# Size Matters: Lorentz Boosted Casimir Effect

Yu-Song Cao111caoyusong15@mails.ucas.ac.cn$~{}^{1,2}$, YanXia Liu222yxliu-
<EMAIL_ADDRESS>Ding-Fang<EMAIL_ADDRESS>
###### Abstract

Many evidences appear in the past decades and show that the negativity of
Casimir energy is responsible for exotic mechanical and gravitational effects.
We study in this work the Lorentz boost of a Casimir cavity, on which little
attention is paid to its momentum in historical works. We find that the vacuum
energy and momentum carried by the cavity transform differently from those of
point particles due to the cavity’s extension feature. However, the mass-shell
condition of the two are identical as long as the cavity is finite along the
moving direction only.

$~{}^{1}$School of Physics and Astronomy, Yunnan University, Kunming 650091,
PR China

$~{}^{2}$School of Physics and Optoelectronic Engineering, Beijing University
of Technology, Beijing, China

##  1 Introduction

Casimir effect is a manifestation of quantum fluctuations so that even two
neutral conductors get attracted to each other when placed in vacuum [1, 2].
This attraction arises from the vacuum energy’s change, which is negative, and
the positivity of such change’s derivative with respect to the two conductors’
separation

$\frac{E}{A}=-\frac{\pi^{2}\hbar c}{720{a}^{3}},~{}\frac{dE}{da}>0.$ (1)

Since in special relativity, negative energy means negative mass, the
negativity of Casimir energy is presumed responsible for many exotics such as
mechanical instability [3, 4, 5, 6, 7], repelling gravitation forces [8, 9,
10, 11, 12, 13, 14, 15, 16, 17] and supporting of traversable wormholes [18,
19, 20, 21, 22, 23, 24].

In references[3, 4, 5], an elegant method was introduced to calculate the
Casimir energy of a rotating Casimir ring in the non-relativistic limit. By
this method, the Klein-Gordon equation was first written down in the lab frame
with time dependent boundary condition. After that, a Galileo transformation
brings the system into the co-moving frame of the cavity so that the mode
function and vacuum energy can be obtained with no difficulty. Finally an
inverse Galileo transformation brings the mode function back into the lab
frame, with which the Casimir energy can be calculated. This way, the author
observes the expected mechanical instability of the Casimir effect. However,
this operation cannot be done the other way because the Klein-Gordon equation
is not invariant under the non-inertial transformation between the static and
rotating frames.

In this work we will apply this method to a uniformly moving Casimir cavity.
This question was previously studied in [25, 26, 27] and recently in [28],
where interests were focused on local quantities such as the energy-momentum
tensor and field correlations. In [29, 30], the inertia of the Casimir cavity
is defined as the motional force against acceleration, which is accompanied by
particle creation. This way, the inertia of the Casimir cavity can not be
identified as the static Casimir energy aprior. As we will see, our results
indicates the similar. Our objective in this work is the transformation
property of the Casimir energy and momentum of the Casimir cavity, both are
non-local quantities. Because both the lab and co-moving frames are inertial,
we expect the the method of [3, 4, 5] can be done in either ways. That is, we
can take the Klein-Gordon equation in co-moving frame as prior and modify the
field equation in lab frame in non-relativistic limit and vice versa. However,
we find that the two ways produce inequivalent results! To understand this
inconsistency, we analyse the system exactly in the framework of special
relativity without Galileo approximation. We find that it is the non-
relativistic approximation, not the method itself that causes the
inconsistency. We notice that the Lorentz contraction of the cavity plays a
crucial role in obtaining the correct result. The Casimir energy and momentum
of the cavity transform differently from that of point particles but their
combination still satisfies the mass-shell condition as long as the cavity is
finitely extended along the moving direction only.

Our paper is organised as follows: In Sec.2, we calculate the Casimir energy
and momentum of one dimensional cavities in the non-relativistic limit
following the routine of [3, 4, 5] and the other way around. We show the
results obtained from the two directions contradict with each other and cannot
be resolved without tampering the whole calculation process. In Sec.3, we
present the self-consistent calculation under full relativistic treatment,
which is the main content of the work. Some discussions will be given there
accordingly. In Sec.4, we show that our calculation can be done on cavities
with more general geometries in higher dimension and present a working
example. In Sec.5, we draw our conclusion and give some discussions.

Throughout this work the mass of the cavity walls are neglected and the units
will be chosen so that $\hbar=c=1$.

##  2 Non-relativistic calculation

###  2.1 Standard treatment

Following the working line of [3, 4, 5], in $(1+1)$ dimension, the field
equation of a Casimir cavity uniformly moving in the lab frame $S$ is given by

$\begin{split}&(\frac{\partial^{2}}{\partial
t^{2}}-\frac{\partial^{2}}{\partial x^{2}})\phi(t,x)=0,\\\
&\phi(t,vt)=\phi(t,vt+L),\end{split}$ (2)

where $v\ll 1$ is the velocity at which the cavity is moving and $L$ is the
cavity length. By applying Galileo coordinate transformation from the lab
frame $S$ to co-moving frame $S^{\prime}$

$\frac{\partial}{\partial t}=\frac{\partial}{\partial
t^{\prime}}-v\frac{\partial}{\partial x^{\prime}},\frac{\partial}{\partial
x}=\frac{\partial}{\partial x^{\prime}}$ (3)

the field equation in $S^{\prime}$-frame becomes

$\begin{split}&[(\frac{\partial}{\partial
t^{\prime}}-v\frac{\partial}{\partial
x^{\prime}})^{2}-\frac{\partial^{2}}{\partial x^{\prime
2}}]\phi(t^{\prime},x^{\prime})=0,\\\
&\phi(t^{\prime},0)=\phi(t^{\prime},L)=0.\end{split}$ (4)

Solutions of Eq.(4) can be mode expanded as

$\phi(t^{\prime},x^{\prime})=\sum_{n}\frac{1}{\sqrt{2\omega^{\prime}_{n}}}(a_{n}e^{-i\omega^{\prime}_{n}t^{\prime}}f_{n}(x^{\prime})+h.c.),$
(5)

where $f_{n}(x^{\prime})$ are the spatial mode function. Substituting this
mode expansion into Eq.(4), we will have

$(1-v^{2})f^{\prime\prime}_{n}-2iv\omega^{\prime}_{n}f^{\prime}_{n}+(\omega^{\prime}_{n})^{2}f_{n}=0,$
(6)

as results

$f_{n}(x^{\prime})=e^{\frac{iv\omega^{\prime}_{n}}{1-v^{2}}x^{\prime}}\sin\frac{n\pi
x^{\prime}}{L},$ (7)

The orthogonality of this mode function set can be verified

$\int_{0}^{L}dxf_{n}(x)f_{m}^{*}(x)=\delta_{n,m}.$ (8)

Eq.(6) gives rise to the corresponding spectra

$\omega^{\prime}_{n}=(1-v^{2})\frac{n\pi}{L}.$ (9)

So the vacuum energy in $S^{\prime}$-frame can be written as

$E_{m}=(1-v^{2})m_{0},$ (10)

where $m_{0}$ is the Casimir energy of a static cavity

$m_{0}=\frac{1}{2}\sum_{n}\frac{n\pi}{L}=-\frac{\pi}{24L}<0.$ (11)

Note here the Casimir energy of the cavity is $E_{m}=(1-v^{2})m_{0}$ instead
of $m_{0}$, which is not expected. However, this does not concern because the
variation is of $\mathcal{O}(v^{2})$ and it would be cancelled by corrections
of special relativity.

