# Traversable wormhole geometries in $f(Q)$ gravity

Zinnat Hassan 0000-0002-6608-2075<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Sanjay Mandal 0000-0003-2570-2335
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of
Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

The current interests in the universe motivate us to go beyond Einstein’s
General theory of relativity. One of the interesting proposals comes from a
new class of teleparallel gravity named symmetric teleparallel gravity, i.e.,
$f(Q)$ gravity, where the non-metricity term $Q$ is accountable for
fundamental interaction. These alternative modified theories of gravity’s
vital role are to deal with the recent interests and to present a realistic
cosmological model. This manuscript’s main objective is to study the
traversable wormhole geometries in $f(Q)$ gravity. We construct the wormhole
geometries for three cases: (i) by assuming a relation between the radial and
lateral pressure, (ii) considering phantom energy equation of state (EoS), and
(iii) for a specific shape function in the fundamental interaction of gravity
(i.e. for linear form of $f(Q)$). Besides, we discuss two wormhole geometries
for a general case of $f(Q)$ with two specific shape functions. Then, we
discuss the viability of shape functions and the stability analysis of the
wormhole solutions for each case. We have found that the null energy condition
(NEC) violates each wormhole model which concluded that our outcomes are
realistic and stable. Finally, we discuss the embedding diagrams and volume
integral quantifier to have a complete view of wormhole geometries.

##  I Introduction

In the last few decades, the evolution of the universe is described by a large
amount of astrophysical observational data such as Laser Interferometer
Gravitational-Wave Observatory (LIGO) [1], Virgo [2], Event Horizon Telescope
(EHT) [3, 4], Advanced Telescope for High-Energy Astrophysics (ATHENA) [5],
International Gamma-Ray Astrophysics Laboratory (INTEGRAL) [6], Imaging x-ray
Polarimetry mission (IXPE) [7], XMM-Newton [8, 9], and Swift [10]. These
observations motivate the research community to explore and develop more
insights and advanced strategies to test the gravity in strong gravitational
fields. The concept of wormholes (WHs) reserved special attention, which are
the hypothetical tunnels connecting different asymptotically flat regions of
the spacetime or universes. These are unusual astrophysical objects provided
with no singularities and horizons [11]. To explore these objects challenges
us in the fundamental physics and draws interest to understand the unique form
of matter called ‘exotic matter’ and how gravity shapes its’ geometry.

In general relativity and modified theories of gravity, WHs are the solutions
of the field equations [12]. These solutions would act as a short-cut way
between two distant universes and be used to make a time machine for which a
stable traversable WH is required [13, 14]. The WH-like solution was first
coined by Einstein and Rosen in their collaborative work [15]. Later on, some
studies showed the WHs features like the Einstein-Rosen bridge coming from the
connection of two Schwarzschild-solutions [16]. Moreover, these studies are
done in the presence of event horizon, which results, anyone trying to escape
the WH throat (the wormhole short-cut path is traversable through a minimal
surface area called WH throat), always falls into the singularity [15]. To
resolve this issue, one can make a prior assumption on the metric. Besides,
one can adopt the Birkhoff theorem approach to put bounds at the WH throat
[13]. In this case, the radial tension might be large enough to exceed the
total mass-energy density, i.e., $\tau_{0}>\rho_{0}c^{2}$ must hold. In
consequence, the energy-momentum tensor violates the null-energy condition
(NEC) at the throat i.e. $T_{\mu\nu}k^{\mu}k^{\nu}$ [17]. Besides, the
existence of this type of exotic matter is possible as the dark energy or
phantom energy shows the same kind of features to explain the accelerated
expansion of the universe [18]. Several wormhole geometries have been studied
using phantom energy, see, e.g. [19].

In this view, there are several approaches have been proposed to relieve the
problem. But, we cannot impose the above condition directly on the matter.
Therefore, we can either consider the exotic form of matter or modified
theories of gravity where the higher-order curvature terms provide the WHs
properties to deal with this problem. In this context, the traversable WHs and
thin-shell WHs with their features have been studied in $f(R)$ gravity [20].
The wormhole geometries in $f(R,T)$ gravity have been studied by presuming
different radial and lateral pressure relations in [21]. Also, they obtained
the solutions for shape functions and discussed their properties with energy
conditions. Moraes and his collaborators studied the WH solutions in
$R^{2}$-gravity and exponential $f(R,T)$ formalism [22]. Moreover, WHs are
discussed widely in teleparallel gravity and other extended theories of
gravity [23]. Besides, one of the interesting work done by M. Zubair et al.
[24], where they have studied the wormhole solution for three separated cases
such as isotropic, anisotropic and barotropic fluids in $f(R,T)$ gravity. Also
they have concluded that anisotropic matter presents a realistic and stable
wormhole model.

This manuscript focused on exploring the WH geometries in symmetric
teleparallel gravity ($f(Q)$ gravity). As the WHs are supported by the exotic
matter, and that is an entirely unsolved problem. This issue motivates us to
study the WH geometries through modified theories where curvature explains the
WHs and retains standard matter. Among several motivations to explore the WHs
in modified theories, we highlight $f(Q)$ gravity, introduced by Jimenez et
al. [25], where the gravitational interaction is described by the non-
metricity term $Q$. Recently, the study on $f(Q)$ gravity has been developed
rapidly in theoretical and observational fields (see details in [26]). Sahoo
and his group studied the cosmic acceleration of the universe in the presence
bulk viscous fluid with the latest Pantheon dataset [27]. Therefore, the study
on WHs in $f(Q)$ gravity may bring new insights into cosmology as it is a
novel approach.

Here, we have studied three types of WH geometries by considering (1) a
relation between the radial and lateral pressure, (2) phantom energy equation
of state (EoS), and (3) a specific shape-function for $b(r)$. We have
calculated and discussed the properties of $b(r)$ for three cases. To do the
stability analysis of the wormhole solutions, we have tested the energy
conditions. Besides this, we have measured the exotic matter for all WH
geometries.

The outlines of this manuscript layered as follows. In Sec. II, we have
presented the basic formulation for $f(Q)$ gravity. And, basic conditions and
remarks required for a traversable wormhole have been discussed in Sec. III.
In Sec IV, we discussed the framework of the traversable wormhole geometries
in $f(Q)$ gravity. We also discussed the energy conditions and three types of
wormhole solutions. In Sec. V, we construct the motion equations for two
different form of $f(Q)$. Then, we discuss three wormhole solutions for the
linear form of $f(Q)$ and two wormhole solutions for a non-linear form of
$f(Q)$ in Sec. VI and VII, respectively. Embedding diagrams for wormhole
solutions have been discussed in Sec VIII. In Sec. IX, we discussed the volume
integral quantifier to measure the exotic matter. Finally we conclude our
outcomes in Sec. X.

##  II Basic Field Equations in $f(Q)$ gravity

Here, we have considered the action for symmetric teleparallel gravity is
given by [25]

$\mathcal{S}=\int\frac{1}{2}\,f(Q)\sqrt{-g}\,d^{4}x+\int\mathcal{L}_{m}\,\sqrt{-g}\,d^{4}x\,$
(1)

where $f(Q)$ represents the function form of Q, $g$ is the determinant of the
metric $g_{\mu\nu}$, and $\mathcal{L}_{m}$ is the matter Lagrangian density.
The non-metricity tensor and its traces can be written as

$Q_{\lambda\mu\nu}=\bigtriangledown_{\lambda}g_{\mu\nu}$ (2)
$Q_{\alpha}=Q_{\alpha}\;^{\mu}\;{}_{\mu},\;\tilde{Q}_{\alpha}=Q^{\mu}\;_{\alpha\mu}$
(3)

Also, the non-metricity tensor helps us to write the superpotential as

$P^{\alpha}\;_{\mu\nu}=\frac{1}{4}\left[-Q^{\alpha}\;_{\mu\nu}+2Q_{(\mu}\;^{\alpha}\;{}_{\nu)}+Q^{\alpha}g_{\mu\nu}-\tilde{Q}^{\alpha}g_{\mu\nu}-\delta^{\alpha}_{(\mu}Q_{\nu)}\right]$
(4)

where the trace of non-metricity tensor [25] has the form

$Q=-Q_{\alpha\mu\nu}\,P^{\alpha\mu\nu}$ (5)

Again, by definition, the energy-momentum tensor for the fluid description of
the spacetime cab be written as

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\,\mathcal{L}_{m}\right)}{\delta
g^{\mu\nu}}$ (6)

Now, one can write the motion equations by varying the action (1) with respect
to metric tensor $g_{\mu\nu}$, which can be written as

$\frac{2}{\sqrt{-g}}\bigtriangledown_{\gamma}\left(\sqrt{-g}\,f_{Q}\,P^{\gamma}\;_{\mu\nu}\right)+\frac{1}{2}g_{\mu\nu}f\\\
+f_{Q}\left(P_{\mu\gamma i}\,Q_{\nu}\;^{\gamma i}-2\,Q_{\gamma
i\mu}\,P^{\gamma i}\;_{\nu}\right)=-T_{\mu\nu},$ (7)

where $f_{Q}=\frac{df}{dQ}$. Also varying (1) with respect to the
connection,one obtains

$\bigtriangledown_{\mu}\bigtriangledown_{\nu}\left(\sqrt{-g}\,f_{Q}\,P^{\gamma}\;_{\mu\nu}\right)=0.$
(8)

##  III Basic Conditions for Traversable wormholes

Now, we consider a general spherically symmetric, static wormhole spacetime of
the Morris Thorne class. This spacetime is generically written as

$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left(1-\frac{b(r)}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\text{sin}^{2}\theta
d\phi^{2}$ (9)

where $\Phi(r)$ is the redshift function of radial co-ordinate $r$
$(0<r_{0}\leq r\leq\infty)$ and its value always finite everywhere to avoid
the event horizons. $b(r)$ is the shape function that determines the shape of
the wormhole. To investigate the wormhole geometry, $b(r)$ has to satisfy the
following conditions:

  * •

Throat condition: $b(r_{0})=r_{0}$ and $b(r)$ should be less than $r$ for
$r>r_{0}$.

  * •

Flaring out condition: $b^{\prime}(r_{0})<1$ i.e.
$\frac{b(r)-rb^{\prime}(r)}{b^{2}(r)}>0$, where ′ represents derivative w.r.t.
$r$.

  * •

Asymptotically Flatness condition: $\frac{b(r)}{r}\rightarrow o$ as
$r\rightarrow\infty$.

