#  Dark Matter Annihilation via Breit-Wigner Enhancement with Heavier Mediator

Yu Cheng<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics and
Astronomy, Shanghai Jiao Tong University, China  Key Laboratory for Particle
Astrophysics and Cosmology (MOE) & Shanghai Key Laboratory for Particle
Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, China
Shao-Feng Ge<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics and
Astronomy, Shanghai Jiao Tong University, China  Key Laboratory for Particle
Astrophysics and Cosmology (MOE) & Shanghai Key Laboratory for Particle
Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, China
Jie Sheng<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics
and Astronomy, Shanghai Jiao Tong University, China  Key Laboratory for
Particle Astrophysics and Cosmology (MOE) & Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai
200240, China  Tsutomu T. Yanagida<EMAIL_ADDRESS>Tsung-Dao Lee
Institute & School of Physics and Astronomy, Shanghai Jiao Tong University,
China  Key Laboratory for Particle Astrophysics and Cosmology (MOE) & Shanghai
Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong
University, Shanghai 200240, China

###### Abstract

We propose a new scenario that both the dark matter freeze-out in the early
Universe and its possible annihilation for indirect detection around a
supermassive black hole are enhanced by a Breit-Wigner resonance. With the
mediator mass larger than the total initial dark matter mass, this
annihilation is almost forbidden at late times. Thus, the stringent cosmic
microwave background and indirect detection constraints do not apply. However,
a supermassive black hole can accelerate the dark matter particles to
reactivate this resonant annihilation whose subsequent decay to photons leaves
a unique signal. The running Fermi-LAT and the future COSI satellites can test
this scenario.

Introduction – More than 80% of the matter in our Universe today is dark
matter (DM) Arbey:2021gdg ; Young:2016ala . One major hunting strategy is the
DM direct detection that uses the DM scattering with nuclear or electron.
Currently, the Xenon-based experiments have reached ton-scale XENON:2023cxc ;
PandaX-4T:2021bab ; LZ:2022lsv . However, for the non-relativistic DM in our
galaxy only those with mass $\gtrsim\mathcal{O}(1)$ GeV carry large enough
kinetic energy to make the nuclear recoil from elastic scattering to overcome
the detection threshold. While DM above GeV scale is already highly
constrained, the sub-GeV region still has large parameter space
Schumann:2019eaa . Thus, light DM is becoming more popular nowadays
Cooley:2022ufh .

In the standard freeze-out scenario, the DM annihilation cross section for
obtaining the observed relic abundance should be around $\langle\sigma
v\rangle\sim 10^{-26}\,$cm3s-1 Steigman:2012nb . The same annihilation into
the Standard Model (SM) particles can still happen at late times. Its
subsequent electromagnetic energy injection into the environment modifies the
ionization history of the Universe and finally affects the observed cosmic
microwave background (CMB). Thus, the DM freeze-out scenario receives
stringent constraint from CMB Bean:2003kd ; Chen:2003gz ; Hansen:2003yj ;
Pierpaoli:2003rz ; Padmanabhan:2005es ; Slatyer:2009yq ; Roszkowski:2017nbc ;
Planck:2018nkj ; Cang:2020exa ; Liu:2023nct ; Kawasaki:2021etm ;
Steigman:2015hda .

One solution is the forbidden-type DM DAgnolo:2015ujb ; Delgado:2016umt ;
DAgnolo:2020mpt ; Wojcik:2021xki  whose annihilation is kinematically
prohibited at late times. Unfortunately, this also makes it difficult to leave
indirect detection signals today. The only place for the forbidden
annihilation to re-open is around a supermassive black hole (SMBH)
Cheng:2022esn ; Cheng:2023hzw . The strong gravitational force can accelerate
the DM particles to overcome the annihilation threshold. The resultant gamma-
ray is then a unique signal that only appears around SMBH but not anywhere
else.

In this paper, we propose an alternative scenario to escape the CMB constraint
and leave indirect detection signals around the SMBH based on the Breit-Wigner
enhancement Ibe:2008ye ; Guo:2009aj  for the DM annihilation. Its original
form of the Breit-Wigner scenario uses a light mediator (mediator mass
$m_{\phi}$ smaller than $2m_{\chi}$) to enhance the annihilation at late times
to explain the electron-positron excess observed by the PAMELA PAMELA:2008gwm
, ATIC Chang:2008aa , and PPB-BETS PPB-BETS:2008zzu  cosmic-ray experiments.
We do the opposite with a heavy mediator, $m_{\phi}>2m_{\chi}$. The DM
annihilation during freeze-out is enhanced by its thermal energy that
compensates the mass difference to reach the $s$-channel resonance pole. When
temperature cools down, the DM annihilation moves away from the resonance pole
and becomes greatly suppressed at late times to escape the CMB constraint.

This new Breit-Wigner scenario with a heavy mediator can reactivate around an
SMBH and the subsequent decay of the final-state SM particles can leave a
unique signature. In addition to the existing gamma-ray telescopes, such as
Fermi-LAT Fermi-LAT:2009ihh  and H.E.S.S. HESS:2018pbp  that can search for DM
signals around the SMBH Sgr $A^{*}$ in the energy range of
$\mathcal{O}(100)\,$MeV to TeV, the upcoming telescope Compton Spectrometer
and Image (COSI) aims at detecting the soft gamma-ray of $0.2\sim 5\,$MeV
Tomsick:2021wed . This opens a new window for testing our new scenario.

DM Production with Breit-Wigner Resonance – In the Breit-Wigner mechanism, the
DM $\chi$ with mass $m_{\chi}$ annihilates into SM particles through an
$s$-channel $\phi$. Although this mediator $\phi$ can have arbitrary spins in
principle, we assume a scalar mediator for simplicity. The general form of the
annihilation cross section with a Breit-Wigner resonance is Ibe:2008ye ;
Guo:2009aj ,

$\displaystyle\sigma=\frac{16\pi\beta_{f}}{s\bar{\beta}_{i}\bar{\beta}_{f}\beta_{i}}\frac{m_{\phi}^{2}\Gamma^{2}_{\phi}}{\left(s-m_{\phi}^{2}\right)^{2}+m_{\phi}^{2}\Gamma^{2}_{\phi}}B_{i}B_{f},$
(1)

where $\Gamma_{\phi}$ is the total decay rate of the mediator $\phi$ while $s$
is the center-of-mass energy squared $s\equiv E_{\rm cm}^{2}$. The
initial/final-state phase space factor
$\beta_{i(f)}\equiv\sqrt{1-4m_{i(f)}^{2}/E_{\rm cm}^{2}}$ is evaluated with
the center-of-mass energy and
$\bar{\beta}_{i(f)}\equiv\sqrt{1-4m_{i(f)}^{2}/m_{\phi}^{2}}$ with the
mediator mass. The branching ratios of the mediator $\phi$ decaying into the
initial- and final-state particles are denoted as
$B_{i(f)}\equiv\Gamma_{i(f)}/\Gamma_{\phi}$. A large boost factor to enhance
the indirect detection signal can be achieved with a light mediator,
$m_{\phi}<2m_{\chi}$, in the original Breit-Wigner scenario Ibe:2008ye ;
Guo:2009aj . Since $s$ is always larger than $m_{\phi}^{2}$, it keeps
decreasing and approaching the pole when temperature cools down.