Applying the coordinate transformation to the mode function in Eq.(7), we will
get the mode function in $S$-frame

$f_{n}(x)=e^{iv\omega_{n}(x-vt)}\sin(\omega_{n}(x-vt)),$ (12)

where $\omega_{n}=\frac{n\pi}{L}$ and the orthogonality relation now takes the
form

$\int_{vt}^{vt+L}dxf_{n}(x)f_{m}^{*}(x)=\delta_{n,m}.$ (13)

The Casimir energy of the cavity in $S$-frame can be calculated in terms of
the energy-momentum tensor

$E_{s}=\int_{vt}^{vt+L}dx\bra{0}T^{00}\ket{0}=(1+2v^{2}+v^{4})m_{0},$ (14)

which is obviously but importantly, time independent, as shown in Fig.1(a).
The figure tells us that the Casimir cavity is mechanically unstable. The
momentum of the cavity can be calculated with similar manners

$P_{s}=\int_{vt}^{vt+L}dx\bra{0}T^{01}\ket{0}=m_{0}(v+v^{3}).$ (14’)

Figure 1: Velocity dependent of Casimir energy. We use Klein-Gordon equation
in (a) lab frame and (b) co-moving frame. Here we set $m_{0}=1$.

###  2.2 The other way around

What would happen if we start with the Klein-Gordon equation in
$S^{\prime}$-frame?

$\begin{split}&(\frac{\partial^{2}}{\partial t^{\prime
2}}-\frac{\partial^{2}}{\partial x^{\prime
2}})\phi(t^{\prime},x^{\prime})=0,\\\
&\phi(t^{\prime},0)=\phi(t^{\prime},L)=0.\end{split}$ (15)

Obviously the Casimir energy in $S^{\prime}$-frame would be $E_{m}=m_{0}$.
When transforming into $S$-frame, the Klein-Gordon equation will become

$\begin{split}&[(\frac{\partial}{\partial t}+v\frac{\partial}{\partial
x})^{2}-\frac{\partial^{2}}{\partial x^{2}}]\phi(t,x)=0,\\\
&\phi(t,vt)=\phi(t,vt+L).\end{split}$ (16)

While the mode functions read

$g_{n}(t,x)=e^{-i\omega_{n}t}\sin[\omega_{n}(x-vt)],$ (17)

whose orthogonality can be verified similarly. Integrating the energy-momentum
tensor we obtain the Casimir energy and momentum in $S$-frame as follows

$\begin{split}&E_{s}=\int_{vt}^{vt+L}dx\bra{0}T^{00}\ket{0}=\frac{1}{2}\sum_{n}\omega_{n}(1+\frac{v^{2}}{2})=m_{0}(1+\frac{v^{2}}{2}),\\\
&P_{s}=\int_{vt}^{vt+L}dx\bra{0}T^{01}\ket{0}=m_{0}v.\end{split}$ (18)

Comparing Eqs.(18) and (14)-(14’), we see that the momentum calculated from
the two roads match to their lowest order in $\mathcal{O}(v)$. But the energy
calculated does not at their lowest order $\mathcal{O}(v^{2})$. This
discrepancy forbids us to ascribe the mismatch to the correction of relativity
effects. This puts the results above as well as those given in references [3,
4, 5] in danger and makes their validity questionable. A full relativistic
treatment of the problem is needed.

##  3 Relativistic calculation

The mismatch between the two results above arises from the improper use of
Galileo transformation on the scalar field equation and boundary conditions.
Remarks from here are ambiguous. Our strategy in this section is to take the
non-relativistic approximation off the table and make sure all calculations be
done relativistically. To start with, we know that the first line of Eq.(15)
holds. Since the Klein-Gordon equation is Lorentz invariant, its first line
experiences no changes in the $S$-frame. When the boundary condition is
concerned, the Lorentz contraction of the cavity must be considered, as
sketched in Fig.(2). In $S$-frame, the field equation and boundary conditions
become

$\begin{split}&(\frac{\partial^{2}}{\partial
t^{2}}-\frac{\partial^{2}}{\partial x^{2}})\phi(t,x)=0,\\\
&\phi(t,vt)=\phi(t,vt+\frac{L}{\gamma})=0,\end{split}$ (19)

where $L$ is the proper length of the cavity and
$\gamma=\frac{1}{\sqrt{1-v^{2}}}$ is the Lorentz contraction factor.

Figure 2: Spacetime diagram of the Casimir cavity. Note that we used
orthogonal coordinate to illustrate the Lorentz contraction factor, which does
not precisely resemble the Lorentz spacetime. The red line is the world line
of the right wall of the cavity while that of the left wall is coincide with
the $t^{\prime}$ axias. The dashed line represents the cavity length measured
in $S^{\prime}$ frame.

The Casimir energy in $S^{\prime}$-frame is simply $E_{m}=m_{0}$. Following
the identical procedure as previous section, the mode function in $S$ frame
can be written as

$\phi_{n}(t,x)=e^{-i\omega_{n}\gamma(t-\beta x)}\sin(\omega_{n}\gamma(x-\beta
t)),$ (20)

where $\beta=v$. This mode functions are compatible with the boundary
condition in Eq.(19). The orthogonality is preserved only when we take the
Lorentz contraction of the cavity into account

$\int_{vt}^{vt+\frac{L}{\gamma}}dx\phi_{n}(t,x)\phi_{m}^{*}(t,x)=\delta_{n,m}.$
(21)

While in $S$-frame, the Casimir energy and momentum of the cavity can thus be
calculated

$\begin{split}&E_{s}=\int_{vt}^{vt+\frac{L}{\gamma}}dx\bra{0}T^{00}\ket{0}=\frac{1}{2}\sum_{n}\omega_{n}\gamma^{2}(1+v^{2})=m_{0}\frac{1+v^{2}}{1-v^{2}},\\\
&P_{s}=\int_{vt}^{vt+\frac{L}{\gamma}}dx\bra{0}T^{01}\ket{0}=\sum_{n}\omega_{n}\gamma^{2}v=2m_{0}\frac{v}{1-v^{2}}.\end{split}$
(22)

Eq.(22) suggests the energy and momentum do not obey the usual transform rule
of Lorentz boost. The key to understand this fact lies in the non-triviality
of the boundary condition of Eq.(19), especially in its velocity dependence.
The form of the mode functions in Eq.(20) suggests that the boundary condition
not only squeezes the eigen-modes with a Lorentz contraction factor, but also
introduce mixes between the time and spatial component. This is a feature does
not appear in calculations with static configuration. The denominator
$1-v^{2}$ suggests that the energy and momentum blows up as the speed of the
cavity approaches that of the light $v\to 1$. This is obsviously what is
expected in a relativistic theory. In Fig.(3) we compare the velocity
dependence of a relativistic point particle’s energy with negative mass and
that of an Casimir cavity with finite extension.