Another important criterion is the proper radial distance $l(r)$, defined as

$\ell(r)=\pm\int_{r_{0}}^{r}\frac{dr}{\sqrt{1-\frac{b(r)}{r}}}$ (10)

To have a proper description of traversable WH, $\ell(r)$ must be finite over
the radial coordinate. Thus, it is a decreasing function. First, it falls down
from the upper universe to the wormhole’s throat, i.e., $\ell=+\infty$ to
$\ell=0$, and then from wormhole’s throat to lower universe, i.e., $\ell=0$ to
$\ell=-\infty$. Besides, $\ell$ should be greater than or equal to the radial
coordinate distance i.e., $\lvert\ell(r)\rvert\geq{r-r_{0}}$. The signature of
$\ell$ represents the lower and upper parts of the wormhole. The positive and
negative sign of $\ell$ denotes the WH’s upper and lower sections, and both
sections are connected by the wormhole’s throat.

For the present interest, let us consider the matter is described by an
anisotropic stress-energy tensor of the form

$T_{\mu}^{\nu}=\left(\rho+P_{t}\right)u_{\mu}\,u^{\nu}-P_{t}\,\delta_{\mu}^{\nu}+\left(P_{r}-P_{t}\right)v_{\mu}\,v^{\nu}$
(11)

where $u_{\mu}$ is the four-velocity, $v_{\mu}$ the unitary space-like vector
in the radial direction, $\rho$ is the energy density, $P_{r}$ is the pressure
in the direction of $u_{\mu}$ (radial pressure) and $P_{t}$ is the pressure
orthogonal to $v_{\mu}$(tangential pressure). Here, $P_{r}$ and $P_{t}$ are
functions of redial component $r$.

##  IV Wormhole Geometries in $f(Q)$ Gravity

In this section, we discuss the different kinds of wormhole solutions with
their self-stability. The trace of the non-metricity tensor $Q$ for the
wormhole metric in (9) takes the form below,

$Q=-\frac{2}{r}\left(1-\frac{b(r)}{r}\right)\left(2\phi^{{}^{\prime}}(r)+\frac{1}{r}\right).$
(12)

Now, by substituting (9) and (11) in (7) one can find the following field
equations

$\left[\frac{1}{r}\left(-\frac{1}{r}+\frac{rb^{{}^{\prime}}(r)+b(r)}{r^{2}}-2\phi^{{}^{\prime}}(r)\left(1-\frac{b(r)}{r}\right)\right)\right]f_{Q}-\frac{2}{r}\left(1-\frac{b(r)}{r}\right)\dot{f}_{Q}-\frac{f}{2}=-\rho,$
(13)
$\left[\frac{2}{r}\left(1-\frac{b(r)}{r}\right)\left(2\phi^{{}^{\prime}}(r)+\frac{1}{r}\right)-\frac{1}{r^{2}}\right]f_{Q}+\frac{f}{2}=-P_{r},$
(14)
$\left[\frac{1}{r}\left(\left(1-\frac{b(r)}{r}\right)\left(\frac{1}{r}+\phi^{{}^{\prime}}(r)\left(3+r\phi^{{}^{\prime}}(r)\right)+r\phi^{{}^{\prime\prime}}(r)\right)-\frac{rb^{{}^{\prime}}(r)-b(r)}{2r^{2}}\left(1+r\phi^{{}^{\prime}}(r)\right)\right)\right]f_{Q}+\\\
\frac{1}{r}\left(1-\frac{b(r)}{r}\right)\left(1+r\phi^{{}^{\prime}}(r)\right)\dot{f}_{Q}+\frac{f}{2}=-P_{t}.$
(15)

Using (13)-(15), one can study different wormhole models and their properties.

###  IV.1 Energy Conditions

Energy conditions are discussed about the physically realistic matter
configuration that developed from the Raychaudhuri equations. The Raychaudhuri
equations state the temporal evolution of expansion scalar ($\theta$) for the
congruences of timelike ($u^{\mu}$) and null ($\eta_{\mu}$) geodesics as [28]

$\frac{d\theta}{d\tau}-\omega_{\mu\nu}\,\omega^{\mu\nu}+\sigma_{\mu\nu}\sigma^{\mu\nu}+\frac{1}{3}\theta^{2}+R_{\mu\nu}u^{\mu}\,u^{\nu}=0$
(16)
$\frac{d\theta}{d\tau}-\omega_{\mu\nu}\,\omega^{\mu\nu}+\sigma_{\mu\nu}\sigma^{\mu\nu}+\frac{1}{2}\theta^{2}+R_{\mu\nu}\eta^{\mu}\eta^{\nu}=0$
(17)

where $\sigma^{\mu\nu}$ and $\omega_{\mu\nu}$ are the shear and the rotation
associated with the vector field $u^{\mu}$ respectively. For attractive nature
of gravity ($\theta<0$) and neglecting the quadratic terms, the Raychaudhuri
equations (16) and (17) satisfy the following conditions

$R_{\mu\nu}u^{\mu}\,u^{\nu}\geq 0$ (18) $R_{\mu\nu}\eta^{\mu}\eta^{\nu}\geq 0$
(19)

As we are working with anisotropic fluid matter distribution, the energy
condition recovered from standard General Relativity (GR) are
$\bullet$ Strong energy conditions (SEC) if $\rho+P_{j}\geq 0$,
$\rho+\sum_{j}P_{j}\geq 0$, $\forall j$.
$\bullet$ Dominant energy conditions (DEC) if $\rho\geq 0$, $\rho\pm P_{j}\geq
0$, $\forall j$.
$\bullet$ Weak energy conditions (WEC) if $\rho\geq 0$, $\rho+P_{j}\geq 0$,
$\forall j$.
$\bullet$ Null energy condition (NEC) if $\rho+P_{j}\geq 0$, $\forall j$.
where $\rho$ and $P$ describe the energy density and pressure, respectively.

##  V Wormhole solutions in different forms of $f(Q)$

In this study we will consider two different form of $f(Q)$, (i) Linear form
of $f(Q)$, and (ii) Non-linear form of $f(Q)$. By using these forms of
$f(Q)$’s, we will get different field equations, and subsequently, we shall
discuss by assuming particular models.

###  V.1 Field equations in $f(Q)=\alpha Q$

To proceed further, we have presumed the linear functional form of $Q$ as

$f(Q)=\alpha Q,$ (20)

where ‘$\alpha$’ is constant, which is the teleparallel gravitational term.
The linear form of $f(Q)$ recovers the symmetric teleparallel equivalent of
general relativity (STEGR), which helps us to compare our WH solutions to its’
fundamental level. Further, the redshift function $\Phi(r)$ in (9) must be
finite and non-vanishing at the throat $r_{0}$. So one can consider
$\Phi(r)=constant$ to achieve the de Sitter and anti-de Sitter asymptotic
behaviour. Therefore the field equations in (13)-(15) reads

$-\frac{b^{{}^{\prime}}(r)}{r^{2}}\alpha=\rho,$ (21)
$\frac{b(r)}{r^{3}}\alpha=P_{r},$ (22)
$\left(\frac{b^{{}^{\prime}}(r)}{2\,r^{2}}-\frac{b(r)}{2\,r^{3}}\right)\alpha=P_{t}.$
(23)

Now, in the next section, we are going to discuss three special cases of
wormhole solutions for our study.

###  V.2 Field equations in $f(Q)=a\,Q^{2}+B$

In this study, we have assumed a particular power-law form of $f(Q)$ i.e.
$f(Q)=a\,Q^{2}+B$, where $a$ and $B$ are constants. Researchers have already
shown that the power-law model can oblige the regular thermal extending
history including the cold dark matter-dominated stage and the radiation. Also
the redshift function $\phi(r)$ is finite everywhere therefore we have
considered $\phi(r)=constant$ in this study. Therefore using the model we
developed the field equations (13)-(15) as follows

$\frac{2a\left(b(r)-r\right)\left(11b(r)-(6\,b^{{}^{\prime}}(r)+7)\,r\right)}{r^{6}}+\frac{B}{2}=\rho$
(24)
$\frac{2a\left(3(b(r))^{2}-4\,r\,b(r)+r^{2}\right)}{r^{6}}-\frac{B}{2}=P_{r}$
(25)
$-\frac{6\,a\left(b(r)-r\right)\left(2b(r)-(b^{{}^{\prime}}(r)+1)r\right)}{r^{6}}-\frac{B}{2}=P_{t}$
(26)

##  VI Wormhole Models With Linear $f(Q)$

In this section, we have considered simplest linear form of $f(Q)$ i.e.
$f(Q)=\alpha Q$. With this form, we shall discuss three special cases of
wormhole solutions.

####  VI.0.1 Wormhole (WH1) solution with $P_{t}=mP_{r}$

In first model, we assume the pressures $P_{r}$ and $P_{t}$ are related as
(for more details see Ref. [21])

$P_{t}=mP_{r}$ (27)

where $m$ is an arbitrary constant.

By using equations (22) and (23) in equation (27), one can obtain

$b(r)=kr^{(1+2m)}$ (28)

where $k$ is an integrating constant. Without loss of generality we consider
$k=1$. For $m<0$, Eqn. (28) retained the asymptotically flatness condition
i.e. $b(r)/r\rightarrow 0$ for $r\rightarrow\infty$. In Fig. 1, we depict the
quantities $b(r),b(r)/r,b(r)-r$ and $b^{{}^{\prime}}(r)$ with varying the
radial component $r$, for $m=-0.25$. One can clearly see that $b(r)-r$ cuts
the $r$-axis at $r_{0}=1$ in Fig. 1. Note that, for a stable wormhole, the
shape function $b(r)$ need to obey flaring, throat, and asymptotically
conditions. Profiles in Fig. 1 shows that, shape function satisfied all the
conditions which are required for a stable traversable wormhole.

Figure 1: The profile of shape function $b(r)$, Flaring out condition
$b^{\prime}(r)<1$, throat condition $b(r)-r<0$, and asymptotically flatness
condition $\frac{b(r)}{r}\rightarrow 0$ as $r\rightarrow\infty$ with varying
$r$, for $m=-0.25$ (for WH1).

Now using equation (27), we can rewrite the energy density $\rho$, radial
pressure $P_{r}$ and lateral pressure $P_{t}$ from (21)-(23) as follows

$\rho=-\alpha(1+2m)\,r^{2(m-1)},$ (29) $P_{r}=\alpha\,r^{2(m-1)},$ (30)
$P_{t}=m\,\alpha\,r^{2(m-1)}.$ (31)

Now from the eqns. (29)-(31), we have

$\rho+P_{r}=-2\,m\,\alpha\,r^{2(m-1)},$ (32)
$\rho+P_{t}=-(m+1)\,\alpha\,r^{2(m-1)},$ (33) $\rho-
P_{r}=-2\alpha(m+1)r^{2(m-1)},$ (34) $\rho-P_{t}=-\alpha(1+3m)r^{2(m-1)}.$
(35)

As we know, energy conditions are the best geometrical tool to test the
cosmological models’ self-stability. So we adopted this technique to test our
models. Moreover, we have presumed the linear functional form of $f(Q)$.
Hence, our models will retain the standard energy conditions.