Our scenario takes instead a heavy mediator, $m_{\phi}>2m_{\chi}$, for the
resonance. In this case, the $s$-channel resonance can be achieved only when
the DM has high enough kinetic energy. To make the resonance feature
transparent, we parametrize the mediator mass as $m_{\phi}^{2}\equiv
4m^{2}_{\chi}(1+\delta)$ with $\delta$ denoting the mass difference and the
non-relativistic $s\equiv 4m_{\chi}^{2}+m_{\chi}^{2}\left(\vec{v}_{\rm
rel}\right)^{2}$ with the relative velocity $\vec{v}_{\rm
rel}\equiv\vec{v}_{1}-\vec{v}_{2}$. Then the cross section expression Eq. (1)
becomes,

$\sigma=\frac{16\pi\beta_{f}}{s\bar{\beta}_{i}\bar{\beta}_{f}\beta_{i}}\frac{\gamma^{2}}{\left(\frac{\vec{v}_{\mathrm{rel}}^{2}/4-\delta}{1+\delta}\right)^{2}+\gamma^{2}}B_{i}B_{f},$
(2)

with the normalized decay width $\gamma\equiv\Gamma_{\phi}/m_{\phi}$. For a
heavy mediator, $\delta>0$. We can see that the cross section reaches the
Breit-Wigner resonance pole when $|\vec{v}_{\rm rel}|^{2}/4=\delta$. For
illustration, the black curve in Fig. 1 shows that the annihilation cross
section with $\delta=0.05$ peaks at exactly $v_{\rm rel}^{2}=0.2$ as expected.

In the early Universe, the DM number density evolution is determined by the
thermally averaged annihilation cross section,

$\langle\sigma v_{\rm
rel}\rangle\equiv\frac{x^{3/2}}{2\pi^{1/2}}\int_{0}^{\infty}dv_{\mathrm{rel}}v_{\mathrm{rel}}^{2}\left(\sigma
v_{\mathrm{rel}}\right)e^{-xv_{\mathrm{rel}}^{2}/4},$ (3)

where $x\equiv m_{\chi}/T_{\chi}$ and the DM phase space distribution has been
approximated by the Maxwell distribution. With Breit-Wigner resonance, the
thermally averaged $\langle\sigma v_{\rm rel}\rangle$ is maximized when the
Maxwell distribution (blue dash-dotted) peak overlaps with the pole (black
solid), $T_{\chi}\simeq\delta\times m_{\chi}$. This happens around $x\sim 20$
in the early Universe with the $\delta=0.05$ adopted by Fig. 1.

When the temperature keeps decreasing, the velocity distribution softens. For
illustration, the DM with temperature $T_{\chi}=m_{\chi}/1000$ (green dotted)
has almost no overlap with the Breit-Wigner resonance (black solid). At the
time of CMB formation, the Universe temperature has already dropped to
$\mathcal{O}(1)\,$eV. The DM kinetic energy is then not enough to activate the
Breit-Wigner enhancement with a heavy mediator. Thus, its annihilation is
highly suppressed and can naturally escape the CMB constraint.

A broader Breit-Wigner peak can increase its overlap with the low-temperature
DM distribution. However, the overlap is still small since the normalized
decay width $\gamma$ (equivalently the cross section) can not be too large in
order to produce the correct relic density.

Figure 1:  The overlap between different DM velocity distributions $f(v_{\rm
rel}^{2})$ and the DM annihilation cross section $\sigma$ as a function of the
relative velocity squared $v^{2}_{\rm rel}$. The black curve clearly shows the
Breit-Wigner resonance with a positive mass difference $\delta=0.05$. The blue
dash-dotted and green dotted curves are the DM velocity distributions when the
DM temperature is just 4% ($x=25$ which is equivalently
$T_{\chi}=m_{\chi}/25$) or 0.1% ($x=1000$ or $T_{\chi}=m_{\chi}/1000$) of its
mass, respectively. The red dashed curve shows the case around an SMBH at the
place with the largest DM annihilation rate per unit radius.

In the freeze-out scenario, the evolution of the DM yield, $Y\equiv
n_{\chi}/s$, follows the Boltzmann equation Kolb:1990vq ,

$\frac{dY}{dx}=-\frac{\tilde{\lambda}}{x^{2}}\left(Y^{2}-Y_{\mathrm{eq}}^{2}\right),$
(4)

with $Y_{\rm eq}\equiv 0.145(g/g_{s*})x^{3/2}e^{-x}$. The thermally averaged
cross section has been redefined as,

$\displaystyle\tilde{\lambda}$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{\pi}{45}}\frac{g_{s*}}{\sqrt{g_{*}}}\left[1+\frac{T}{3}\frac{\mathrm{d}}{\mathrm{d}T}\ln\left(g_{s*}\right)\right]m_{\chi}M_{\rm
pl}\left\langle\sigma v\right\rangle,$ (5)

where $M_{\rm pl}=1.22\times 10^{19}$ GeV is the Planck mass and $g_{*}$
($g_{\mathrm{s}*}$) the effective relativistic energy (entropy) degrees of
freedom. When the DM is in thermal equilibrium, its yield $Y$ tracks $Y_{\rm
eq}$. As temperature cools down, the DM number density drops exponentially and
it begins to freeze out once the annihilation rate is comparable to the Hubble
rate, $n_{\chi}\braket{\sigma v}\simeq H$. We take the freeze-out criterion
$Y-Y_{\rm eq}\simeq Y_{\rm eq}$ to estimate the freeze-out time Kolb:1990vq ,

$x_{f}\equiv\ln\frac{0.038gM_{\rm pl}m_{\chi}\langle\sigma
v\rangle}{g_{*}^{1/2}x_{f}^{1/2}}.$ (6)

This iterative equation gives that the DM freeze-out point is roughly
$x_{f}\sim 25$. By solving the differential equation
$dY/dx=-\tilde{\lambda}Y^{2}/x^{2}$ from the freeze-out point to infinity, we
can get the DM yield today,

$Y_{\chi}=\frac{Y(x_{f})}{1+Y(x_{f})\int^{\infty}_{x_{f}}\frac{\tilde{\lambda}}{x^{2}}dx}.$
(7)

The DM relic density is related to $Y_{\chi}$ as,

$\rho_{\chi}=m_{\chi}s_{0}Y_{\chi},\quad\Omega_{\chi}h^{2}=\frac{\rho_{\chi}}{\rho_{c}/h^{2}},$
(8)

where $s_{0}=2891.2$ cm-3 is the entropy density today and
$\rho_{c}=1.05\times 10^{5}h^{2}$ GeV/cm3 is the critical density of the
Universe.