When non-relativistic approximation is made on Eq.(22), we find that

$\begin{split}&E_{s}=m_{0}(1+2v^{2}),\\\ &P_{s}=2m_{0}v,\end{split}$ (23)

This suggests that the Casimir cavity should not be treated as point particles
in any circumstances. Researchers focus on the energy expression may conclude
that the inertia of the cavity is four times to its energy, while those focus
on the momentum expression may argue that the inertia is two times of the
Casimir energy. This implies that the definition of inertia of Casimir cavity
need be carefully examined.

Even though the Casimir energy and momentum has their own ways of
transformation under boost, their combination still satisfy the mass-shell
condition,

$E_{s}^{2}-P_{s}^{2}=m_{0}^{2}.$ (24)

This is somewhat surprising! It should be noted that Eq.(24) is also valid for
configurations in higher dimensions, as long as the the cavity geometry is
infinitely extended on directions perpendicular to its moving direction, as
sketched in Fig.(4).

Figure 3: The yellow line represents velocity dependence of the energy of a
negative mass particle obeying Lorentz transformation while the blue line
represents that of Casimir cavity.

##  4 Higher dimensions

The calculations conducted here can be shifted to cavities in higher
dimensions routinely. Here we show calculations for the $(2+1)$ dimensional
rectangular cavity, which sets the tone for general cases.

In $S^{\prime}$-frame, the field equation reads

$\begin{split}&(\frac{\partial^{2}}{\partial t^{\prime
2}}-\frac{\partial^{2}}{\partial x^{\prime 2}}-\frac{\partial^{2}}{\partial
y^{\prime 2}})\phi(t,x^{\prime},y^{\prime})=0,\\\
&\phi(t^{\prime},0,y^{\prime})=\phi(t^{\prime},a,y^{\prime})=\phi(t^{\prime},x^{\prime},0)=\phi(t^{\prime},x^{\prime},b)=0,\end{split}$
(25)

where $a$ and $b$ is the proper length of the cavity along $x$ and $y$ axis,
respectively. The Casimir energy in this frame can be derived
straightforwardly [5]

$E_{m}=\frac{1}{2}\sum_{n,m}\omega_{n,m}=\frac{\pi}{48}-\frac{\zeta(3)b}{16\pi
a^{2}}+\frac{\pi}{a}G(\frac{b}{a}),$ (26)

where
$\omega_{n,m}=\sqrt{k_{n}^{2}+p_{m}^{2}},k_{n}=\frac{n\pi}{a},p_{m}=\frac{m\pi}{b}$
and the function $G(x)$ is defined in [5].

In $S$ frame, our field equation and boundary condition become

$\begin{split}&(\frac{\partial^{2}}{\partial
t^{2}}-\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial
y^{2}})\phi(t,x,y)=0\\\
&\phi(t,vt,y^{\prime})=\phi(t,vt+\frac{a}{\gamma},y^{\prime})=\phi(t,x^{\prime},0)=\phi(t,x^{\prime},b)=0,\end{split}$
(27)

so the mode expansion functions read

$\phi_{n,m}(t,x,y)=e^{-i\omega_{n,m}\gamma(t-\beta x)}\sin(k_{n}\gamma(x-\beta
t))\sin(p_{m}y),$ (28)

whose orthogonal relation can be verified similarly. By the calculation logic
of (20)-(22), the Casimir energy and momentum in $S$-frame can be obtained as
follows

$\begin{split}&E_{s}=\int_{vt}^{vt+\frac{a}{\gamma}}dx\int_{0}^{b}dy\bra{0}T^{00}\ket{0}=\gamma^{2}(1+\beta^{2})\sum_{n,m}(\frac{\omega_{n,m}}{2}+\frac{k_{n}^{2}}{2\omega_{n,m}}),\\\
&P_{s}=\int_{vt}^{vt+\frac{a}{\gamma}}dx\int_{0}^{b}dy\bra{0}T^{01}\ket{0}=\gamma^{2}\beta\sum_{n,m}(\frac{\omega_{n,m}}{2}-\frac{k_{n}^{2}}{2\omega_{n,m}}),\end{split}$
(29)

Obviously this has no similar and simple form as that in Eq.(22). The mass-
shell condition also becomes hard to verify if it is still the same as that of
point particles. However, if such a condition is imposed as necessary, then we
will get a highly non-trivial subtraction or regularisation rule for the
infinite summation in (29).

Figure 4: Configuration finite (a) only and (b) not only along moving
direction.

##  5 Conclusion and discussion

In this work we showed that applying the method of references [3, 4, 5] to the
calculation of Casimir energy and momentum of uniformly moving cavities under
non-relativistic limit will lead to conflicting results and demonstrated the
way to fix this inconsistence. We find that the Casimir energy and momentum
transform differently from differently from that of usual particles under
Lorentz boost. However, the mass-shell condition of the two remains identical.
We showed that our calculations can be generalised to higher dimensional
Casimir cavities and yield non-trivial subtraction or regularisation rules for
the infinite summation involved in the vacuum energy definition.

The most dramatic feature in our result is that the Casimir energy and
momentum transform according to Eq.(22) instead of the usual Lorentz boost
rule. This is somewhat counterintuitive as one may expect that the
renormalised energy-momentum tensor in the presence of finite boundary obey
Lorentz transformation rule
$T^{\mu\nu}\rightarrow\Lambda^{\mu}_{\rho}\Lambda^{\nu}_{\sigma}T^{\rho\sigma}$
as derived in [28]. Thus the energy density will receive a factor $\gamma^{2}$
under Lorentz boost, where one of them will be cancelled by the Lorentz
contraction factor of the cavity so that the final expression has Lorentz
transformation rules the same as usual point particles. We note that the
boundary condition of Eq.(19) affects solutions of the field equation non-
trivially, as we explained above. Here we make a more detailed pass over this
issue. Comparing Eq.(19) with Eq.(15) we can see that although the Klein-
Gordon equation is invariant under Lorentz boost, the boundary condition is
not. Since the solution of partial differential equations is very sensitive to
the boundary condition, we expect the solution of Eq.(19) and Eq.(15) to be
very different. Therefore, more concern should be taken when dealing with the
field equation in the presence of boundaries.