Figure 2: Profile of energy density, $\rho(r)$ w.r.t. $r$ and $\alpha$ with
$m=-0.25$ for WH1.

Figs. 2 and 3, depicts the behavior of the energy conditions. It can be seen
from Fig. 2 that the energy density, $\rho$ is positive throughout the
spacetime. From Fig. 3, one can observed that NEC for the radial pressure
violates i.e. $\rho+P_{r}<0$, whereas NEC for the lateral pressure i.e.
$\rho+P_{t}\geq 0$ obeys. Also, DEC is satisfied i.e. $\rho-P_{r}\geq 0$ and
$\rho-P_{t}\geq 0$. These profiles of energy conditions aligned with the
properties of exotic matter which is responsible for a traversable wormhole.
From equations (29)-(31), the strong energy condition (SEC) yields
$\rho+P_{r}+2P_{t}=0$. This similar result obtained in [29].

(a) $\rho+P_{r}$ (b) $\rho+P_{t}$

(c) $\rho-P_{r}$ (d) $\rho-P_{t}$

Figure 3: Profiles of WEC, NEC and DEC w.r.t. $r$ and $\alpha$ with $m=-0.25$
for WH1.

####  VI.0.2 Wormhole (WH2) solution with $P_{r}=\omega\rho$

In this model, we assume a relation between $\rho$ and $P_{r}$ as a linear
equation of state (EoS) such as [30, 31, 32, 33].

$P_{r}=\omega\rho$ (36)

where $\omega$ is as equation of state (EoS).
Therefore, using equations (21) and (22) in Eq. (36), one can obtain the
following shape function

$b(r)=cr^{-\frac{1}{\omega}}$ (37)

where $c$ is an integrating constant.
Note that, to satisfy the asymptotically flatness condition, i.e.
$\frac{b(r)}{r}\rightarrow 0$ as $r\rightarrow\infty$, $\omega$ should be less
than -1 i.e. $\omega<-1$.

The profiles of the necessary conditions for a shape function i.e. throat
condition, flare out condition and asymptotically flatness conditions are
depicted in Fig. 4. From Fig. 4, one can see that the shape function $b(r)$ is
in the increasing direction as $r$ increases. For $r>r_{0}$, $b(r)-r<0$, which
represents the consistent of throat condition for wormholes and from the same
figure 4, it is clear that $b(r)-r$ cuts the $r$-axis at $r_{0}=1$. Besides,
the flaring out condition satisfied at $r_{0}$ i.e.
$b^{{}^{\prime}}(r_{0})=b^{{}^{\prime}}(1)\approx 0.667<1$. Asymptotically
flatness condition is also satisfied, i.e. $\frac{b(r)}{r}\rightarrow 0$ as
$r\rightarrow\infty$ satisfied. Therefore, from Fig. 4, one can notice that
the shape function satisfies all the required conditions for a traversable WH.

Figure 4: The behavior of shape function $b(r)$, flaring out condition
$b^{\prime}(r)<1$, throat condition $b(r)-r<0$, and asymptotically flatness
condition $\frac{b(r)}{r}\rightarrow 0$ as $r\rightarrow\infty$ for
$c=1,\omega=-2$ for WH2.

Now, we discuss the traversable wormhole spacetime supported by the phantom
energy with the presence of exotic matter. For phantom energy EoS,
$P_{r}=\omega\rho$ with $\omega<-1$ and using the shape function (37) in field
equations (21)-(23), we get the stress-energy tensor components are given by

$\rho=\frac{c\,\alpha}{\omega}\,r^{-\left(\frac{1}{\omega}+3\right)},$ (38)
$\rho+P_{r}=c\,\alpha\left(\frac{1}{\omega}+1\right)\,r^{-\left(\frac{1}{\omega}+3\right)},$
(39)
$\rho+P_{t}=\frac{1}{2}\,c\,\alpha\left(\frac{1}{\omega}-1\right)\,r^{-\left(\frac{1}{\omega}+3\right)},$
(40)

Now, the null energy condition (NEC) at the throat is given by

$\rho+P_{r}\mid_{r_{0}}=c\,\alpha\left(\frac{1}{\omega}+1\right)\,r_{0}^{-\left(\frac{1}{\omega}+3\right)}.$
(41)

In this case, it is clear that $\omega$ should not be equal to $-1$ i.e.
$\omega\neq{-1}$, so we consider $\omega<-1$ to imply the violation of NEC at
the throat, i.e. the throat of the WH needs to open with phantom energy.

The DEC for this model is

$\rho-
P_{r}=c\,\alpha\left(\frac{1}{\omega}-1\right)\,r^{-\left(\frac{1}{\omega}+3\right)},$
(42) $\rho-
P_{t}=\frac{1}{2}c\,\alpha\left(\frac{3}{\omega}+1\right)\,r^{-\left(\frac{1}{\omega}+3\right)}.$
(43)

Figure 5: The profile of energy density, $\rho(r)$ w.r.t. $r$ and $\alpha$
with $\omega=-2,\,c=1$ for WH2.

We draw the graphical behavior of energy conditions using equations (38)-(43)
are shown in Figs. 5 and 6. For this model, we have considered a matter
content that is related to phantom equation of state, i.e., $\omega<-1$. In
Fig. 5, we have shown the behavior of energy density and it takes positive
values for all over the range. It can be seen from the Fig. 6 that DEC is
satisfied i.e. $\rho-P_{r}\geq 0$ and $\rho-P_{t}\geq 0$ but we observe that
NEC is violated due to $\rho+P_{r}<0$. The violation of NEC is proof of the
presence of the exotic matter, which might be needed for the wormhole
geometry.

(a) $\rho+P_{r}$ (b) $\rho+P_{t}$

(c) $\rho-P_{r}$ (d) $\rho-P_{t}$

Figure 6: Profiles of WEC, NEC and DEC w.r.t. $r$ and $\alpha$ with
$\omega=-2,c=1$ for WH2.

####  VI.0.3 Wormhole (WH3) solution with
$b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{n}$

In this subsection, we have considered the specific shape function,
$b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{n}$ and for this choice the
corresponding stress energy components from Eqns. (21)-(23) are obtained as
follows

$\rho=-n\,\alpha\,\frac{r^{n-3}}{r_{0}^{n-1}}$ (44)
$\rho+P_{r}=\left(1-n\right)\,\alpha\,\frac{r^{n-3}}{r_{0}^{n-1}}$ (45)
$\rho+P_{t}=-\frac{1}{2}\left(n+1\right)\alpha\,\frac{r^{n-3}}{r_{0}^{n-1}}$
(46)

Now, at the throat i.e. at $r=r_{0}$, Eq. (45) reduce to

$\rho+P_{r}\mid_{r_{0}}=\frac{\left(1-n\right)\alpha}{r_{0}^{2}}$ (47)

The dominant energy conditions (DEC) for this model are

$\rho-P_{r}=-\left(n+1\right)\,\alpha\,\frac{r^{(n-3)}}{r_{0}^{(n-1)}}$ (48)
$\rho-
P_{t}=\left(\frac{1}{2}-\frac{3n}{2}\right)\,\alpha\,\frac{r^{(n-3)}}{r_{0}^{(n-1)}}$
(49)

Taking into account that the condition $\frac{b(r)}{r}\rightarrow 0$ as
$r\rightarrow\infty$ will satisfy when $n<1$. In Fig. 7, we have shown the
behavior of $b(r),\,b(r)-r,\,\frac{b(r)}{r}$ and $b^{\prime}(r)$ respectively
and in this case $b(r)-r$ cuts the $r$-axis at $r_{0}$= 2, which is the throat
radius of wormhole (see Fig. 8). From Fig. 7, one can see that $b(r)$
satisfies all the necessary conditions for a traversable wormhole.

Figure 7: Characteristic of the shape functions for WH3. Figure 8: The profile
shows two-dimensional embedding diagrams for WH1, WH2, and WH3 (here, WH1 and
WH2 overlap each other).

Moreover, from Fig. 9, we see that energy density, $\rho$ is always positive
throughout the spacetime. In Fig. 10 (b), (c) and (d), we observe that the
energy condition tends to zero as the radial component $r$ goes on for the
negative range of $\alpha$. But, NEC violates as $\rho+P_{r}<0$.

Figure 9: The energy density, $\rho(r)\geq 0$ with $n=-0.5$, $r_{0}=2$ for
WH3.

(a) $\rho+P_{r}$ (b) $\rho+P_{t}$

(c) $\rho-P_{r}$ (d) $\rho-P_{t}$

Figure 10: Profiles of WEC, NEC and DEC w.r.t. $r$ and $\alpha$ with $n=-0.5$,
$r_{0}=2$ for WH3.

##  VII Wormhole Models With Non-linear $f(Q)$

In this section, by presuming a polynomial form of $f(Q)=a\,Q^{2}+B$, we
continue trying to discuss the fundamental equations for the anisotropic
fluid. For the choice of $P_{t}=mP_{r}$ and $P_{r}=w\rho$, the differential
equations in (24)-(26) becomes highly non-linear and very complicated to solve
and analyze these types of systems (see Ref. [35]) . Therefore, we consider
two different specific shape functions in finding wormhole solutions.

####  VII.0.1 Wormhole (WH4) solution with
$b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{n}$

Here, in this subsection, we have assumed the specific shape function given by
$b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{n}$ and for this choice the
corresponding stress energy components from Eqns. (24)-(26) are obtained as
follows

$\rho=\frac{2a\left((6n-11)r_{0}\left(\frac{r}{r_{0}}\right){}^{n}+7r\right)\left(r-r_{0}\left(\frac{r}{r_{0}}\right){}^{n}\right)}{r^{6}}+\frac{B}{2}$
(50)
$\rho+P_{r}=\frac{4a\left(r-r_{0}\left(\frac{r}{r_{0}}\right){}^{n}\right)\left((3n-7)r_{0}\left(\frac{r}{r_{0}}\right){}^{n}+4r\right)}{r^{6}}$
(51)
$\rho+P_{t}=\frac{2a\left(r-r_{0}\left(\frac{r}{r_{0}}\right){}^{n}\right)\left((3n-5)r_{0}\left(\frac{r}{r_{0}}\right){}^{n}+4r\right)}{r^{6}}$
(52) $\rho-
P_{r}=\frac{4ar_{0}\left((3n-7)r-(3n-4)r_{0}\left(\frac{r}{r_{0}}\right){}^{n}\right)\left(\frac{r}{r_{0}}\right){}^{n}+12ar^{2}+Br^{6}}{r^{6}}$
(53) $\rho-
P_{t}=\frac{2ar_{0}\left(9(n-3)r-(9n-17)r_{0}\left(\frac{r}{r_{0}}\right){}^{n}\right)\left(\frac{r}{r_{0}}\right){}^{n}+20ar^{2}+Br^{6}}{r^{6}}$
(54)

The behavior of shape function is depicted in Fig. 11. It can be seen from the
Fig. 11 that shape function, $b(r)$ shows the increasing behavior for
different values of $n$. Furthermore, to check the asymptotically flatness
condition we plot $\frac{b(r)}{r}$ verses $r$ which gives
$\frac{b(r)}{r}\rightarrow 0$ as $r\rightarrow\infty$. Here $b(r)-r$ cuts the
$r$-axis at $r_{0}=1$, which is the throat radius (see Fig. 19) for this WH4
when $n<1$. All of the above results shows the traversablity of WH4.