Figure 2:  The parameter space to generate the correct relic density
$\Omega_{\chi}h^{2}=0.12$ for the Breit-Wigner resonance scenario with a heavy
mediator. We fix $B_{i}B_{f}=0.01$ while varying the DM mass $m_{\chi}$
(horizontal axis) and the normalized decay width $\gamma$ (vertical axis).
Then, the mass difference $\delta$ is uniquely determined by the DM relic
density whose value can be read off according to the blue color bar on the
right side.

There are four independent parameters, $m_{\chi}$, $\delta$, $\gamma$, and
$B_{i}B_{f}$. However, these parameters are not totally free since the
inequality $\Gamma^{2}_{\phi}=(\Gamma_{i}+\Gamma_{f})^{2}\geq
4\Gamma_{i}\Gamma_{f}$ requires $B_{i}B_{f}\leq 0.25$. The parameter values to
produce the correct relic density $\Omega_{\rm\chi}h^{2}=0.12$ are shown in
Fig. 2. For illustration, we fix the product of branching ratios
$B_{i}B_{f}=0.01$ and vary the DM mass $m_{\chi}$ as well as the normalized
decay width $\gamma$. Then, the observed DM relic density uniquely determines
the mass difference $\delta$. We can first see that a larger normalized decay
width $\gamma$ requires a larger DM mass which is the same feature as in the
standard case. Second, a larger mass difference $\delta$ prefers a larger
normalized decay width $\gamma$. This is because a big mass difference makes
the DM difficult to touch the Breit-Wigner resonance with a heavy mediator
which can be compensated by a larger coupling strength that is encoded in
$\gamma$.

The Right-Handed Neutrino DM Model – One interesting model to further
illustrate our scenario is a right-handed neutrino (RHN) DM Cox:2017rgn ;
Borah:2018smz ; Arcadi:2020aot . The heavy right-handed Majorana neutrinos are
widely considered as a key ingredient beyond the SM. They explain not only the
observed tiny neutrino masses via the seesaw mechanism Minkowski:1977sc ;
Yanagida:1979as ; Yanagida:1979gs ; Gell-Mann:1979vob , but also the baryon
asymmetry of our Universe through leptogenesis Fukugita:1986hr . Normally, we
assume three heavy RHNs. However, two heavy Majorana neutrinos are already
sufficient to explain baryon asymmetry and the observed neutrino mass squared
differences from oscillation Frampton:2002qc ; Raidal:2002xf ; Glashow:2003nk
; Barger:2003gt ; Ge:2010js . The remaining right-handed neutrino $N$ can
serve as a DM candidate. A byproduct is that we have an anthropic argument
Weinberg:1987dv  for the presence of three families of quarks and leptons
Ibe:2016yfo .

We assume a $Z_{2}$ parity acting only on this RHN DM $N_{\chi}$ to make it
stable Cox:2017rgn . To ensure that $N_{\chi}$ is in equilibrium at the early
Universe, we introduce a scalar mediator $\phi$ that couples to $N_{\chi}$
through a Majorana-type Yukawa coupling $\phi N_{\chi}^{T}\epsilon N_{\chi}$.
Notice here that $N_{\chi}$ is a right-handed two-component Weyl fermion.
Since the RHN DM $N_{\chi}$ is assumed to carry odd parity under $Z_{2}$,
$\phi$ is even and can couple with a pair of the SM Higgs bosons, $H$ and
$H^{\dagger}$, via $\phi H^{\dagger}H$. Thus, the interaction Lagrangian is,

$\mathcal{L}_{\rm int}=(y\phi N_{\chi}^{T}\epsilon N_{\chi}+h.c.)+\lambda
m_{\phi}\phi H^{\dagger}H.$ (9)

With a heavy mediator, $m_{\phi}>2m_{\chi}$, the DM decay receives two
contributions, $\Gamma_{\phi}=\Gamma_{\phi\rightarrow
N_{\chi}N_{\chi}}+\Gamma_{\phi\rightarrow f\bar{f}}$,

$\displaystyle\Gamma_{\phi\rightarrow N_{\chi}N_{\chi}}$
$\displaystyle=y^{2}\frac{m_{\phi}}{4\pi}\left(1-\frac{4m_{\chi}^{2}}{m_{\phi}^{2}}\right)^{3/2},$
(10a) $\displaystyle\Gamma_{\phi\rightarrow f\bar{f}}$
$\displaystyle=(\lambda^{\prime})^{2}\frac{m_{\phi}}{8\pi}\left(1-\frac{4m_{f}^{2}}{m_{\phi}^{2}}\right)^{3/2},$
(10b)

where $\lambda^{\prime}\equiv m_{f}\sin\theta/v\approx\lambda
m_{f}/(2m_{\phi})$ is the Yukawa coupling between the mediator $\phi$ and the
SM fermions. The parameter, $\theta\equiv\frac{1}{2}\arctan[\lambda
m_{\phi}v/(m_{\phi}^{2}-m_{h}^{2})]$, is the mixing angle between $\phi$ and
the SM Higgs particle $h$. The DM decay branching ratios are,
$B_{i}\equiv\Gamma_{\phi\rightarrow N_{\chi}N_{\chi}}/\Gamma_{\phi}$ and
$B_{f}\equiv\Gamma_{\phi\rightarrow f\bar{f}}/\Gamma_{\phi}$ with
$\Gamma_{i}=\Gamma_{\phi\rightarrow N_{\chi}N_{\chi}}$ and
$\Gamma_{f}=\Gamma_{\phi\rightarrow f\bar{f}}$.