In [5], the author addressed his concern on the applicability of
regularisation schemes in non-inertial frame. In this work since we are
working in the inertial frames the whole time, thus there is no concern for us
on this issue. However, in this work, as well as in [3, 4, 5], we assumed the
applicability of the regularisation schemes on the time-space mixed mode
functions as prior. We think further study is required on this issue.

Nevertheless, we believe that our work provides insight on the topic of the
regularisation of quantum fluctuations with moving boundaries, which is to be
further studied. In the meantime, our results suggests that due the finite
size feature the Casimir cavity cannot be treated as relativistic particles
with negative mass. This may stimulate further directions on the gravitational
effects of Casimir energy.

## Acknowledgements

Y.-S. Cao gives his sincere thanks to A. Flachi and M. Edmonds for helpful
discussion and suggestions. This research was funded by the National Natural
Science Foundation of China under Grant No. $11875082$ and No. $12204406$ .

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11institutetext: University of Technology Sydney, Australia 11email:
<EMAIL_ADDRESS>
22institutetext: Northeastern University, China 22email<EMAIL_ADDRESS>
# Propagation-aware Social Recommendation by Transfer Learning

Haodong Chang 11 0000-0002-5015-1793 Yabo Chu 22 0000-0002-1694-9179

###### Abstract

Social-aware recommendation approaches have been recognized as an effective
way to solve the data sparsity issue of traditional recommender systems. The
assumption behind is that the knowledge in social user-user connections can be
shared and transferred to the domain of user-item interactions, whereby to
help learn user preferences. However, most existing approaches merely adopt
the first-order connections among users during transfer learning, ignoring
those connections in higher orders. We argue that better recommendation
performance can also benefit from high-order social relations. In this paper,
we propose a novel Propagation-aware Transfer Learning Network (PTLN) based on
the propagation of social relations. We aim to better mine the sharing
knowledge hidden in social networks and thus further improve recommendation
performance. Specifically, we explore social influence in two aspects: (a)
higher-order friends have been taken into consideration by order bias; (b)
different friends in the same order will have distinct importance for
recommendation by an attention mechanism. Besides, we design a novel
regularization to bridge the gap between social relations and user-item
interactions. We conduct extensive experiments on two real-world datasets and
beat other counterparts in terms of ranking accuracy, especially for the cold-
start users with few historical interactions.

###### Keywords:

Recommender system Social Connections Transfer Learning Social-aware
Recommendation.

(Haodong Chang and Yabo Chu contributed equally to this work)

##  1 Introduction

Nowadays, recommender systems play an essential role in providing effective
recommendations to users with items of interest. The key of success is to
learn precise user and item embeddings, where Collaborative Filtering (CF) is
the most traditional method [1, 2] to learn from user historical records, such
as ratings, clicks, and reviews. However, for many users, it is lack of
interaction data to provide accurate recommendations. The data sparsity
problem limits the performance of CF-based models.

Figure 1: An example to illustrate different friends’ influence. User $b$ and
$c$ have the same color as user $u$ indicates that they have similar
preferences with user $u$, and $a$ have different preferences with user $u$.
Then the system make a recommendation by considering these friends’
preference. “Mission Impossible” is in the front of the recommended list.
”Avatar” is ranked behind the other two films.

With the prevalence of online social networks, social connections have been
widely leveraged to alleviate the data sparsity problem, and formed the line
of research called social-aware recommendation. Transfer learning [3, 4] is a
useful approach to learn the common knowledge shared between a source domain
and a target domain, and then transfer the common knowledge to enhance the
model learning in target domain. Transfer learning is also applied in social-
aware recommendation to learn user preference from social connections and then
transfer to item domain, leading to more fine-refined user preference and thus
better recommendation performance. However, most existing methods only adopt
the first-order connections while ignoring the high-order connections. For
example, in Figure 1, users $a$ and $b$ are both friends of user $u$ in the
first order, user $c$ lies in the second order. Users $u,b,c$ share similar
interests, while user $a$ has different interests with $u$. In this case, user
$c$ (in the 2nd order) will have more positive influence on learning
preference of user $u$ than first-order friend user $a$.

Therefore, we argue that high-order friends are informative and can also help
learn user preference, especially considering the fact that users may not have
many direct connections with other users. It is valuable to find more relevant
social friends to deal with the data sparsity problem in social networks.
Therefore, we adopt the trust propagation in our model to mine informative
knowledge hidden in high-order social relations. Specifically, social
influence have been considered in two aspects. Firstly, friends in different
orders will affect the learning of user preference. Different order has
distinct bias towards preference learning. To the authors’ best knowledge, we
are the first to take into account order bias in modelling high-order social
influence. Secondly, friends in the same order will have different importance
for preference learning. We apply the attention mechanism to adaptively learn
the importance of friends in the same order. Moreover, we propose a novel
regularization term to formulate the relationship between domain-specific and
cross-domain (common) knowledge to reduce the risk of model overfitting.

To summarize, the main contributions of this paper are as follows:

  * •

We apply transfer learning to learn the sharing common knowledge between
social and item domains, and leverage social propagation to take into account
high-order social influence for better recommendation.

  * •

We propose a new factor ‘order bias’ to distinguish social influence in high
orders from low orders. We design a novel regularization term to formulate the
relationship between domain-specific and cross-domain (common) knowledge and
thus to avoid overfitting.

  * •

We conduct extensive experiments on two real-world datasets Ciao and Yelp, and
demonstrate the effectiveness of our approach in ranking accuracy.

##  2 Related Work

####  2.0.1 Social-aware Recommendation

Most previous social-aware recommendation works are based on homogeneity and
social influence theory, that is, users who are connected tend to have similar
behavioral preferences, and people with similar behavioral preferences are
more likely to establish connections. The meaning reflected in the
recommendation model is that the user’s feature vector should be as close as
possible to the vector space’s similar user’s feature vector. For example, [5]
assumed that users are more likely to have seen items consumed by their
friends, and extended BPR[2] by changing the negative sampling strategy.
TrustSVD[6] believed that not only the user’s explicit rating data and social
relationships should be modeled, but the user’s implicit behavior data and
social relationships should also be considered. Therefore, implicit social
information is introduced based on the SVD++[7] model. Recent research has
used deep neural networks as classifiers, yielding significant accuracy. E.g.,
SAMN[8] leverages attention mechanism to model both aspect- and friend-level
differences for social-aware recommendations. However, these methods use
direct social connections and ignore high-order social relationships, which
has a wealth of information.