Figure 11: Characteristic of the shape functions for WH4 Figure 12: The plot
of $\rho$ versus $r$ with $n=0.1\ (Red)$, $n=0.3\ (Purple)$, $n=0.5\
(Orange)$, $n=0.8\ (Black)$, $a=0.1$, $B=0.2$ and $r_{0}=1$ for WH4.

(a) $\rho+P_{r}$ (b) $\rho+P_{t}$

Figure 13: The plots of $\rho+P_{r}$ and $\rho+P_{t}$ versus $r$ with $n=0.1\
(Red)$, $n=0.3\ (Purple)$, $n=0.5\ (Orange)$, $n=0.8\ (Black)$, $a=0.1$,
$B=0.2$ and $r_{0}=1$ for WH4.

(a) $\rho-P_{r}$ (b) $\rho-P_{t}$

Figure 14: The plots of $\rho-P_{r}$ and $\rho-P_{t}$ versus $r$ with $n=0.1\
(Red)$, $n=0.3\ (Purple)$, $n=0.5\ (Orange)$, $n=0.8\ (Black)$, $a=0.1$,
$B=0.2$ and $r_{0}=1$ for WH4.

The behavior of energy conditions (ECs) has shown in Figs. 12-14. To verify
the profiles of ECs, we have fixed the parameters as $a=0.1$, $B=0.2$ and
$r_{0}=1$. As we know, energy density has to be positive everywhere, and its
behaviors for WH4 are presented in Fig. 12 for different choices of $n$. Also,
from Fig. 13, we observed that $\rho+P_{r}<0$ and $\rho+P_{t}>0$, implying the
violation of NEC at wormhole throat. Besides, we have shown the graph of DEC
in Fig. 14. The violation of NEC for different values of the parameter $n$ is
the proof for the presence of exotic matter at the wormhole throat, which may
be required for the wormhole geometry.

####  VII.0.2 Wormhole (WH5) solution with
$b(r)=\gamma\,r_{0}\,\left(1-\frac{r_{0}}{r}\right)+r_{0}$

Here, we have considered a specific shape function given by
$b(r)=\gamma\,r_{0}\,\left(1-\frac{r_{0}}{r}\right)+r_{0}$ and for this choice
the corresponding stress energy components from Eqns. (24)-(26) are obtained
as follows

$\rho=\frac{2a\left(r-r_{0}\right)\left(r-\gamma
r_{0}\right)\left(7r^{2}-11(\gamma+1)r_{0}r+17\gamma
r_{0}^{2}\right)}{r^{8}}+\frac{B}{2}$ (55)
$\rho+P_{r}=\frac{4a\left(r-r_{0}\right)\left(r-\gamma
r_{0}\right)\left(4r^{2}-7(\gamma+1)r_{0}r+10\gamma r_{0}^{2}\right)}{r^{8}}$
(56) $\rho+P_{t}=\frac{2a\left(r-r_{0}\right)\left(r-\gamma
r_{0}\right)\left(4r^{2}-5(\gamma+1)r_{0}r+8\gamma r_{0}^{2}\right)}{r^{8}}$
(57) $\rho-
P_{r}=\frac{12ar^{4}+4ar_{0}\left(r_{0}\left(2(\gamma(2\gamma+9)+2)r^{2}+\gamma
r_{0}\left(7\gamma
r_{0}-11(\gamma+1)r\right)\right)-7(\gamma+1)r^{3}\right)+Br^{8}}{r^{8}}$ (58)
$\rho-
P_{t}=\frac{20ar^{4}+2ar_{0}\left(r_{0}\left((\gamma(17\gamma+70)+17)r^{2}+\gamma
r_{0}\left(26\gamma
r_{0}-43(\gamma+1)r\right)\right)-27(\gamma+1)r^{3}\right)+Br^{8}}{r^{8}}$
(59)

The behavior of shape function is depicted in Fig. 15. It can be seen from the
Fig. 15 that shape function, $b(r)$ shows the increasing behavior for
different values of $n$. We plot $\frac{b(r)}{r}$ with respect to $r$ to check
the asymptotically flatness condition which gives $\frac{b(r)}{r}\rightarrow
0$ as $r\rightarrow\infty$. Here, in this case $b(r)-r$ cuts the $r$-axis at
$r_{0}=1.05$, which is the throat radius (see Fig. 15) for this WH5. Flare out
condition is also satisfied as $b^{{}^{\prime}}(r)<1$ for $r>r_{0}$. All the
above profiles for the shape function of WH5 satisfies the necessary
properties for a traversable wormhole.

Figure 15: Characteristic of the shape functions for WH5 Figure 16: The plot
of $\rho$ versus $r$ with $\gamma=0.05\ (Red)$, $\gamma=0.15\ (Purple)$,
$\gamma=0.25\ (Orange)$, $\gamma=0.35\ (Black)$, $a=0.1$, $B=0.2$ and
$r_{0}=1.05$ for WH5.

(a) $\rho+P_{r}$ (b) $\rho+P_{t}$

Figure 17: The plots of $\rho+P_{r}$ and $\rho+P_{t}$ versus $r$ with
$\gamma=0.05\ (Red)$, $\gamma=0.15\ (Purple)$, $\gamma=0.25\ (Orange)$,
$\gamma=0.35\ (Black)$, $a=0.1$, $B=0.2$ and $r_{0}=1.05$ for WH5.

(a) $\rho-P_{r}$ (b) $\rho-P_{t}$

Figure 18: The plots of $\rho+P_{r}$ and $\rho+P_{t}$ versus $r$ with
$\gamma=0.05\ (Red)$, $\gamma=0.15\ (Purple)$, $\gamma=0.25\ (Orange)$,
$\gamma=0.35\ (Black)$, $a=0.1$, $B=0.2$ and $r_{0}=1.05$ for WH5.

Figs. 16-18 shows the behavior of energy conditions for WH5. Here, we have
chosen the parameters as $a=0.1$, $B=0.2$ and $r_{0}=1.05$. In Fig. 16, we
have shown the graph of energy density, $\rho$ versus $r$, which shows a
positive behavior throughout the space-time for different values of the
parameter $\gamma$. Fig. 17 shows the behavior of NEC, and one can be seen
from the figure that NEC is violated as $\rho+P_{r}<0$, which implying the
violation of WEC. The violation of NEC is the primary requirement for a
traversable wormhole. Moreover, we have shown the graph of DEC in Fig. 18.
These results tend to be determined that the wormhole solutions that have been
obtained are acceptable in this symmetric teleparallel gravity.

Figure 19: The profile shows two dimensional embedding diagram for WH4 and WH5

##  VIII Embedding Diagram

In this section, we are going to discuss the embedding diagrams that would
help us to understand the wormhole spacetime in (9). On the explicit interest
in geometry, we composed some constraints on the coordinate system. We
considered the equatorial slice $\theta=\pi/2$ for a fixed time i.e.
$t=$constant. For this, Eq. (9) reduces to

$ds^{2}=\left(1-\frac{b(r)}{r}\right)^{-1}+r^{2}d\phi^{2}.$ (60)

This reformed metric can be embedded into three-dimensional Euclidean space
with cylindrical coordinate $r,\,\ \phi$ and $z$ as

$ds^{2}=dz^{2}+dr^{2}+r^{2}d\phi^{2}.$ (61)

Now, comparing Eq. (60) and (61), we can find the embedding surface $z(r)$,
and we obtained a slope as

$\frac{dz}{dr}=\pm\sqrt{\frac{r}{r-b(r)}-1}$ (62)

In Eq. (62), we used the solutions for $b(r)$, which are evaluated for three
WH model to draw the embedding surfaces. In Fig. 20, we have shown the
embedding surfaces for WH1, WH2, WH3, WH4 and WH5 concerning five models. The
values of free parameters are the same as those used to discuss the energy
conditions for respective models.

(a) WH1 (b) WH2

(c) WH3 (d) WH4

(e) WH5

Figure 20: Embedding diagrams.

##  IX Volume Integral Quantifier

Volume integral quantifier provides the information about the ”total amount of
exotic matter” required for wormhole maintenance. To do this, one may compute
the definite integral $\int T_{\mu\nu}U^{\mu}\,U^{\nu}$ and $\int
T_{\mu\nu}k^{\mu}\,k^{\nu}$, where $U^{\mu}$ is the four-velocity [34]. For
spherically symmetry and average null energy condition (ANEC) violating matter
related to the radial component is defined as

$Iv=\oint[\rho+P_{r}]dV$ (63)

where $dV=r^{2}\sin\theta\,dr\,d\theta\,d\phi$
it can also be written as

$Iv=8\pi\int_{r_{0}}^{\infty}\,(\rho+P_{r})r^{2}\,dr$ (64)

Now suppose that the wormhole enlarge from the throat, $r_{0}$, with a cutoff
of the stress energy tensor at certain radius $a$, then it reduces to

$Iv=8\pi\int_{r_{0}}^{a}\,(\rho+P_{r})r^{2}\,dr$ (65)

where $r_{0}$ is the throat of the wormhole, which is the minimum value of
$r$. The main point of this discussion is when limiting $a\rightarrow
r_{0}^{+}$, one can verify that $Iv\rightarrow 0$. From Figs. 21,22 and 25 we
found that for each wormhole solution one may construct wormhole solutions
with small quantities of exotic matter which need to open the wormhole throat.