The coupling $\lambda$ with the SM Higgs boson should maintain $\phi$ in
thermal equilibrium at the early Universe. In other words, the decay rate of
$\phi$ to the SM fermions $\Gamma_{\phi\rightarrow f\bar{f}}$ should be larger
than the Hubble rate $H\propto T_{f}^{2}/M_{\rm pl}$. Suppose $\phi$ decouples
around $T_{f}\sim m_{\phi}/25$, thermal equilibrium requires
$\lambda\gtrsim(10^{-8}\sim 10^{-7})$ depending on the mediator mass
$m_{\phi}$. Further through the first Yukawa term in Eq. (9) and the resultant
$\phi\rightarrow N_{\chi}+N_{\chi}$ decay, $N_{\chi}$ can also get in thermal
equilibrium. The parameter space of this RHN DM model to give the correct
relic density can be directly read off from Fig. 2.

Reactivation around SMBH – In the current Universe, the DM particles have
already become non-relativistic, which means almost no DM particles can gain
enough kinetic energy to touch the Breit-Wigner resonance pole. Thus, the DM
annihilation cross section is highly suppressed to leave almost no signal in
indirect detection. However, a strong gravitational source, such as an SMBH,
can accelerate the DM particles to reactivate their annihilation Cheng:2022esn
. For illustration, we show the velocity distribution at the radius where the
DM annihilation rate per unit radius, $(4\pi
r^{2}\rho_{\chi}^{2}/m_{\chi}^{2})\langle\sigma v\rangle$, is maximized as the
red curve in Fig. 1. We can see that the DM velocity distribution near SMBH
also has a big overlap with the Breit-Wigner resonance peak. Therefore, the DM
will gradually annihilate into the SM fermions around SMBH and the subsequent
decays produce gamma rays as observable signal. We take the SMBH in our galaxy
center, Sgr $A^{*}$, as an example for analysis.

The differential gamma ray flux from the DM annihilation and subsequent decay
is,

$\displaystyle\frac{dF_{\gamma}}{dE_{\gamma}}=\int_{0}^{1}dV_{r}dV_{c}\mathcal{P}_{\mathrm{r}}\left(V_{r},V_{c}\right)(\sigma
v)_{\rm cm}\frac{dN_{\gamma}}{dE_{\gamma}}(V_{r},V_{c}),$ (11)

with the joint probability distribution,

$\mathcal{P}_{\mathrm{r}}\left(V_{r},V_{c}\right)=\frac{x^{2}\gamma_{r}^{2}(\gamma_{r}^{2}-1)(1+\gamma_{r})V^{2}_{c}}{2K_{2}^{2}(x)(1-V_{c}^{2})^{2}}e^{-x\sqrt{\frac{2+2\gamma_{r}}{1-V_{c}^{2}}}}.$
(12)

The temperature parameter $x=x(r)$ now has radius $(r)$ dependence through the
DM temperature $T_{\chi}=\frac{1}{2}m_{\chi}v_{d}^{2}(r)$ where $v_{d}$ is the
DM velocity dispersion and $dN_{\gamma}/dE_{\gamma}$ the boosted photon
spectrum from $\phi$ decay. Note that the cross section $(\sigma v)_{\rm cm}$
is defined in the center-of-mass frame of DM collision. For the mass range
$m_{\phi}>10$ GeV, the dominate channel is $\phi\rightarrow b\bar{b}$. We use
the PPPC4DMID package Cirelli:2010xx  to generate the photon spectra and
further boost it to the lab frame Elor:2015bho ; Elor:2015tva .

The observed gamma ray flux is an integrated total result over the radius $r$
from $r_{0}\equiv 4GM$ to $r_{b}$,

$\frac{d\Phi_{\gamma}}{dE_{\gamma}}=\frac{1}{4\pi
D^{2}}\frac{1}{2m_{\chi}^{2}}\int_{4GM}^{r_{b}}4\pi
r^{2}dr\rho^{2}(r)\frac{dF_{\gamma}}{dE_{\gamma}}(r).$ (13)

Here, $\rho(r)$ is the DM density around the SMBH Fields:2014pia ;
Alvarez:2020fyo . For the innermost capture region, $r<r_{0}$, all particles
are attracted to fall into the SMBH so that there is no DM. When the
gravitational influence of SMBH no longer dominates for $r>r_{b}\equiv
0.2GM/v_{0}^{2}$ where $v_{0}$ is the DM velocity dispersion, the DM halo
simply follows the generalized NFW profile Navarro:1995iw . In between, a DM
spike forms. The full DM profile around a SMBH is,

$\rho(r)=\begin{cases}0,&r<r_{0},\text{ (Capture Region),}\\\
\frac{\rho_{\mathrm{sp}}(r)\rho_{\mathrm{in}}(t,r)}{\rho_{\mathrm{sp}}(r)+\rho_{\mathrm{in}}(t,r)},&r_{0}\leq
r<r_{b},\text{ (Spike), }\\\ \rho_{b}(r_{b}/r)^{\gamma_{c}}&r_{b}<r<D,\text{
(Halo). }\end{cases}$ (14)

The handy spike profile $\rho_{\rm sp}(r)\equiv\rho_{b}(r_{b}/r)^{\gamma_{\rm
sp}}$ with $\rho_{b}\equiv 0.3\,$GeV/cm${}^{3}\times(D/r_{b})^{\gamma_{c}}$ is
a function of the distance $D=8.5$ kpc between the Milky Way galaxy center and
our solar system. The two power indices $\gamma_{sp}$ and $\gamma_{c}$ are
adjustable parameters with $0.9\leq\gamma_{c}\leq 1.2$ Diemand:2008in ;
Navarro:2008kc . The annihilation plateau density, $\rho_{\rm ann}\equiv
m_{\chi}/\braket{\sigma v}t$, determined by the SM annihilation cross section
is scaled by the power index $\gamma_{\rm in}=1/2$ to give the inner profile
$\rho_{\rm in}(r,t)\equiv\rho_{\rm ann}(t)(r/r_{\rm in})^{-\gamma_{\rm in}}$.

In principle, we should include all the contributions along the line of sight.
However, the annihilation outside spike is negligible due to lack of the
Breit-Wigner enhancement. So we take the integration upper limit in Eq. (13)
as the outer boundary $r_{b}$ of spike.