There are also some studies considering trust propagation to get high-order
information. DeepInf[9] models the high-order to predict the social influence.
[10] proposed a DiffNet neural model with a layer-wise influence diffusion
part to model how users’ trusted friends recursively influence users’ latent
preferences. The further work[11] jointly model the higher-order structure of
the social and the interest network. However, they need to use text or image
information for data enhancement, which may lack a certain degree of
versatility. Moreover, existing methods ignore the influence of different
order’s friends on users.

Our work differs from the above studies as the designed model uses attention
mechanism to aggregate different friends’ influence in each order adaptively.
And the influence of order are considered as order bias. Order bias could
adjust the friend’s influence depend on the friend’s order.

####  2.0.2 Transfer Learning

Transfer learning deals with the situation where the data obtained from
different resources are distributed differently. It assumes the existence of
common knowledge structure that defines the domain relatedness and
incorporates this structure in the learning process by discovering a shared
latent feature space in which the data distributions across domains are close
to each other. [12] pointed out that parts of the source domain data are
inconsistent with the target domain observations, which may affect the
construction of the model in the target domain. Based on that, some
researchers [3, 13] designed selective latent factor transfer models to better
capture the consistency and heterogeneity across domains for recommendation.
However, in these works, the transfer ratio needs to be properly selected
through human effort and can not change dynamically in different scenarios.

There are also some studies considering the adaption issue in transfer
learning. [14] proposed to adapt the transfer-all and transfer-none schemes by
estimating the similarity between a source and a target task. [15] designed a
completely heterogeneous transfer learning method to determine different
transferability of source knowledge. However, these methods mainly focus on
task adaptation or domain adaption. [4] propose to adapt each user’s two kinds
of information (item interactions and social connections) with a finer
granularity, which allows the shared knowledge of each user to be transferred
in a personalized manner. [16] propose a novel dual transfer learning-based
model that significantly improves recommendation performance across other
domains. Nevertheless, these methods still ignore the following two
issue:1)High-order information is very helpful to improve the recommendation
performance. 2)Sparse data in rating domain and social domain can lead to
overfitting problems.

Our method innovatively leverage the high-order information for transfer
learning. And we propose a novel regularization so that the user
representation about the common knowledge can be reconstructed to the user
representation in the social and item domains, which could reduce the risk of
overfitting due to the lack of data.

##  3 Our Proposed Model

###  3.1 Notations

Suppose we have a user set $\mathcal{U}$ and an item set $\mathcal{V}$, let M
denote the number of users and N denote the number of items. Symbols u, t
denote two different users, and v denotes an item. $\mathcal{F}_{u}$
represents the friend set of user $u$. In social rating networks, users can
form social connections with other users and interact with items, resulting in
two matrices: user-user social matrix and user-item interaction matrix. The
user-item interaction matrix is defined as
$\textbf{R}=[\emph{r}_{uv}]_{M\times N}$ from users’ historical behaviors.
$r_{uv}=1$ indicates that user $u$ has an observed interaction (purchases,
clicks) with item $v$. Similarly, we define the user-user social matrix
$\textbf{X}=[\emph{x}_{ut}]_{M\times M}$ from social networks. $x_{ut}=1$
indicates that user $u$ trusts user $t$. We represent user $u$’s embedding in
three parts: $\textbf{c}_{u}$, $\textbf{s}_{u}$ and $\textbf{i}_{u}$, where
$\textbf{c}_{u}$ denotes the latent factors shared between the item domain and
social domain, i.e., the common knowledge; $\textbf{s}_{u}$ and
$\textbf{i}_{u}$ are user latent factors corresponding to the social domain
and item domain. The purpose of item recommendation is to generate a list of
ranked items that meet user $u$’s preference.

Figure 2: An overview of our PTLN model. ‘ProB’ represents the propagation
block introduced in Figure 3.

###  3.2 Model Overview

The overall structure of our Propagation-aware Transfer Learning Network
(PTLN) is illustrated in Figure 2. It includes three types of input: 1) the
user embedding of user u and u’s each order friends, 2) the social user
embedding of u’s first order friends, 3) the item embedding of the item which
u has interacted. The outputs of our model are the predicted probability
$\hat{r}_{uv}$ that how user $u$ will like item $v$, and the predicted
probability $\hat{x}_{ut}$ that how user $u$ will trust another user $t$. The
main architecture of PTLN contains two components: propagation layer and
prediction layer.

The propagation layer propagates over social networks to incorporate the
influence of high-order social friends, and then aggregate social influence of
friends in different orders. Besides, the order itself is also considered as
order bias, indicating the influence bias of general friends in a specific
order. In the prediction layer, we adopt attention mechanism to consider the
domain relationships to better transfer the domain-specific knowledge and the
shared knowledge for each task. Moreover, we adopt an efficient whole-data
based training strategy [4], and involves a novel regularization term in loss
function to optimize the model.

###  3.3 Propagation Layer

In this part, we aim to explore the high-order social influence based on the
idea that a user may share similar preferences with her friends. As shown in
Figure 2, the propagation layer are constructed in a multi-block structure.
Each block’s input is the user embedding of target user $u$ and that of $u$’s
friends at this order. The output is the new user embedding which includes
high-order friends’ influence. The new user embedding in each aspect is
calculated as same in propagation block, therefore we take the process of
calculating the new user embedding in common knowledge aspect as an example to
explain the details of the formula. The new user embedding is learned in below
four steps:

Figure 3: The details of K-th propagation block in common knowledge aspect.

1)Calculate Similarity Embedding User’s social connection will indirectly
influence the user’s preference in different degrees. As discussed in the
introduction, the similarity between two connected users can be used as an
essential basis for revealing the degree of influence. Thus we adopt attention
mechanism to assign the non-uniform weights to each friend according to the
similarity between the user and her friends. we firstly calculate the
similarity embedding between user $u$ and her k-th order friend $t$ in common
knowledge aspect as follow:

$\textbf{sim}_{(u,t)}^{C}=\textbf{c}_{u}^{0}\odot\textbf{c}_{t}^{0}$ (1)

where $\textbf{sim}_{(u,t)}^{C}\in\mathbb{R}^{D_{1}}$ denotes the similarity
embedding between user $u$ and her k-th order friend $t\in\mathcal{F}_{u}^{k}$
in common knowledge aspect. The superscript $0$ indicates the illustrated
subject is initial. $\mathcal{F}_{u}^{k}$ represents the k-th order friend set
of user $u$. the operation $\odot$ denotes the element-wise product of
vectors.