Figure 21: 3D Plot for volume integral quantifier associated with model 1 and
here it is clear that when $a\rightarrow r_{0}^{+}$ then $Iv\rightarrow 0$
i.e. minimize the violation of NEC would be possible. For this model we
consider $n=-0.25$, $r_{0}=1$ and $a=3$. Figure 22: 3D Plot for volume
integral quantifier associated with model 2 and here it is clear that when
$a\rightarrow r_{0}^{+}$ then $Iv\rightarrow 0$. For this model we consider
$\omega=-2$, $r_{0}=1$ and $a=3$. Figure 23: 3D Plot for volume integral
quantifier associated with model 3 and here it is clear that when
$a\rightarrow r_{0}^{+}$ then $Iv\rightarrow 0$ and we consider $n=-0.5$,
$r_{0}=2$ and $a=3$. Figure 24: 3D Plot for volume integral quantifier
associated with model 4 and here it is clear that when $a\rightarrow
r_{0}^{+}$ then $Iv\rightarrow 0$ and we consider $a=0.1$ and $B=0.2$. Figure
25: 3D Plot for volume integral quantifier associated with model 5 and here it
is clear that when $a\rightarrow r_{0}^{+}$ then $Iv\rightarrow 0$ and we
consider $a=0.1$ and $B=0.2$.

##  X Concluding Remarks

In this manuscript, we have discussed Morris-Throne wormholes, i.e., static
and spherically symmetric traversable wormholes in the framework of symmetric
teleparallel gravity (i.e., $f(Q)$ gravity), where the gravitational
interaction is described by the non-metricity term $Q$. $f(Q)$ gravity is a
recently developed modified theory, so many investigations are going on to
explore the current interests of the cosmological scenarios. And, the study of
WH solutions in $f(Q)$ is a novel approach. Besides this, to have a
traversable WH, the NEC has to be violated. That is possible in the presence
of exotic matter in WH-throat, which is physically unrealistic. To explore a
realistic model, it is better to minimize the usage of exotic matter. In this
view, we explored three traversable WH solutions by considering a linear
functional form of $Q$ (i.e., $f(Q)\propto Q$) and two WH solutions for non-
linear form of $f(Q)$. Also, we considered a constant redshift function, i.e.,
$\Phi^{\prime}(r)=0$ throughout our calculation, which simplifies our
calculations and provides some exciting WH solutions.

For the first case, we have considered a relation between the radial and
lateral pressure. We calculated the solution for shape function $b(r)$ in the
power-law form. The parameter ${}^{\prime}n^{\prime}$ has to be negative to
satisfy the asymptotically flatness condition. Keeping this in mind, we have
tested all the necessary requirements of a shape function and the energy
conditions. We found that the null energy condition is violated. For the
second WH solution, we have considered the phantom energy equation of state
(EoS), which also violates the NEC. We have explored the WH solution by taking
the range of $\omega$ as $\omega<-1$. It is interesting to note here that the
energy density $\rho>0$ throughout the spacetime and the phantom energy may
support traversable wormholes. In the third model, we have considered a
specific shape function for $b(r)$. For this, we have discussed its’ stability
through the energy conditions and in this case, also NEC is violated. Thus,
the violation of NEC for each model defines the possibilities of the presence
of exotic matter at the wormhole’s throat.

Moreover, we have discussed two WH geometries by considering a quadratic
Lagrangian $f(Q)$ with two specific shape functions. To test the
traversability and stability for these two models, we have tested the flaring
out condition and energy conditions. Both the models successfully passed all
the tests. The advantage of these types of models is that it minimizes the
usage of the unknown form of matter called ”exotic matter” to have a
traversable wormhole in comparison to the models for linear $f(Q)$ case (as it
mimics the fundamental interaction of gravity, i.e., GR). Furthermore, in the
case of the non-linear form of $f(Q)$, the profiles of NEC are violated near
the throat of the WHs and then satisfied (see Fig. 13 and 17). Whereas, for
linear $f(Q)$ case, it violated throughout the evolution of $r$ (see Fig. 3,
6, and 10). It is worthy of exploring the wormhole geometries in the non-
linear framework of $f(Q)$ gravity.

Finally, we have discussed the volume integral quantifier to measure the
exotic matter required for a traversable WH. We estimate that a small amount
of exotic matter is required to have a traversable WH for our three solutions.
This results are aligned with [36].

Therefore, it is safe to conclude that in $f(Q)$ gravity, we found suitable
geometries for traversable WH that violates the NEC at its’ throat. It will be
interesting to explore the wormhole geometries in a more generalized $f(Q)$
form in the future, which may help to construct the traversable wormhole
without presence of exotic matter.

## Acknowledgements

S.M. acknowledges Department of Science & Technology (DST), Govt. of India,
New Delhi, for awarding INSPIRE Fellowship (File No. DST/INSPIRE
Fellowship/2018/IF180676). PKS acknowledges CSIR, New Delhi, India for
financial support to carry out the Research project [No.03(1454)/19/EMR-II,
Dt. 02/08/2019]. We are very much grateful to the honorable referee and the
editor for the illuminating suggestions that have significantly improved our
work in terms of research quality and presentation.

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# Learning Representations for Hyper-Relational Knowledge Graphs

Harry Shomer1 Wei Jin1 Juanhui Li1 Yao Ma2 Jiliang Tang 1
1 Department of Computer Science, Michigan State University, Michigan, USA
2 Department of Computer Science, New Jersey Institute of Technology, New
Jersey, USA
{shomerha, jinwei2, lijuanh1<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Knowledge graphs (KGs) have gained prominence for their ability to learn
representations for uni-relational facts. Recently, research has focused on
modeling hyper-relational facts, which move beyond the restriction of uni-
relational facts and allow us to represent more complex and real-world
information. However, existing approaches for learning representations on
hyper-relational KGs majorly focus on enhancing the communication from
qualifiers to base triples while overlooking the flow of information from base
triple to qualifiers. This can lead to suboptimal qualifier representations,
especially when a large amount of qualifiers are presented. It motivates us to
design a framework that utilizes multiple aggregators to learn representations
for hyper-relational facts: one from the perspective of the base triple and
the other one from the perspective of the qualifiers. Experiments demonstrate
the effectiveness of our framework for hyper-relational knowledge graph
completion across multiple datasets. Furthermore, we conduct an ablation study
that validates the importance of the various components in our framework. The
code to reproduce our results can be found at
https://github.com/HarryShomer/QUAD.

##  1 Introduction

Figure 1: An example of a hyper-relational KG. The blue circles represent
entities and the arrows represent relations. The dashed arrows represent
qualifier relations while the solid arrows are base relations. The ??? entity
represents the potential occupation of Richard Bachman (Novelist).

Knowledge graphs (KGs) are a collection of facts represented in a structured
and graphical format. Facts are represented as a triple $(h,r,t)$ that
connects two entities $h$ and $t$ (i.e. nodes) with a relation $r$. KGs have
become very popular recently with applications in language representation Liu
et al. (2020), question answering Huang et al. (2019), and recommendation Wang
et al. (2019b).

In traditional triple-based KGs, facts are represented as binary relations
between entities, which often fall short in representing the complex nature of
the data. To address this shortcoming, hyper-relational KGs are introduced by
moving from representing uni-relational facts to representing facts with N-ary
relations. In hyper-relational KGs, triples are often associated with
relation-entity pairs, which are known as qualifiers. Qualifiers help qualify
a given fact by providing more supporting information and are defined as an
(entity, relation) pair that belongs to a triple. An example is shown in
Figure 1. In Figure 1 the triple (Stephen King, Author Of, The Running Man)
contains a single qualifier pair (Under Pseudonym, Richard Bachman). This
qualifier pair helps provide more context to the base triple by telling us
that Stephen King published the novel under the pseudonym Richard Bachman.
Research on hyper-relational KGs Guan et al. (2019); Galkin et al. (2020); Yu
and Yang (2021); Wang et al. (2021) have focused on learning representations
for such hyper-relational graphs and examining how the addition of qualifier
pairs can help boost the performance of knowledge graph completion.

However, the majority of existing approaches only consider the impact from the
qualifiers on base triples while overlooking the flow of information from the
base triples to the qualifiers. This can lead to suboptimal performance
especially when a large amount of qualifiers are presented. To illustrate its
importance, we use Figure 1 as an example. In (Richard Bachman, Occupation,
Novelist), suppose the occupation of Richard Bachman is missing and we are
trying to predict (Richard Bachman, Occupation, ?). If there is no information
spreading from the base triple to the qualifier entity Richard Bachman,
predicting the missing link would be very hard as no knowledge of Stephen King
is transferred to the qualifier entity. Prior work such as NaLP Guan et al.
(2019) and HINGE Rosso et al. (2020) both struggle to achieve this transfer of
information due to the simplicity of their convolutional-based frameworks.
StarE Galkin et al. (2020) ignores such flow of information from the base
triples to the qualifiers. Although transformer-based frameworks Yu and Yang
(2021); Wang et al. (2021) model the mutual influence between base triples and
qualifiers via self attention, they inevitably ignore the structured nature of
KGs.

Therefore, in this work, we aim to investigate the novel problem of learning
representations for hyper-relational KGs by encouraging the mutual influence
between base triples and qualifiers. Essentially, we are faced with the
following challenge: how to properly enhance the influence from base triples
to qualifiers while maintaining effective impact from qualifiers to base
triples. To address it, we propose a novel framework - QUalifier Aggregated
Hyper-RelationAl KnowleDge Graphs (QUAD), which encourages influence in both
directions. Specifically, QUAD utilizes two aggregators from different
perspectives - a base aggregator and a qualifier aggregator. The base
aggregator aggregates information for the base entities from the qualifiers
while the qualifier aggregator performs the aggregation from the qualifier
perspective. Inside the qualifier aggregator, we further propose the concept
of a “qualifier triple" that allows us to easily aggregate information from a
base triple through the qualifier relation. Following both aggregations, the
entity and relation representations are passed to the decoder to perform
knowledge base completion. We show that our framework can achieve strong
performance on multiple benchmark datasets. Our contributions can be
summarized as follows:

  * •

We propose a novel architecture for hyper-relational knowledge graphs by
introducing a graph encoder that aggregates information from the perspective
of the qualifier entities.

  * •

We further show that several representative hyper-relational KG methods can be
unified as the special cases of QUAD.

  * •

Extensive experiments demonstrate the effectiveness of our framework against
numerous baselines on multiple hyper-relational knowledge graphs.

Figure 2: An overview of QUAD. It takes the set of entity and relation
embeddings and encodes them using two aggregators. We then mask both the base
and qualifier separately and pass the the statement to the decoder to predict
the most likely entity $\hat{e}$ for the mask.