Requiring the predicted gamma-ray flux to be below the Fermi-LAT data (4FGL
J1745.6-2859 around Sgr $A^{*}$ Fermi-LAT:2019yla ; Ballet:2020hze  from
August 4, 2008 to may 26, 2023) at 95% C.L. for each bin, the allowed model
parameter space is shown in Fig. 3 as colored regions. To make an
illustration, we have taken $\gamma_{c}=1.2$. For $\gamma_{\rm sp}=1.8$, the
mass difference $\delta$ can be either large or small. A small $\delta$
increases the annihilation cross section but decreases the DM spike density.
On the other hand, a large $\delta$ makes it difficult for DM particles to
touch the resonance pole and annihilate. Both cases lead to smaller photon
flux. With increasing $\gamma_{\rm sp}$ (deeper color), the DM density profile
becomes sharper and the constraint becomes stronger.

Figure 3:  The allowed parameter space not exceeding the constraints from each
data bin of the Fermi-LAT gamma-ray observation at 95 % C.L. With fixed
$\gamma_{c}=1.2$ and $B_{i}B_{f}=0.01$, the three colored regions correspond
to three typical values $\gamma_{sp}=(1.8,1.9,2.0)$, respectively.

Conclusions and Discussions – We propose for DM annihilation a new Breit-
Wigner enhancement scenario with the mediator mass larger than two times of
the DM mass. While the DM annihilation is highly suppressed to evade the CMB
constraint when the temperature cools down, the DM freeze-out in the early
Universe and its reactivation around an SMBH is enhanced by the $s$-channel
resonance with large enough kinetic energy. For illustration, we construct a
UV complete RHN DM model for the mass range from $\mathcal{O}(0.1)$ GeV to
$\mathcal{O}(100)$ GeV.

If sourced by DM annihilation, the observed positron excess requires a DM mass
above $10\,$GeV. Although the DM annihilation in our scenario can not reach
the resonance pole to give a large boost factor at late stages, astrophysics
sources can explain the positron excess equally wellBartels:2015aea ;
Malyshev:2009tw . So our DM scenario with a heavy mediator is consistent with
all these observations.

## Acknowledgements

Yu Cheng and Jie Sheng would like to thank Prof. Shigeki Matsumuto for useful
discussions and hospitality during their stay at Kavli IPMU where this paper
was partially completed. SFG is supported by the National Natural Science
Foundation of China (12375101, 12090060 and 12090064) and the SJTU Double
First Class start-up fund (WF220442604). T. T. Y. is supported by the China
Grant for Talent Scientific Start-Up Project and by Natural Science Foundation
of China (NSFC) under grant No. 12175134, JSPS Grant-in-Aid for Scientific
Research Grants No. 19H05810, and World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan. Both SFG and T. T. Y. are affiliated
members of Kavli IPMU, University of Tokyo.

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# Modeling Complex Dependencies for Session-based Recommendations via Graph
Neural Networks

Qian Zhang  Qilu University of Technology (Shandong Academy of
Sciences)JinanChina<EMAIL_ADDRESS>and  Wenpeng Lu  Qilu University
of Technology (Shandong Academy of Sciences)JinanChina<EMAIL_ADDRESS>
(2018)

###### Abstract.

Session-based recommendations (SBRs) capture items’ dependencies from the
sessions to recommend the next item. In recent years, Graph neural networks
(GNN) based SBRs have become the mainstream of SBRs benefited from the
superiority of GNN in modeling complex dependencies. Based on a strong
assumption of adjacent dependency, any two adjacent items in a session are
necessarily dependent in most GNN-based SBRs. However, we argue that due to
the uncertainty and complexity of user behaviors, adjacency does not
necessarily indicate dependency. However, the above assumptions do not always
hold in actual recommendation scenarios, so it can easily lead to two
drawbacks: (1) false dependencies occur in the session because there are
adjacent but not really dependent items, and (2) the missing of true
dependencies occur in the session because there are non-adjacent but actually
dependent items. These drawbacks significantly affect item representation
learning, degrading the downstream recommendation performance. To address
these deficiencies, we propose a novel review-refined inter-item graph neural
network (RI-GNN), which utilizes topic information extracted from the reviews
of items to improve dependencies between items. Experiments on two public
real-world datasets demonstrate that RI-GNN outperforms SOTA methods111The
implementation is available at https://github.com/Nishikata97/RI-GNN..

Recommender system, Session-based recommendation, Graph neural network,
Adjacent dependency

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation emai; June 03–05, 2018; Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Information systems Recommender
systems

##  1\. Introduction

In recent years, session-based recommendations (SBRs) have attracted extensive
attention (Hidasi et al., 2016; Wang et al., 2021b) for its strong capability
to capture users’ dynamic and short term preference. SBRs recommend the next
item to a user by modeling the sequential dependencies over items within
sessions.

Driven by the development of deep learning, many neural network based SBRs
have been developed. Among them, recurrent neural network (RNN) and graph
neural network (GNN) (Wang et al., 2021c) based approaches have shown good
performance. RNN-based methods attempt to capture sequential dependencies
between items within the sessions, which is based on the assumption that there
is a strict chronological order inside the session (Hidasi et al., 2016).
However, this assumption does not always hold in the real-world scenarios
since users’ behaviours are usually uncertain and dynamic and thus not all
interacted items in one session are sequentially dependent. Benefiting from
the capability of GNN in learning complex dependencies, many GNN-based SBRs
(Guo et al., 2021; Qiu et al., 2019; Wang et al., 2020; Wu et al., 2019) have
been proposed. Leveraging the flexibility of graph structure used in GNN, the
problem of strict chronological order confusing RNN-based methods is thus
alleviated.

However, GNN-based SBRs often rely heavily on the strong assumption of
adjacent dependency, namely, the adjacent items within one session are
necessarily dependent. This is determined by its particular work mechanism.
Specifically, most GNN-based SBRs first convert a given session consisting of
a sequence of interacted items into a session graph by mapping each item to a
node and the adjacency relation between any two items to an edge (Wu et al.,
2019) to indicate the dependency between them, as shown in Fig. 1(b). This
common practice of constructing session graphs often leads to two significant
deficiencies: (1) the introduction of false dependencies between adjacent but
actually independent items in a session, e.g., the item $v_{1}$ (i.e., a bird
cage) and item $v_{2}$ (i.e., cat food) in the session described in Fig. 1(a),
and (2) the missing of true dependencies between items which are non-adjacent
but actually dependent in a session, e.g., the item $v_{1}$ and item $v_{3}$
(i.e., a bird) in the session described in Fig. 1(a). In practice, both false
dependencies and true dependencies mentioned above are not uncommon in the
real-world cases (Wang et al., 2019). Obviously, these two deficiencies
significantly downgrade the accurate learning of inter-item dependencies
embedded in session data and thus reduce the performance of the downstream
next-item recommendations. Therefore, it is critical to refine the
dependencies between items by identifying and keeping all true dependencies
while removing the false ones.