2)Calculate Attention score After obtaining similarity embedding from k-th
order friends, the attention are calculated by a trainable weighted matrix
$\textbf{W}\in\mathbb{R}^{D_{1}\times 1}$. For each aspect, the trainable
weighted matrix are unique. The k-th order friend $t$’s attention in common
knowledge aspect $\mathcal{A}_{(u,t)}^{*(C)}$ is defined as:

$\mathcal{A}_{(u,t)}^{*(C)}=\mathbf{W}_{C}^{\mathrm{\emph{T}}}\mathbf{sim}_{(u,t)}^{C}$
(2)

where $\mathbf{W}_{C}$ is the trainable weighted matrix to the common
knowledge aspect.

Then we use the softmax function to normalize the friend’s attention score:

$\mathcal{A}_{(u,t)}^{C}=\frac{exp(\mathcal{A}_{(u,t)}^{*(C)})}{\sum_{z\in\mathcal{F}_{u}^{k}}exp(\mathcal{A}_{(u,z)}^{*(C)})}$
(3)

where $\mathcal{A}_{(u,t)}^{C}$ is the final attention of friend $t$ which
indicates the degree of $t$’s influence on user $u$.

3)Aggregate Friend’s Influence We leverage the attention score to aggregate
the k-th order friend’s influence, so that the friend influence embedding we
get is obtained by dynamically absorbing the influence of her friends at this
order.

$\textbf{f}_{(C,u)}^{k}=\sum_{t\in\mathcal{F}_{u}^{(k)}}\mathcal{A}_{(u,t)}^{k}\textbf{c}_{t}^{0}$
(4)

where $\textbf{f}_{(C,u)}^{k}\in\mathbb{R}^{D_{1}}$ represents the $u$’s
friend influence embedding at k-th order.

4)Update User Embedding When generating the friend influence embedding, we
merely consider the similarity between the user and friend’s preference
ignoring the influence of the friend’s order, as discussed in the
introduction. Therefore we propose a concept of order bias to model the
influence bias of general friends in a specific order. we consider that the
order bias can dynamically adapt to the friend influence according to the
order. With the friend influence embedding and order bias, the user embedding
will be updated as follow:

$\textbf{c}_{u}^{k}=\textbf{c}_{u}^{0}+\textbf{f}_{(C,u)}^{k}+\textbf{o}^{k}$
(5)

The generated embedding $\textbf{c}_{u}^{k}$ is the new user embedding in k-th
order. $\textbf{o}_{k}\in\mathbb{R}^{D_{1}}$ indicates the order bias of k-th
order.

After propagating with k times, we obtain k new user embedding from first
order to k-th order. We will use all new user embedding achieved in each order
with initial user embedding to generate final user embedding $\textbf{c}_{u}$
as follow:

$\textbf{c}_{u}=\sum_{k}\textbf{c}_{u}^{k}$ (6)

###  3.4 Prediction Layer

Transfer Learning framework can transfer the shared knowledge from the source
domain to the target domain which is a promising method of using cross-domain
data to solve problems. [3] points that the degree of relationship between
domains is varied according to the user. Thus, we apply the attention
mechanism to use the domain-specific knowledge and common knowledge for better
learning the feature embedding which represent social domain preference and
item domain preference. For a user, if the two domains are less related, the
shared knowledge (c) will be penalized and the attention network will learn to
utilize more domain-specific knowledge (s or i) instead. Formally, the item
domain attention and the social domain attention are defined as:

$\alpha_{(C,u)}^{*}=\textbf{h}_{\alpha}^{\mathrm{\emph{T}}}\delta(\textbf{W}_{\alpha}\textbf{c}_{u}+\textbf{b}_{\alpha});\alpha_{(I,u)}^{*}=\textbf{h}_{\alpha}^{\mathrm{\emph{T}}}\delta(\textbf{W}_{\alpha}\textbf{i}_{u}+\textbf{b}_{\alpha})$
(7)
$\beta_{(C,u)}^{*}=\textbf{h}_{\beta}^{\mathrm{\emph{T}}}\delta(\textbf{W}_{\beta}\textbf{c}_{u}+\textbf{b}_{\alpha});\beta_{(S,u)}^{*}=\textbf{h}_{\beta}^{\mathrm{\emph{T}}}\delta(\textbf{W}_{\beta}\textbf{s}_{u}+\textbf{b}_{\alpha})$
(8)

Weight matrices $\textbf{W}\in\mathbb{R}^{D_{1}\times
D_{2}},\textbf{h}\in\mathbb{R}^{D_{1}}$ and bias units b serve as parameters
of the two-layer attention network. $\alpha$ and $\beta$ are related to the
item domain and social domain, respectively. $D_{2}$ denotes the dimension of
attention network, and $\delta$ is the nonlinear activation function $ReLU$.

Then, the final attention scores are normalized with a softmax function:

$\displaystyle\alpha_{(C,u)}$
$\displaystyle=\frac{exp(\alpha_{(C,u)}^{*})}{exp(\alpha_{(C,u)}^{*})+exp(\alpha_{(I,u)}^{*})}=1-\alpha_{(I,u)};\beta_{(C,u)}$
$\displaystyle=\frac{exp(\beta_{(C,u)}^{*})}{exp(\beta_{(C,u)}^{*})+exp(\beta_{(S,u)}^{*})}=1-\beta_{(S,u)}$
(9)

$\alpha_{(C,u)}$ and $\beta_{(C,u)}$ denote the weights of common knowledge c
for item domain and social domain, respectively, which determine how much to
transfer in each domain. After obtaining the above attention weights, the
feature embedding of user $u$ for the two domains are calculated as follows:

$\textbf{p}_{u}^{I}=\alpha_{(I,u)}\textbf{i}_{u}+\alpha_{(C,u)}\textbf{c}_{u};\textbf{p}_{u}^{S}=\beta_{(S,u)}\textbf{s}_{u}+\beta_{(C,u)}\textbf{c}_{u}$
(10)

The generated two feature embeddings $\textbf{p}_{u}^{I}$ and
$\textbf{p}_{u}^{S}$ represent the user’s preferences for items and other
users after transferring the shared knowledge between the two domains.