##  2 Related Work

###  2.1 Knowledge Graph Embedding

Knowledge graph embeddings (KGE) use embeddings to represent the latent
features of entities and relations in a KG. Many KGE techniques also employ a
score function that produce a score for a given triple $(h,r,t)$ measuring the
plausibility of the triple being true Ji et al. (2021). Many different
frameworks have been proposed in the literature. Bordes et al. (2013) proposed
using a translational scoring function to model triples. Yang et al. (2015)
use a bilinear scoring function to score triples utilizing a diagonal matrix
to model relations. RotatE Sun et al. (2018) scores a triple as a translation
in a complex space and ConvE Dettmers et al. (2018) does so using a
convolutional neural network. Other works have focused on modeling knowledge
graphs using graph neural networks. RGCN Schlichtkrull et al. (2018) extends
GCN Kipf and Welling (2017) by using a relation specific weight matrix when
aggregating an entity’s neighbors. However, RGCN suffers from over-
parameterization when there are many relations. To alleviate this concern,
CompGCN Vashishth et al. (2019) proposes to use direction specific weight
matrices when aggregating instead of relational weight matrices.

###  2.2 Hyper-Relational KGs

Several methods have been proposed that model hyper-relational facts as N-ary
facts. Wen et al. (2016) propose m-Transh, a method that builds on TransH Wang
et al. (2014) by transforming each hyper-relational fact using a star-to-
clique conversion. RAE Zhang et al. (2018) builds upon m-TransH Wen et al.
(2016) by further considering the relatedness between two entities. NaLP Guan
et al. (2019) uses a convolutional-based framework to compute a relatedness
vector for a triple and its qualifier pairs that can be used for prediction.

More recently, there has been some work that model hyper-relational KGs from a
purely hyper-relational viewpoint. Rosso et al. (2020) propose doing so using
a convolutional framework. For a given hyper-relational fact, the base triple
is convolved by itself and with each specific qualifier pair. The resulting
feature vectors are then combined and used for prediction. StarE Galkin et al.
(2020) extends CompGCN Vashishth et al. (2019) by encoding the qualifiers for
a specific triple and combining it with the base relation of the triple. Using
StarE Galkin et al. (2020) as their foundation, Yu and Yang (2021) replace the
GNN aggregation with layer normalization layers Ba et al. (2016) to improve
performance. Additionally they mask the qualifier entities when training as a
form of self-supervised learning (SSL). Lastly, Wang et al. (2021) propose
GRAN, a transformer based architecture that employs edge-specific attention
biases and masks all entities and relations in the sequence. A missing element
in previous work is the lack of a clear flow of information from the base
triples to the qualifiers. To address this, we propose a graph encoder to
aggregate information from the perspective of the qualifiers, thereby creating
better qualifier encodings.

##  3 Preliminaries

###  3.1 Knowledge Graphs

Let $\mathcal{G}=\\{\mathcal{V},\mathcal{R},\mathcal{E}\\}$ be a knowledge
graph with nodes (i.e. entities) $\mathcal{V}$, edges $\mathcal{E}$, and
relations $\mathcal{R}$. For $e\in\mathcal{E}$, it represents a directed edge
where two entities $u\in\mathcal{V}$ and $v\in\mathcal{V}$ are connected by a
relation $r\in\mathcal{R}$. We also denote the edge as a triple $(v,r,u)$.
Further, we use $\mathcal{N}_{v}$ to denote the neighboring entities and
relations of a node $v$.

CompGCN Vashishth et al. (2019), a popular method for modeling knowledge
graphs, utilizes a direction specific weight matrix $W_{\lambda(r)}$ and a
function $\phi$ that combines the neighboring entity $\mathbf{h}_{u}$ and
relation $\mathbf{h}_{r}$ of a given edge

$\mathbf{h}_{v}^{(k+1)}=f\left(\sum_{(u,r)\in\mathcal{N}_{v}}W_{\lambda(r)}^{k}\>\phi(\mathbf{h}_{u}^{k},\mathbf{h}_{r}^{k})\right),$
(1)

where $f$ is a non-linear function (e.g. ReLU) and $\lambda(r)$ represents the
direction of relation that can be one of: a standard, inverse, or self-loop
relation. Several function are proposed for modeling the interaction of the
embedding in $\phi$ including subtraction, multiplication, and the cross
correlation Vashishth et al. (2019). The relation embedding is updated through
a transformation by a weight matrix $\mathbf{W}_{rel}$,

$\mathbf{h}_{r}^{k+1}=\mathbf{W}_{rel}\>\mathbf{h}_{r}^{k}.$ (2)

###  3.2 Hyper-Relational Knowledge Graphs

A hyper-relational knowledge graph can be seen as an extension of a standard
knowledge graph where there is a set of qualifier pairs
$\\{(qv_{i},qr_{i})\\}$, where $qv\in\mathcal{V}$ and $qr\in\mathcal{R}$,
associated with each triple $(v,r,u)$. For simplicity we use $q$ to represent
the set of neighboring qualifier pairs $(qv_{i},qr_{i})\in Q_{(v,r,u)}$
associated with a triple $(v,r,u)$. We can therefore represent a hyper-
relational fact as $(v,r,u,q)$. We refer to hyper-relational facts as
statements. Furthermore, we can represent the neighborhood for a qualifier
entity $qv$ as $(v,r,u,qr)\in\mathcal{N}_{qv}$.

A representative example for modeling hyper-relational KGs is StarE Galkin et
al. (2020). It proposes a formulation based on CompGCN to incorporates an
embedding $\mathbf{h}_{q}$ that is an encoded representation of the qualifier
pairs for the base triple $(v,r,u)$,

$\displaystyle\mathbf{h}_{v}=f\left(\sum_{(u,r)\in\mathcal{N}_{v}}W_{\lambda(r)}\>\phi\left(\mathbf{h}_{u},\gamma(\mathbf{h}_{r},\mathbf{h}_{q})\right)\right).$
(3)

The embedding $\mathbf{h}_{q}$ is combined with the relation embedding
$\mathbf{h}_{r}$ through a function $\gamma$ that performs a weighted sum. Yu
and Yang (2021) show that replacing the graph aggregation with layer
normalization can achieve comparable if not better performance than other
frameworks. Furthermore they show that masking the qualifier entities during
training can raise the test performance on knowledge graph completion.

###  3.3 Knowledge Graph Completion

Knowledge graph completion (i.e. link prediction) masks one of the two
entities belonging to the triple and attempts to predict the correct entity.
For example, given $(v,r,u)$ we would try to predict the correct entity for
both $(v,r,?)$ and $(?,r,u)$. This is defined similarly for hyper-relational
KGs where we are also provided with the qualifier information for the triple.
Therefore, given $(v,r,u,q)$ we would try to predict the correct entity for
both $(v,r,?,q)$ and $(?,r,u,q)$.

##  4 The Proposed Framework

In this section, we propose a new framework QUAD for learning representations
of hyper-relational knowledge graphs. An overview of QUAD is shown in Figure 2
that consists of two main components. The first component, detailed in Section
4.1, encodes the graph using two separate aggregations while the second
component detailed in Section 4.2 decodes a given hyper-relational fact for a
particular downstream task. In detail, we first pass the initial entity and
relation embeddings to the encoder, which is comprised of two graph encoders
that aggregate information from the base entities and qualifier entities,
respectively. Once encoded, we create a separate sample for each statement
masking each of the base and qualifier entities. Each sample then gets passed
to the decoder to predict the most likely entity $\hat{e}$ for the mask.

###  4.1 The Encoder

In this subsection, we define the encoder used in QUAD. The encoder is
composed of two neighborhood aggregations: one that aggregates information for
the base entities and one that does so for the qualifier entities. We refer to
these two aggregators as the base aggregator and qualifier aggregator,
respectively. The initial entity $E$ and relation $R$ embeddings are first
passed to the base aggregator and then the qualifier aggregator. The encoded
entity embedding $\hat{E}$ and relation embedding $\hat{R}$ can be expressed
as follows:

$\displaystyle E^{\prime},R^{\prime}$ $\displaystyle=\text{Base-
Agg}(E,R,\mathcal{G}),$ (4) $\displaystyle\hat{E},\hat{R}$
$\displaystyle=\text{Qual-Agg}(E^{\prime},R^{\prime},\mathcal{G}),$ (5)

where $\text{Base-Agg}(\cdot)$ and $\text{Qual-Agg}(\cdot)$ are the
aggregation functions in base aggregator and qualifier aggregator,
respectively. In the following, we introduce the details of both aggregators.

####  4.1.1 The Base Aggregator

The base aggregator aims to aggregate information for the base entities.
Concretely, it takes $E$ and $R$ as input and aggregates the neighborhood
information for a given base entity $v$ using a function $\psi_{v}$:

$\mathbf{h}_{v}=\text{Aggregate}(\psi_{v}(\mathbf{h}_{u},\mathbf{h}_{r},\mathbf{h}_{q}),\forall(u,r,q)\in\mathcal{N}_{v}),$
(6)

where $\mathbf{h}_{q}$ is the encoded representation for all the qualifier
pairs belonging to the base triple $(v,r,u)$. We utilize CompGCN Vashishth et
al. (2019) as the Aggregate($\cdot$) function. We can then rewrite Eq. (6) as
follows,

$\mathbf{h}_{v}=f\left(\sum_{(u,r)\in\mathcal{N}(v)}\mathbf{W}_{\lambda(r)}\psi_{v}\left(\mathbf{h}_{u},\mathbf{h}_{r},\mathbf{h}_{q}\right)\right),$
(7)

where the encoded qualifier representation $\mathbf{h}_{q}$ is defined as in
StarE Galkin et al. (2020) as:

$\mathbf{h}_{q}=\mathbf{W}_{q}\sum_{\left(qr,qv\right)\in
Q_{(v,r,u)}}\phi\left(\mathbf{h}_{qr},\mathbf{h}_{qv}\right),$ (8)

with $\mathbf{W}_{q}$ as the projection matrix, $\mathbf{h}_{qr}$ as the
qualifier relation embedding, and $\mathbf{h}_{qv}$ as the qualifier entity
embedding. Note that the relation $\mathbf{h}_{r}$ is updated through a linear
transformation as shown in Eq. (2). StarE implements $\psi_{v}$ in Eq. (7) by
combining the encoded qualifier information $\mathbf{h}_{q}$ with the triple’s
relation, i.e.,
$\psi_{v}(\mathbf{h}_{u},\mathbf{h}_{r},\mathbf{h}_{q})=\phi\left(\mathbf{h}_{u},\gamma(\mathbf{h}_{r},\mathbf{h}_{q})\right)$
as in Eq. (3). However, it is restricted by the assumption that qualifier
information should be incorporated into the base relation embedding. Instead,
we remove this restriction and combine the qualifier information with the
output of the composition function $\phi$ as follows:

$\psi_{v}(\mathbf{h}_{u},\mathbf{h}_{r},\mathbf{h}_{q})=\alpha\odot\phi(\mathbf{h}_{u},\mathbf{h}_{r})+(1-\alpha)\odot\mathbf{h}_{q},$
(9)

where $\alpha\in[0,1]$ is a hyperparameter that balances the contribution of
the base triple information and the encoded qualifiers. In this way, the
qualifier information encoded in $\mathbf{h}_{q}$ can directly interact with
both the base entity and relation.

