two elements that constituted the subject of
study of the recommender system were the influence of the order of
recommendation on user exploratory behavior and the impact of the choice among
two well-established recommendation algorithms, pre-selected by the company,
on engagement. In order to evaluate such aspects, we acquired data of real
interaction with the application between 15 October 2018 and 1 December 2018,
running an A/B/C test with three different implementations of a recommender
system (including a random baseline) and measuring clicks made on
recommendations.

We found a direct association between the order in the recommendation ribbon
and the number of clicks. Apps on the left gather more interest that those on
the right hand side. The friction introduced by the fact that reaching the
last recommendations requires either swiping or pressing on the arrows on the
side does not seem to have a negative impact.

The A/B/C testing analysis used to compare the two recommendation approaches,
one that recommends apps based on popularity and an implementation of
collaborative filtering, vs the random recommender used as baseline offered
two main results in regards to engagement: The first one is that the popular
algorithm beats the other two both in terms of number of games and the total
time spent playing the recommended contents. The second is that the current
implementation of collaborative filtering does not seem to add value, as it
offers the same performance as the baseline. A likely explanation to this
outcome is that there are some games that due to their nature are very
likeable, and for this reason they end up being “the popular ones”. These
games will end up being suggested by the recommender based on popularity, and
if children start to play them, in most cases they will quickly become
engaged. This process could be adding some bias in the computed engagement
metrics in favor of the popular recommendation strategy. If we consider
performance indicators, the approach based on popularity seems to be the best
alternative in terms of precision and $F$1 score, but collaborative filtering
offers the highest accuracy.

##  5 Summary and Conclusions

In this paper we have described and evaluated the behavior of a recommender
system in the scope of an Ed-Tech application aimed at children aged 2–10 and
their families. The smart library gathers thousands of data points based on
user interaction and uses that information to generate tailored reports and
recommendations. The latter aspect is managed by a recommender system designed
for the purpose of easing navigation through the library and enhancing
exploration. We evaluated two aspects of the system: the influence of the
order of recommendations on user exploratory behavior, and the impact of the
choice of the recommendation algorithm on engagement.

The analysis led to the following conclusions: first, the order in the
recommendation ribbon matter. Apps in the left positions got more clicks, and
the effort to access hidden recommendations did not seem to have a negative
impact. The second one is that the popular algorithm resulted in more
engagement than the other two. Finally, the current implementation of
collaborative filtering does not seem to add value. Future works would include
thorough studies on alternatives to the current implementation of
collaborative filtering or the optimization of its parameters; studying new
possibilities based on the interface to drive the attention of the user to the
recommended applications; or the implementation of recommendation strategies
based on competences aimed at fostering the development of the user according
to predefined preferences established by parents and educators.

D.Q. surveyed the state of the art; A.B. and D.Q. conceived and designed the
experiments; A.B. performed the experiments, A.B. and D.Q. analyzed the data;
A.B. and D.Q. wrote the paper.

This research has received funding from the European Union’s Horizon 2020
Research and Innovation Programme under grant agreement No. 756826.

The Authors declare no conflict of interest.

References

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