For predicting the scores of each item and user, we adopt a neural form MF
[17] to utilize the user’s feature embedding. For each task, a specific output
layer is employed. The scores of user $u$ for item $v$ are calculated as
follow:

$\hat{r}_{uv}=\textbf{W}_{I}(\textbf{p}_{u}^{I}\odot\textbf{q}_{v});\hat{x}_{ut}=\textbf{W}_{S}(\textbf{p}_{u}^{S}\odot\textbf{g}_{t})$
(11)

$\textbf{q}_{v}$ and $\textbf{g}_{t}$ denotes the latent factor vector of item
$v$ and user $t$ as a friend, respectively. The operation $\odot$ denotes the
element-wise product of vectors

Whole-data based strategy leverages the full data with a potentially better
coverage. Thus we adopt an efficient whole-data train strategy [4] to optimize
our model. For each task, the loss functions are defined as follow:

$\displaystyle\tilde{\mathcal{L}}_{I}(\Theta)$
$\displaystyle=\sum^{D_{1}}_{i=1}\sum^{D_{1}}_{j=1}\bigg{(}(h_{I,i}h_{I,j})\bigg{(}\sum_{u\in\mathcal{B}}p_{u,i}^{I}p_{u,j}^{I}\bigg{)}\bigg{(}\sum_{v\in\mathcal{V}}c_{v}^{I-}q_{v,i}q_{v,j}\bigg{)}\bigg{)}$
(12)
$\displaystyle+\sum_{u\in\mathcal{B}}\sum_{v\in\mathcal{V}^{+}}\bigg{(}(1-c_{v}^{I-})\hat{r}_{uv}^{2}-2\hat{r}_{uv}\bigg{)}$
$\displaystyle\tilde{\mathcal{L}}_{S}(\Theta)$
$\displaystyle=\sum^{D_{1}}_{i=1}\sum^{D_{1}}_{j=1}\bigg{(}(h_{S,i}h_{S,j})\bigg{(}\sum_{u\in\mathcal{B}}p_{u,i}^{S}p_{u,j}^{S}\bigg{)}\bigg{(}\sum_{t\in\mathcal{U}}c_{t}^{S-}g_{t,i}g_{t,j}\bigg{)}\bigg{)}$
(13)
$\displaystyle+\sum_{u\in\mathcal{B}}\sum_{t\in\mathcal{U}^{+}}\bigg{(}(1-c_{t}^{S-})\hat{x}_{ut}^{2}-2\hat{x}_{ut}\bigg{)}$

$I$ and $S$ are related to the item domain and social domain. $D_{1}$ is the
latent factor number. The scalar $h$,$p$,$q$,$g$ denote the element of their
corresponding vectors $h$,$p$,$q$,$g$. $i$ and $j$ denote the index of element
in the vector. $U^{+}$ and $V^{+}$ denote the items $v$ have interacted and
the friends that directly connect. $B$ is batch of users. $c_{v}^{I-}$ and
$c_{t}^{S-}$ are the weight of negative instances in two domains.

Both rating and social information are very sparse which could lead to the
overfitting problem. We consider that there has an implicit correlation
between common knowledge and domain-specific knowledge. This assumption
motivates us to propose a novel regularization term to against the overfitting
problem:

$\displaystyle\tilde{\mathcal{L}}_{Reg}(\Theta)=\sum_{k}(\|\textbf{i}^{k}-\theta_{\alpha}^{k}\textbf{c}^{k}\|^{2}+\|\textbf{s}^{k}-\theta_{\beta}^{k}\textbf{c}^{k}\|^{2})$
(14)

Where $\theta$ represents the weight of common knowledge c. $\alpha$ and
$\beta$ are related to the item domain and social domain.

After that, we integrate both the sub-tasks loss and the novel regularization
term into an overall objective function as follow:

$\mathcal{L}(\Theta)=\tilde{\mathcal{L}}_{I}(\Theta)+\lambda_{1}\tilde{\mathcal{L}}_{S}(\Theta)+\lambda_{2}\tilde{\mathcal{L}_{Reg}}(\Theta)+\lambda_{3}\|\Theta\|^{2}$
(15)

$\Theta$ represents the parameters of our model.
$\lambda_{1}$,$\lambda_{2}$,and $\lambda_{3}$ are the parameters to adjust the
weight proportion of each term.

##  4 Experiments

Table 1: Performance of all the comparison methods on the Ciao and Yelp
datasets. The last column “Avg Imp” indicates the average improvement of PTLN
over the corresponding baseline on average. N indicates top-N task.

Baselines Metrics | Precision | Recall | NDCG | MRR |
---|---|---|---|---|---
Ciao | N=5 | N=10 | N=15 | N=5 | N=10 | N=15 | N=5 | N=10 | N=15 | N=5 | N=10 | N=15 | Avg Imp
BPR | 0.0208 | 0.017 | 0.0141 | 0.0272 | 0.0496 | 0.0631 | 0.0289 | 0.036 | 0.0402 | 0.0479 | 0.0538 | 0.056 | 60.84%
NCF | 0.0217 | 0.0176 | 0.0149 | 0.0392 | 0.057 | 0.0721 | 0.0294 | 0.0385 | 0.0441 | 0.0508 | 0.057 | 0.0596 | 45.92%
SAMN | 0.0266 | 0.0225 | 0.0195 | 0.0482 | 0.0743 | 0.0959 | 0.0405 | 0.0506 | 0.0575 | 0.0562 | 0.0632 | 0.0653 | 17.47%
EATNN | 0.0295 | 0.0233 | 0.0195 | 0.0528 | 0.0763 | 0.094 | 0.0454 | 0.054 | 0.0598 | 0.071 | 0.0787 | 0.0816 | 6.53%
PTLN | 0.0307 | 0.0244 | 0.0203 | 0.0571 | 0.0818 | 0.1006 | 0.0494 | 0.0585 | 0.0646 | 0.0755 | 0.083 | 0.0866 |

Baselines Metrics | Precision | Recall | NDCG | MRR |
---|---|---|---|---|---
Yelp | N=5 | N=10 | N=15 | N=5 | N=10 | N=15 | N=5 | N=10 | N=15 | N=5 | N=10 | N=15 | Avg Imp
BPR | 0.0349 | 0.0282 | 0.0235 | 0.0317 | 0.0497 | 0.0601 | 0.0507 | 0.0537 | 0.0548 | 0.0832 | 0.092 | 0.0949 | 16.03%
NCF | 0.0337 | 0.0292 | 0.0266 | 0.0503 | 0.0626 | 0.0711 | 0.0429 | 0.0465 | 0.0487 | 0.0721 | 0.081 | 0.0843 | 24.74%
SAMN | 0.0333 | 0.0283 | 0.0276 | 0.0496 | 0.0568 | 0.0695 | 0.045 | 0.047 | 0.0511 | 0.0745 | 0.0822 | 0.0865 | 12.96%
EATNN | 0.0327 | 0.029 | 0.0266 | 0.0507 | 0.0579 | 0.0666 | 0.0462 | 0.049 | 0.0516 | 0.0749 | 0.0835 | 0.087 | 12.18%
PTLN | 0.0356 | 0.0307 | 0.0274 | 0.0558 | 0.0647 | 0.0714 | 0.0543 | 0.057 | 0.0587 | 0.0889 | 0.0978 | 0.1009 |

###  4.1 Experimental Settings

####  4.1.1 Dataset

We experimented with two public datasets: Ciao[4] and Yelp[18]. Ciao provides
a large amount of rating information and social information, while users make
friends with others and express their experience through the form of reviews
and ratings on Yelp. The two datasets were constructed following previous work
[4, 18]. Each dataset contains users’ ratings of the items they have
interacted with and the social connections between users. To address the Top-N
recommendation task, we remove all ratings that less than 4 for all datasets
and keep others with a score of 1. This preprocessing method aims at
recommending the item list that users liked, and is widely used in existing
works [4, 8, 19].