Figure 1. A toy example for the construction of adjacency-driven inter-item
graph.

In practice, in addition to the session information, the review information
associated with items can reveal dependencies between them to some degree. For
example, the reviews associated with item $v_{1}$ and $v_{3}$ shown in Fig. 1
(a) are closely related and fall into the same topic. This actually provides
extra information to enable the possibility to refine dependencies between
items in sessions. To this end, we propose review-refined inter-item graph
neural network (RI-GNN) to address the two deficiencies mentioned above in
this paper. By leveraging the topic information from reviews written for
items, RI-GNN can not only reduces the false dependencies between adjacent but
actually independent items, but also well captures true dependencies between
non-adjacent but dependent items which are usually ignored or weaken by
existing GNN-based SBRs.

The main contributions of this work are summarized below:

  * •

We propose and discuss a novel and important research question: _does
adjacency necessarily indicate dependency between items in sessions_? We
perform a preliminary exploration with the hope of shedding some light in this
area.

  * •

We propose a novel review-refined inter-item graph neural network, called RI-
GNN, for session-based recommendations. To the best of our knowledge, this is
the first work for leveraging reviews to enhance the dependency learning for
SBR on anonymous sessions.

  * •

We propose a novel method for constructing a novel review-refined inter-item
graph for each session. In the graph, reviews for items are employed to filter
out the false dependencies between some adjacent items, and recall the true
dependencies between non-adjacent items missed by existing methods.

##  2\. Related Work

###  2.1. Session-based Recommendation

Existing methods for SBR can be summarized into: (1) Markov chain-based SBR;
(2) RNN-based SBR; (3) Attention-based SBR; and (4) GNN-based SBR.

(1) _Markov chain-based SBR_. Early researches on SBR rely Markov chain to
model the short-term dependencies to predict the next item. For example, FPMC
(Rendle et al., 2010) combined Markov chain and matrix factorization to model
sequential behavior between two adjacent items and recommend next item.
However, Markov chain-based methods only focus on first-order dependencies
between adjacent items, while neglecting the high-order dependencies between
long-distance items. (2) _RNN-based SBR_. Due to the powerful ability in
modeling sequential data, RNN-based methods are applied widely to SBR (Hidasi
et al., 2016; Li et al., 2017; Liu et al., 2018). GRU4Rec (Hidasi et al.,
2016) first applied RNN to SBR, which adopted gated recurrent unit (GRU) to
model the dependencies within sessions. However, RNN-based SBR also suffers
the similar problem in Markov chain-based methods, which always biases to
short-distance items while missing the information from the long-distance
items in sessions. (3) _Attention-based SBR_. The attention mechanism is
applied to further improve SBR (Wang et al., 2021a, 2018) by identifying
important items within sessions. NARM (Li et al., 2017) first integrated the
attention mechanism into SBR to extract the the user’s main purpose in the
current session. STAMP (Liu et al., 2018) proposed a short-term memory
priority model based on multi-layer perceptron and attention mechanism, which
captured the long-term and short-term interests of users. However, the
attention mechanism only focuses on few important items that belong to the
user’s main purpose, while neglecting the other purposes indicated by few
inferior items. (4) _GNN-based SBR_. Due to the superiority of GNN on modeling
transition dependencies between items, GNN-based methods have been applied
widely to SBR. SR-GNN (Wu et al., 2019) modeled all sessions via directed
graphs, utilized GNN to capture the dependencies between items with sessions,
and extracted the long-term and short-term interest of users to suggest the
next item. FGNN (Qiu et al., 2019) proposed to capture the sequence order and
latent order in session graph, which devised the weighted attention graph
layer to learn item embeddings and session embeddings for more accurate next
item recommendation. GCE-GNN (Wang et al., 2020) proposed to learn the
transitions between items from local and global perspectives simultaneously,
so as to make better recommendations by leveraging the information from other
sessions. Although GNN-based methods have achieved great success on SBR, all
of these approaches construct the session graph according to the adjacent
items, which ignore the dependencies from the non-adjacent items.

###  2.2. Review-based Recommendation

Considering the great value of user reviews on items, some works strive to
model reviews to improve the performance of SR (Li et al., 2019; Zheng et al.,
2017). DeepCoNN (Zheng et al., 2017) employed two parallel cooperative neural
networks to learn user behaviors by exploiting reviews written by the user and
learn item properties from the reviews written for the item, then utilized
factorization machine to predict item ratings. RNS (Li et al., 2019) proposed
a review-driven neural sequential recommendation, which learned user’s long-
term preference according to her historical reviews. Although the existing
methods improve the recommendation performance, most of them are devised for
the task of rating prediction instead of session-based recommendation.
Although RNS is proposed for sequential recommendation, it requires to collect
all reviews written by a user according to the user’s explicit ID. This means
that it is unable to work well on the anonymous session-based recommendation.
Neither of the existing works really solve the problem of session-based
recommendations based on review information.

##  3\. Preliminary

###  3.1. Problem Statement

Let $V=\left\\{v_{1},v_{2},\ldots,v_{m}\right\\}$ represent the whole item
set. Each anonymous session $s=[v_{1},v_{2},\ldots,v_{n}](v_{i}\in V)$ is an
ordered list of items, where all the items in $s$ are interacted by an
anonymous user in a chronological order. We embed each item into the same
embedding space and let $\mathbf{h}_{v_{i}}\in\mathbb{R}^{d}$ denote the
embedding of item $v_{i}$, where $d$ is the dimensionality. To accurately
identify item dependencies within the session, we utilize review information
to enhance item representations. Given an item $v_{i}$, all of its reviews are
collected to form the review document $D_{i}$, where each word is represented
by the corresponding embedding with the dimension $d_{w}$. For the session-
based recommendation problem, the goal is to predict the top-$N$ items that
the user is most likely to click in the next step.

###  3.2. Graph Construction

In this subsection, we first introduce the _adjaceny-driven inter-item
graph_(AIG), which is widely adopted by existing GNN-based approaches (Wang et
al., 2020; Wu et al., 2019), and then we present a novel graph, i.e., _review-
refined inter-item graph_(RIG). RIG is used as an additional graph to
complement AIG rather than to replace it by enhancing the learning of true
dependencies.