####  4.1.2 The Qualifier Aggregator

The previously introduced base aggregator only considers the aggregation of
information from the qualifiers to the base triple but not vice versa.
Encoding base triple information in the qualifiers is advantageous as it can
help learn better representations for the qualifiers. For example, in Figure 1
the base aggregation doesn’t consider the flow of information from the triple
(Stephen King, Author Of, The Running Man) to the the qualifier entity Richard
Bachman resulting in limited information about the author being encoded in the
qualifier entity. This makes it difficult to infer new facts where Richard
Bachman is a base entity. To encode more information for a qualifier entity,
we can aggregate information from its neighbors, i.e., the base triples it
belongs to and the qualifier relation connecting them to those triples. Using
our previous example, the qualifier entity Richard Bachman would aggregate
information from the base triple (Stephen King, Author Of, The Running Man)
and the qualifier relation Under Pseudonym.

Hence, the Qual-Agg($\cdot$) function is designed to aggregate the
neighborhood information for the qualifier entity $qv$ as follows:

$\displaystyle\mathbf{h}_{qv}=\>\text{Aggregate}($
$\displaystyle\psi_{q}(\mathbf{h}_{v},\mathbf{h}_{r},\mathbf{h}_{u},\mathbf{h}_{qr}),$
$\displaystyle\forall(v,r,u,qr)\in\mathcal{N}_{qv}),$ (10)

where $\mathbf{h}_{qr}$ is the qualifier relation embedding and the function
$\psi_{q}$ is used to combine the neighboring embeddings. For $\psi_{q}$, we
hope that the base triple embeddings
$(\mathbf{h}_{v},\mathbf{h}_{r},\mathbf{h}_{u})$ should be considered as a
whole. Since the qualifiers serve as additional context for explaining the
whole triple, we should aggregate information from the whole triple instead of
treating $\mathbf{h}_{v},\mathbf{h}_{r},\mathbf{h}_{u},\mathbf{h}_{qr}$
individually. To achieve this goal, we first consider that a qualifier entity
$qv$ is linked to some base triple $t=(v,r,u)$ by the qualifier relation $qr$.
This can be seen as analogous to a standard triple where the base triple is
the head, the qualifier relation is the relation, and the qualifier entity is
the tail entity. For convenience, we can write this in a triple notation as
$\left(t,qr,qv\right)$, which we refer to as a qualifier triple. An example
found in Figure 1 would be $t=$ _(Stephen King, Author Of, The Running Man)_ ,
$qr=\emph{Under Pseudonym}$ and $qv=\emph{Richard Bachman}$. Using these
ideas, we can view $\psi_{q}$ as a function of the base triple $t$ and the
qualifier relation $qr$:

$\psi_{q}(\mathbf{h}_{v},\mathbf{h}_{r},\mathbf{h}_{u},\mathbf{h}_{qr})=\phi(\mathbf{h}_{t},\mathbf{h}_{qr}),$
(11)

where the embedding $\mathbf{h}_{t}$ is the encoded representation of the base
triple $t$ and $\phi$ is defined similarly to the composition function used in
CompGCN Vashishth et al. (2019). The base triple representation
$\mathbf{h}_{t}$ is formulated as the linear projection of the concatenated
embeddings of the base triple:

$\mathbf{h}_{t}=\text{Linear}\left(\text{Concat}\left(\mathbf{h}_{v},\mathbf{h}_{r},\mathbf{h}_{u}\right)\right).$
(12)

###  4.2 The Decoder

Using the encoded representations of the entities and relations, i.e.,
$\hat{E}$ and $\hat{R}$, each hyper-relational fact is passed to a decoder to
make the final prediction. To decode each fact we utilize a transformer
Vaswani et al. (2017) that employs an architecture similar to CoKE Wang et al.
(2019a) extended to include qualifier information. For a given input sample
$S$, we mask the entity token we are trying to predict. Eq. (13) is an example
where we mask the object entity:

$S=(\mathbf{h}_{v},\>\mathbf{h}_{r},\>[Mask],\>\mathbf{h}_{qr1},\>\mathbf{h}_{qv1},\>\cdots).$
(13)

After being passed through the transformer, we extract the masked embedding
and pass it through a fully-connected layer and score it against all possible
entities. This is then passed through a sigmoid function with the highest
scoring entity being chosen as our prediction.

####  4.2.1 Model Training

As the downstream task is to perform knowledge graph completion for entities
in the base triples, we mask the head and tail entities for the fact
$(\mathbf{h}_{v},\mathbf{h}_{r},\mathbf{h}_{u},q)$ where $q$ is the set of
qualifier pairs associated with the triple. Then we try to predict the masked
entities and minimize the loss $\mathcal{L}_{\text{base}}$ using binary cross-
entropy loss.

To further enhance the learned representations of the embeddings and exploit
the qualifier information, we include an auxiliary task that masks and
attempts to predict the qualifier entities for each fact as proposed in Hy-
Transformer Yu and Yang (2021). We refer to this loss as
$\mathcal{L}_{\text{qual}}$ and minimize it using the binary cross-entropy
loss. Since the downstream task is solely to predict the missing base
entities, we introduce a hyperparameter $\beta\in[0,1]$ that balances the
contribution of the qualifier entity loss. The loss can now be written as
thus:

$\mathcal{L}=\mathcal{L}_{\text{base}}+\beta\mathcal{L}_{\text{qual}}.$ (14)

###  4.3 Parallel Architecture

We also consider a version of QUAD that combines the entity representations
encoded by the two aggregation schemes in parallel instead of sequentially.
Under this setting, both aggregate functions take the initial entity embedding
matrix $\mathcal{E}$ as input. The encoded entity representations outputted by
the two encoders are then combined via a weight matrix $\mathbf{W}_{p}$ and
passed to the decoder. The relation embeddings are still passed sequentially
as we found that this performed best. We believe that this version of our
framework may be better at balancing the contribution of the base and
qualifier aggregation for some datasets. We formulate the parallel encoding
scheme as follows where $E_{base}$ and $R_{base}$ are the output of
$\text{Base-Agg}(\cdot)$ and $E_{qual}$ the output of $\text{Qual-
Agg}(\cdot)$:

$\displaystyle E_{base},R_{base}=\text{Base-Agg}(E,R,\mathcal{G}),$ (15)
$\displaystyle E_{qual},\hat{R}=\text{Qual-Agg}(E,R_{base},\>\mathcal{G}),$
(16)
$\displaystyle\hat{E}=\mathbf{W}_{p}\;\text{Concat}\left(E_{base},\>E_{qual}\right).$
(17)

###  4.4 Relationship to Other Frameworks

Table 1: Knowledge Graph Completion Results for Multiple Datasets

Method WD50K (13.6) Wikipeople (2.6) JF17K (45.9) MRR H@1 H@10 MRR H@1 H@10
MRR H@1 H@10 NaLP-Fix 0.177 0.131 0.264 0.420 0.343 0.556 0.245 0.185 0.358
HINGE 0.243 0.176 0.377 0.476 0.415 0.585 0.449 0.361 0.624 StarE 0.349 0.271
0.496 0.491 0.398 0.648 0.574 0.496 0.725 Hy-Transformer 0.356 0.281 0.498
0.501 0.426 0.634 0.582 0.501 0.742 QUAD 0.348 0.270 0.497 0.466 0.365 0.624
0.582 0.502 0.740 QUAD (Parallel) 0.349 0.275 0.489 0.497 0.431 0.617 0.596
0.519 0.751

Table 2: Knowledge Graph Completion Results on WD50K Splits

Method WD50K (33) WD50K (66) WD50K (100) MRR H@1 H@10 MRR H@1 H@10 MRR H@1
H@10 NaLP-Fix 0.204 0.164 0.277 0.334 0.284 0.423 0.458 0.398 0.563 HINGE
0.253 0.190 0.372 0.378 0.307 0.512 0.492 0.417 0.636 StarE 0.331 0.268 0.451
0.481 0.420 0.594 0.654 0.588 0.777 Hy-Transformer 0.343 - - 0.515 - - 0.699
0.637 0.812 QUAD 0.349 0.286 0.470 0.515 0.456 0.623 0.703 0.638 0.820 QUAD
(Parallel) 0.346 0.287 0.459 0.510 0.454 0.615 0.693 0.628 0.812

In this subsection we provide a unified view on several popular hyper-
relational KG methods by showing that they are subsets of our framework.

####  4.4.1 StarE

StarE Galkin et al. (2020) uses a GNN to encode the KG and a transformer to
perform knowledge graph completion. The GNN encoder is detailed in Eq. (3). We
can show that StarE is a special case of QUAD where:

  * •

Information is only aggregated from the perspective of the base entities.
$\text{Qual-Agg}(\cdot)$ is therefore equal to the identity function.

  * •

The function $\psi_{v}$ in the base aggregator is defined as
$\phi(\mathbf{h}_{u},\alpha\odot\mathbf{h}_{r}+(1-\alpha)\odot\mathbf{h}_{q})$.

  * •

StarE only masks the base entities so $\beta=0$.

####  4.4.2 Hy-Transformer

Hy-Transformer Yu and Yang (2021) modifies StarE by replacing the GNN encoder
with layer normalization layers. They also introduce an auxiliary task that
masks the qualifier entities during training. We demonstrate that Hy-
Transformer is a special case of QUAD where:

  * •

Both aggregate functions are equal to layer normalization layers for the
entity and relation embeddings such that: $\hat{\mathcal{E}}=\text{Layer-
Norm}(\mathcal{E})$ and $\hat{\mathcal{R}}=\text{Layer-Norm}(\mathcal{R})$.

  * •

The loss balancing hyperparameter $\beta=1$.

####  4.4.3 GRAN

GRAN Wang et al. (2021) uses a transformer with edge-specific attention
biases. We demonstrate that GRAN is a special case of QUAD where:

  * •

Both aggregate functions are equal to the identity function.

  * •

The loss balancing hyperparameter $\beta=1$.

  * •

The transformer (decoder) considers edge-specific biases. The transformer used
in QUAD is equivalent to the GRAN-complete model where the biases for the key
and value matrices are set to zero $(e_{ij}^{K},e_{ij}^{V})=(0,0)$.

##  5 Experiment

In this section, we conduct experiments to demonstrate the effectiveness of
our proposed framework QUAD. We first introduce the experimental settings and
then compare the results of QUAD against the baselines on numerous benchmark
datasets. Next we perform an ablation study to determine the importance of
each component in our framework. Lastly, we perform some additional
experiments to assess the impact of the loss balancing term on the
performance.