Baselines To evaluate the performance of Top-K recommendation,we compare our
PTLN with the following methods: BPR [2]: A classic and widely used ranking
algorithm for recommendation. It is implemented by learning pairwise relation
of rated and unrated items for each user rather than direct learning to
predict ratings. NCF [17]: A neural CF model combines element-wise and hidden
layers of the concatenation of user and item embedding to capture their high-
order interactions. SAMN [8]: A state-of-the-art deep learning method
leverages attention mechanism to model both aspect- and friend-level
differences for the social-aware recommendation. EATNN [4]: A state-of-the-art
method uses attention mechanisms to adaptively capture the interplay between
item domain and social domain for each user.

Evalutation Metrics We adapt four popular metrics Precision, Recall, NDCG(
Normalized Discounted Cumulative Gain), and MRR(Mean Reciprocal Rank) for
evaluation. Specifically, NDCG is a position-aware ranking metric, which
assigns a higher score to hits at higher positions. MRR considers the ranking
position of the first correct item in the recommended list. The higher value
of these evaluation metrics, the better performance of the recommender system.

Parameter Setting The parameters for all baseline methods were initialized as
in the corresponding papers and were then carefully tuned to achieve optimal
performance. The learning rate for all models were tuned among [0.0005,
0.0001, 0.005, 0.001, 0.05, 0.01]. To prevent overfitting, we tuned the
dropout ratio in [0.5,0.7,0.9]. The batch size was tested in
[16,32,64,128,256], the embedding size $D_{1}$ and the dimension of attention
network $D_{2}$ were tested in [32,64,128,256]. For our PTLN model, $D_{1}$
and $D_{2}$ were set to 128 and 32 on Ciao and set to 64 and 32 on Yelp. The
learning rate was set to 0.0005 when using the Yelp and 0.01 when using Ciao.
The dropout ratio $\rho$ was set to 0.7 on both datasets.

###  4.2 Performance Comparison

We investigate the Top-N performance with N set to [5,10,15], according with
the real recommendation scenario. We observe the results in Table 1:

Figure 4: Performance with different PLTN variants on Ciao and Yelp.

  1. 1.

Methods incorporating social information generally perform better than non-
social method. SAMN, EATNN, and PTLN perform better than BPR and NCF. This
result is consistent with previous work which indicates that social
information reflects users’ interest and is helpful in the recommendation.

  2. 2.

Our method PTLN achieves the best performance on the two datasets and
significantly outperforms all baseline methods. Specifically, compared to
EATNN which is the best baseline that uses attention mechanisms to adaptively
capture the interplay between item domain and social domain for each user.
PTLN improves over EATNN about 6.53% on Ciao and 12.18% on Yelp. The
substantial improvement of our model over the baselines could be attributed to
two reasons: 1) our model considers the propagation of social domain
knowledge, item domain knowledge and common knowledge, which allows the latent
factor to be modeled with a finer granularity; 2) we consider the difference
of friends’ influence and the order bias.

Table 2: Performance with different propagation depth K on Ciao and Yelp.

Ciao | Pre@10 | Recall@10 | NDCG@10 | MRR@10 | Yelp | Pre@10 | Recall@10 | NDCG@10 | MRR@10
---|---|---|---|---|---|---|---|---|---
K=1 | 0.0242 | 0.0802 | 0.0559 | 0.0791 | K=1 | 0.0306 | 0.0619 | 0.0559 | 0.0966
K=3 | 0.0242 | 0.0799 | 0.0557 | 0.0798 | K=3 | 0.0301 | 0.0617 | 0.0558 | 0.0973
K=2 | 0.0244 | 0.0818 | 0.0585 | 0.0835 | K=2 | 0.0307 | 0.0647 | 0.057 | 0.0978

Analyze of the propagation depth K The number of propagation layers K reflects
the extent to which the model uses social information and the degree to which
social information influences the model. Table 2 shows the results of
different K values for both datasets. When K increases from 1 to 2 ,the
performance increases, while the performance drops when K=3. We empirically
conclude that when the depth equals two is enough for the social
recommendation.

###  4.3 Ablation Study

####  4.3.1 Impact of the order bias

A key characteristic of our proposed model is the order bias which considers
the order of user’s friend in general. PTLN-O denotes a variant model of PTLN
without using order bias. We can see that order bias has dramatically improved
performance in Figure 4. We speculate a possible reason is that order bias can
dynamically adjust the output after fusing so that the updated user embedding
of this order can better reflect the preference of the user after being
influenced by friends.

####  4.3.2 Impact of the attention mechanism

Another critical characteristic of our proposed model is we considering the
diversity of friends’ influence by attention mechanism. PTLN-A directly
aggregate the friends’ influence and user’s embedding without any attention
learning process. From Figure 4, we can see that our model has a notable
improvement in performance on the Ciao dataset when considering the difference
of friends’ influence. However, results on the Yelp dataset is not as
significant as Ciao. This observation implies that the usefulness of
considering the importance strength of different elements in the modeling
process varies, and our proposed friend-level attention modeling could adapt
to different datasets’ requirements.

####  4.3.3 Impact of the novel regularization

To evaluate the effectiveness of the proposed correlative regularization, we
compare PTLN-R, a variant model of PTLN without using novel regularization,
with PLTN, in Figure 4. The PTLN model performs better than the PTLN-R,
proving that our novel regularization can make the algorithm more stable.

##  5 Conclusions

In this paper, we present a novel social-aware recommendation model PTLN to
address the sparsity problem of data. The core component of our model is
propagation layers that learn user embedding of each order by leveraging high-
order information from the social domain, item domain, and common knowledge
between the two domains. Attention mechanism and the concept of order bias are
further employed to better distinguish the influence of different user
friends. The proposed PTLN consistently and significantly outperforms the
state-of-the-art recommendation models on different evaluation metrics,
especially on the dataset with complicated social relationships and fewer item
interactions which verified our hypothesis about the varying degrees of
different friends’ influence.

## References

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