Adjaceny-driven inter-item graph (AIG). AIG captures important sequential
patterns based on pair-wise adjacent items within the current session, which
is first proposed by SR-GNN (Wu et al., 2019). AIG converts each session _s_
into a directed graph
$\mathcal{G}_{s}^{adj}=(\mathcal{V}_{s},\mathcal{E}_{s}^{adj})$, where
$\mathcal{V}_{s}\subseteq V$ denotes the node set, $\mathcal{E}_{s}^{adj}$
denotes the edge set. The weight of each edge is set as the value of the
occurrences of the edge divided by the outdegree of the edge’s start node.
This means that the more frequent occurrences of the edge, the stronger
dependency between the items connected by it. The connection matrix
$\mathbf{A}_{s}\in\mathbb{R}^{n\times 2n}$ describes how nodes are connected
with each other in the graph, and $\mathbf{A}_{s,i:}\in\mathbb{R}^{1\times
2n}$ are the two columns of blocks in $\mathbf{A}_{s}$, corresponding to node
$v_{i}$. $\mathbf{A}_{s}=\mathbf{A}_{s}^{\text{(out)
}}\|\mathbf{A}_{s}^{\text{(in) }}$, where $\mathbf{A}_{s}^{\text{(out) }}$ and
$\mathbf{A}_{s}^{\text{(in) }}$ are the outgoing and incoming adjacency matrix
respectively. $\|$ indicates the concatenation operation, and $n$ is the
length of session $s$.

Review-refined inter-item graph (RIG). The aforementioned AIG faces two
deficiencies caused by the adjacent but independent items and the nonadjacent
but dependent items. To address them, we utilize reviews to refine the session
graph, and thus devise the review-refined inter-item graph (RIG).

Figure 2. Construction of AIG.

We convert the session $s$ into a directed graph
$\mathcal{G}_{s}^{re}=(\mathcal{V}_{s},\mathcal{E}_{s}^{re})$, where
$\mathcal{V}_{s}\subseteq V$ indicates the node set, $\mathcal{E}_{s}^{re}$
denotes the edge set. As shown in Fig. 2, there are two steps for RIG to
recognize the cross-item dependencies (i.e., the dependencies between non-
adjacent items) to obtain the correct edges. _Firstly_ , once AIG is
generated, in order to filter out the noise edges and refine the session
graph, we only reserve the edges between items sharing the same topic in the
AIG, i.e., the same user purpose, and remove the other edges. For recognizing
the topic information conveniently, we collect all reviews written for the
item into the document set $D$ together, and utilize LDA (Blei et al., 2003)
to extract their topics. Once obtaining the topics of each item, we filter out
the edges between items that do not belong to the same topic. _Secondly_ , for
each ordered pair of nodes $(v_{t_{1}},v_{t_{2}})$ in the session sequence, we
add the directed edge $(v_{t_{1}}\rightarrow v_{t_{2}})$ if item $v_{t_{1}}$
shares the same topic with item $v_{t_{2}}$ and $t_{1}<t_{2}$. This makes the
current item directly connect to all the following dependent items in the
session for capturing the cross-item dependencies. Finally, we obtain the
refined edge set $\mathcal{E}_{s}^{re}$. Similar to AIG, we calculate the
connection matrix $\mathbf{B}_{s}\in\mathbb{R}^{n\times 2n}$ for RIG, where
$\mathbf{B}_{s}=\mathbf{B}_{s}^{\text{(out) }}\|\mathbf{B}_{s}^{\text{(in)
}}$.

##  4\. Architecture of RI-GNN Model

The architecture of our proposed RI-GNN is shown in Fig. 3, which mainly
consists of five components, i.e., _adjacency-driven inter-item graph (AIG)
learning layer_ , _review-refined inter-item graph (RIG) learning layer_ ,
_multi-stacking layer_ , _session representation learning layer_ and
_prediction layer_.

Figure 3. Architecture of our proposed RI-GNN model.

###  4.1. Adjaceny-driven Inter-item Graph Learning Layer (AIL)

AIL aims to capture the sequential dependencies between items based on AIG
within the current session. Next, we will present how to learn the sequential
dependencies between adjacent pair-wise items, as follows:

(1)
$\small\begin{split}\mathbf{a}_{s,i}^{t}&=\mathbf{A}_{s,i:}^{s}\left[\mathbf{h}_{{v}_{1}}^{t-1},\ldots,\mathbf{h}_{{v}_{n}}^{t-1}\right]^{\top}\mathbf{H}+\mathbf{b_{1}},\\\
\mathbf{z}_{s,i}^{t}&=\sigma\left(\mathbf{W}_{z}\mathbf{a}_{s,i}^{t}+\mathbf{U}_{z}\mathbf{h}_{{v}_{i}}^{t-1}\right),\\\
\mathbf{r}_{s,i}^{t}&=\sigma\left(\mathbf{W}_{r}\mathbf{a}_{s,i}^{t}+\mathbf{U}_{r}\mathbf{h}_{{v}_{i}}^{t-1}\right),\\\
\widetilde{\mathbf{h}}_{v_{i}}^{t}&=\tanh\left(\mathbf{W}_{o}\mathbf{a}_{s,i}^{t}+\mathbf{U}_{o}\left(\mathbf{r}_{s,i}^{t}\odot\mathbf{h}_{{v}_{i}}^{t-1}\right)\right),\\\
\mathbf{h}_{{v}_{i}}^{t}&=\left(1-\mathbf{z}_{s,i}^{t}\right)\odot\mathbf{h}_{v_{i}}^{t-1}+\mathbf{z}_{s,i}^{t}\odot\tilde{\mathbf{h}}_{v_{i}}^{t},\end{split}$

where $\mathbf{a}_{s,i}^{t}\in\mathbb{R}^{2d}$ is the current state at time
step $t$, which aggregates the adjacent items’ embeddings for item $v_{i}$ in
AIG. $[\mathbf{h}_{v_{1}}^{t-1},\ldots,\mathbf{h}_{v_{n}}^{t-1}]$ is the list
of item embeddings in session $s$ at previous time step $t-1$,
$\mathbf{A}_{s,i:}\in\mathbb{R}^{1\times 2n}$ are the two columns of blocks in
$\mathbf{A}_{s}$ corresponding to item $v_{i}^{s}$,
$\mathbf{H}\in\mathbb{R}^{d\times 2d}$,
$\mathbf{W}_{z},\mathbf{W}_{r},\mathbf{W}_{o}\in\mathbb{R}^{d\times 2d}$,
$\mathbf{U}_{z},\mathbf{U}_{r},\mathbf{U}_{o}\in\mathbb{R}^{d\times d}$,
$\mathbf{b_{1}}\in\mathbb{R}^{2d}$ are trainable parameters, $\sigma(\cdot)$
is the sigmoid function, and $\odot$ is the element-wise multiplication
operator, $\mathbf{z}_{s,i}^{t}\in\mathbb{R}^{d}$ and
$\mathbf{r}_{s,i}^{t}\in\mathbb{R}^{d}$ are the update and reset gates
respectively, $\mathbf{h}_{{v}_{i}}^{t}$ is the final state at the time step
$t$. We mark the final representation of item $v_{i}$ in AIL as
$\mathbf{h}_{v_{i}}^{adj}$.