###  5.1 Experimental Settings

####  5.1.1 Datasets

We consider three datasets for our experiments including JF17K Wen et al.
(2016), Wikipeople Guan et al. (2019), and WD50K Galkin et al. (2020). An
issue with WD50K and Wikipeople is that only a small percentage of triples
contain qualifiers, being 13.6% for WD50K and 2.6% for Wikipeople. We
therefore also measure the performance on the WD50K splits introduced by
Galkin et al. (2020) that contain a higher percentage of triples with
qualifiers. The three splits are WD50K (33), WD50K (66), and WD50K (100) with
the number in parentheses representing the percentage of triples with
qualifiers.

####  5.1.2 Baselines

We compare the results of our framework with other prominent hyper-relational
baselines including NaLP-Fix Rosso et al. (2020), HINGE Rosso et al. (2020),
StarE Galkin et al. (2020), and Hy-Transformer Yu and Yang (2021). Note that
we do not include GRAN Wang et al. (2021) as one baseline since (1) similar to
Hy-Transformer, it is also a transformer-based method; and (2) in addition to
the auxiliary task, GRAN also masks the relations and we can incorporate such
component to the proposed framework that we leave as one future work.

####  5.1.3 Evaluation Metrics

To evaluate the performance on the test set we report the mean reciprocal rank
(MRR) and the percentage of top 1 and 10 hits (H@1 and H@10) when performing
knowledge graph completion. We utilize the filtered setting introduced by
Bordes et al. (2013).

###  5.2 Performance Comparison on Benchmarks

####  5.2.1 Performance on WD50K, Wikipeople and JF17K

In this subsection, we evaluate QUAD on the benchmark datasets and compare its
performance to the aforementioned baselines. We first evaluate performance on
the WD50K, Wikipeople, and JF17K datasets. The method QUAD is the original
formulation of our framework while the method QUAD (Parallel) is the
alternative formulation presented in Section 4.3. The results are shown in
Table 1. For each dataset we include the percentage of triples with qualifiers
in parentheses.

Evaluating the results in Table 1 we observe that the performance of QUAD
varies by dataset. For JF17K it achieves the best performance for all three
metrics including a 2.4% increase in MRR over the second best performing
model. On Wikipeople its performance is similar Hy-Trasnformer while for WD50K
it is slightly below state of the art. This is due to both WD50K and
Wikipeople containing a low percentage of triples with qualifier pairs with
13.6% and 2.6%, respectively. On JF17K, which has a much higher percentage of
qualifiers at 45.9%, QUAD is able to outperform the baseline models. We
therefore believe that datasets with a higher ratio of qualifiers is where
QUAD shows its value.

####  5.2.2 Varying the ratio of qualifiers

To test this hypothesis, we evaluate QUAD on the WD50K subsets introduced in
Section 5.1.1. The percentage of triples with qualifier pairs for the three
subsets is approximately 33%, 66%, and 100%, respectively. The results are
presented in Table 2. From the results, we see state-of-the-art performance
across all the three datasets for each of the three evaluation metrics.
Interestingly, as opposed to the results shown in Table 1 the original (non-
parallel) version of QUAD performs the best. We believe that the strong
performance on the datasets with a more substantive percentage of qualifiers
shows the utility of our framework and that the relatively poorer performance
on the Wikipeople and WD50K datasets result from them containing too few
triples with qualifiers.

###  5.3 Ablation Study

In this subsection, we conduct an ablation study to determine the importance
for each component in QUAD. We do so by considering three versions of QUAD:
(1) One that doesn’t mask the qualifier entities, (2) one without the
qualifier aggregation component and (3) one without (1) and (2). Evaluating
the results on the three ablated frameworks will help us ascertain both the
individual and cumulative effect those two components have on the performance.
Of most importance is the impact of the novel qualifier aggregation introduced
in this paper. We report the results of this study on the WD50K (100) dataset
under the original (non-parallel) setting. The results can be found in Table
3.

Table 3: Ablation Study on WD50K (100) Method | MRR | H@1 | H@10
---|---|---|---
w/o Qual Agg & Mask | 0.658 | 0.591 | 0.781
w/o Qual Mask | 0.677 | 0.613 | 0.794
w/o Qual Agg | 0.696 | 0.628 | 0.820
QUAD | 0.703 | 0.638 | 0.820

Evaluating the results in Table 3 we see that the removal of either or both of
the two components leads to a degradation in performance. Ablating the
qualifier aggregation, the main contribution of our framework, causes a 1%
reduction in MRR and a 1.6% drop in H@1. Furthermore, removing just the
qualifier masking results in a 3.7% reduction in MRR and ablating both
components leads to the largest drop with a 6.4% drop in MRR. These results
validate the importance of the qualifier aggregation as its removal leads to a
degradation in performance.

###  5.4 Effect of Loss Balancing Hyperparameter

In this subsection, we study the effect that the loss balancing hyperparameter
$\beta$ has on the performance. To do so we train and evaluate multiple
versions of our framework with values of $\beta$ in the set
$\\{0,0.25,0.5,0.75,1\\}$. We note that a value of $\beta=0$ is equivalent to
no qualifier masking while a value of $\beta=1$ is equivalent to the loss used
in Hy-Transformer Yu and Yang (2021). We perform this study on the three WD50K
splits and JF17K to determine the impact the loss balancing hyperparameter has
on a variety of datasets. We utilize the parallel setting when training on
JF17K and our original formulation on the WD50K splits. For simplicity we
report only the MRR performance in Table 4.

Table 4: Loss Hyperparameter Study (MRR) $\mathbf{\beta}$ |  WD50K (33) |  WD50K (66) |  WD50K (100) |  JF17K
---|---|---|---|---
0 |  0.33 |  0.491 |  0.677 |  0.592
0.25 |  0.346 |  0.507 |  0.690 |  0.596
0.50 |  0.348 |  0.514 |  0.697 |  0.592
0.75 |  0.348 |  0.514 |  0.701 |  0.589
1 |  0.349 |  0.515 |  0.703 |  0.583

The results in Table 4 shows multiple trends. For WD50K (100) there is a clear
relationship with the value of $\beta$ and the performance. The higher the
value of $\beta$ the better the performance. This is likely due to the fact
that every triple in the dataset has at least one qualifier pair. For both
WD50K (33) and (66) the MRR improves as we increase $\beta$ from 0 to 0.50 but
we see little to no improvement when increasing the value from 0.50 to 1.
Lastly, for JF17K the MRR is maximized when $\beta=0.25$. Furthermore, for
increasing values of $\beta$ beyond 0.25 we see a noticeable drop in
performance resulting in a MRR 2.2% lower at $\beta=1$ compared to
$\beta=0.25$. This shows the importance of balancing the magnitude of the
auxiliary loss, as depending on the dataset balancing them equally ($\beta=1$)
can potentially lead to a large degradation in performance.

##  6 Conclusion

In this paper we introduce our framework QUAD for learning representations for
hyper-relational knowledge graphs. Our framework is motivated by better
encoding qualifier information for a given hyper-relational fact. To this
point we design a novel qualifier aggregation module to learn better encoded
representations for the qualifiers. We also introduce a hyperparameter to
balance the contribution of the auxiliary task in the loss. Experiments show
that our framework performs well on several benchmark datasets as compared to
competitive baselines. Further experiments validate the importance of the
various components in our framework and the need to balance the auxiliary
loss. For future work we plan on exploring how additional auxiliary
information such as text description or literals help learn better
representations of hyper-relational KGs.

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##  Appendix A Infrastructure

All experiments were done on one 32G Tesla V100 GPU and implemented using
Pytorch Geometric Fey and Lenssen (2019).

##  Appendix B Datasets

We conducted experiments on three datasets. These are JF17K Wen et al. (2016),
Wikipeople Guan et al. (2019), and WD50K Galkin et al. (2020). Table 5 details
the statistics for each.

Table 5: Datasets. We note the approximate percentage of triples with qualifiers in parentheses. Dataset | Statements | #Entities | #Relations
---|---|---|---
Wikipeople (2.6) | 369,866 | 34,839 | 375
JF17K (45.9) | 100,947 | 28,645 | 501
WD50K (33) | 102,107 | 38,124 | 475
WD50K (66) | 49,167 | 27,347 | 494
WD50K (100) | 31,314 | 18,792 | 279
WD50K (13.6) | 236,507 | 47,156 | 532

##  Appendix C Hyperparameters

QUAD is trained for 500 epochs with a learning rate of 1e-4. Furthermore the
embedding dimension is 200 and the number of layers for the Base-Agg encoder
is 2. For the transformer we set the number of layers to 2, the number of
heads to 4, dropout to 0.1, and the hidden dimension is tuned from {512, 768}.
The batch size is tuned from {128, 256}, $\alpha$ is tuned from [0, 1] in
steps of 0.1, beta is tuned from {0.25, 0.5, 0.75, 1}, the number of layers
for the Qual-Agg from {1, 2}, the encoding dropout from {0.1, 0.2, 0.3}, the
learning rate decay from {None, 0.996, 0.9975, 0.999}, and the label smoothing
from {0.1, 0.2, 0.4, 0.6, 0.8}. The Adam optimizer is used in all the
experiments Kingma and Ba (2014). For the composition function $\phi$ we
utilize the RotatE scoring function Sun et al. (2018). Under the parallel
setting we tune an additional dropout layer after we combine the
representation of the two encoders from {0.2, 0.3}.

Table 6 holds the hyperparameters for the general datasets and Table 7 for the
three WD50K splits. The values are the same for the parallel and non-parallel
versions of QUAD with the exception of the inclusion of the parallel dropout
hyperparameter.

Table 6: Hyperparameter values for the general datasets. Hyperparameter | JF17K | Wikipeople | WD50K
---|---|---|---
Batch Size | 128 | 128 | 128
Alpha | 0.8 | 0.8 | 0.7
Beta | 0.25 | 0 | 0.5
Qual Agg Layers | 2 | 2 | 1
Encoder Dropout | 0.1 | 0.1 | 0.2
Transformer Dim | 768 | 512 | 768
LR Decay | 0.999 | 0.999 | 0.9975
Label Smoothing | 0.6 | 0.2 | 0.2
Parallel Dropout | 0.2 | 0.2 | 0.2
Table 7: Hyperparameter values for the WD50K splits. Hyperparameter | (33) | (66) | (100)
---|---|---|---
Batch Size | 128 | 128 | 256
Alpha | 0.7 | 0.7 | 0.6
Beta | 1 | 1 | 1
Qual Agg Layers | 1 | 1 | 1
Encoder Dropout | 0.2 | 0.2 | 0.2
Transformer Dim | 768 | 768 | 768
LR Decay | 0.9975 | 0.9975 | None
Label Smoothing | 0.2 | 0.2 | 0.1
Parallel Dropout | 0.3 | 0.3 | 0.3