###  4.2. Review-refined Inter-item Graph Learning Layer (RIL)

In order to filter out the noise information and improve the representations
of items, we next present how to propagate features on RIG to encode item
dependencies from reviews. This layer is built based on the architecture of
graph neural network, and we generate attention weights based on the
similarity of reviews between items by exploiting the idea of graph attention
network (Velickovic et al., 2018).

For each item’s review document $D_{i}$ obtained from _Section 3.2_ , we first
convert it into a representation vector $\mathbf{E}_{i}\in\mathbb{R}^{l\times
d_{w}}$ through word embeddings. In order to better extract item features from
the review representation $\mathbf{E}_{i}$, we utilize the self-attention
method proposed by Transformer (Vaswani et al., 2017):

(2)
$\small\operatorname{Attention}(\mathbf{Q},\mathbf{K},\mathbf{V})=\operatorname{softmax}\left(\frac{\mathbf{Q}\mathbf{K}^{\mathsf{T}}}{\sqrt{d}}\right)\mathbf{V},$

where $\mathbf{Q}$ is the queries, $\mathbf{K}$ is the keys, $\mathbf{V}$ is
the values and $\sqrt{d}$ is the scale factor. We adopt multi-head attention
to enable the model to jointly focus on information from different
representation subspaces from different positions. For item $v_{i}$, the
detailed operations are described as below:

(3)
$\small\begin{split}&\text{head}_{\textit{k}}=\operatorname{Attention}\left(\mathbf{E}_{i}\mathbf{W}_{k}^{\mathbf{Q}},\mathbf{E}_{i}\mathbf{W}_{k}^{\mathbf{K}},\mathbf{E}_{i}\mathbf{W}_{k}^{\mathbf{V}}\right),\\\
\mathbf{r}_{i}=&\text{MultiHead}(\mathbf{E_{i}})\left.=\text{
Concat(head}_{\textit{1}},\ldots,\text{head}_{\textit{h}}\right)\mathbf{W}_{1},\end{split}$

where $\mathbf{r}_{i}$ is the review representation of item $v_{i}$ extracted
by the multi-head attention, $\mathbf{W^{Q}}\in\mathbb{R}^{d_{w}\times
d_{q}}$, $\mathbf{W^{K}}\in\mathbb{R}^{d_{w}\times d_{k}}$,
$\mathbf{W^{V}}\in\mathbb{R}^{d_{w}\times d_{v}}$, and
$\mathbf{W_{1}}\in\mathbb{R}^{{h}d_{v}\times d_{w}}$ are the learnable
parameters. In our experiments, we set the number of parallel attention heads
$h$ to 3, set the dimensions of $d_{q}$, $d_{k}$, $d_{v}$, $d_{w}$ to 100,
100, 100, 300.

In order to distinguish the importance of neighbor items for obtaining the
representation of current item, we adopt attention mechanism and calculate the
attention weight by cosine similarity:

(4) $\small\begin{split}\pi\left(v_{i},v_{j}\right)=\left\\{\begin{aligned}
\operatorname{sim}\left(\mathbf{r}_{i},\mathbf{r}_{j}\right),&\text{ if
}TP_{i}=TP_{j}\\\ 0,&\text{ if }TP_{i}\neq
TP_{j}\end{aligned}\right.,\end{split}$

where $\pi\left(v_{i},v_{j}\right)$ estimates the importance weight of
different neighbor items, $\operatorname{sim()}$ is the cosine similarity
function, ${\mathbf{r}}_{i}$ and ${\mathbf{r}}_{j}$ are multi-head review
representation of item $v_{i}$ and item $v_{j}$ respectively, $TP_{i}$ and
$TP_{j}$ are the topics of item $v_{i}$ and item $v_{j}$ respectively. Next,
we can obtain the final item representation by the linear combination of
neighbor items:

(5)
$\small\begin{split}\hat{\pi}(v_{i},v_{j})&=\frac{\exp\left(\pi\left(v_{i},v_{j}\right)\right)}{\sum_{v_{k}\in\mathcal{N}_{v_{i}}^{re}}\exp\left(\pi\left(v_{i},v_{k}\right)\right)},\quad\mathbf{h}_{v_{i}}^{re}=\sum_{v_{j}\in\mathcal{N}_{v_{i}}^{re}}\hat{\pi}\left(v_{i},v_{j}\right)\mathbf{h}_{v_{j}},\end{split}$

where $\hat{\pi}(v_{i},v_{j})$ is attention coefficient normalized by softmax,
which means the different contribution of neighbor item $v_{j}$ to the current
item $v_{i}$. $\mathcal{N}_{v_{i}}^{re}$ is the neighbor set of item $v_{i}$
in the _RIG_ , $\mathbf{h}_{v_{j}}$ is the representation of the neighbor item
$v_{j}$ of item $v_{i}$, $\mathbf{h}_{v_{i}}^{re}$ is the final representation
of item $v_{i}$ in the RIL.

###  4.3. Multi-stacking Layer

In order to fully capture the deep dependencies between items, inspired by the
work of Chen et al. (Chen and Wong, 2020), we stack multiple AIL and RIL
layers, which can capture the complex dependencies (i.e., both adjacent-item
and cross-item) within the session, described as below:

(6)
$\small\mathbf{h}_{0,v_{i}}^{adj}\rightarrow\mathbf{h}_{1,v_{i}}^{re}\rightarrow\ldots\rightarrow\mathbf{h}_{l,v_{i}}^{*}\rightarrow\ldots\rightarrow\mathbf{h}_{k,v_{i}}^{*},$

where $\mathbf{h}_{l,v_{i}}^{*}$ denotes the representation of item $v_{i}$
which is the output of layer $l$, $l\in(2,k)$, $k$ is the hyper-parameter, and
$*$ indicates either $adj$ or $re$.

To fully utilize all features captured by all layers, we apply dense
connections (Huang et al., 2017) in our work. The input of each layer consists
of the output representations of all previous layers. More specifically, the
input of the $l$-th layer is
$[\mathbf{h}_{0,v_{i}}^{adj}\|\mathbf{h}_{1,v_{i}}^{re}\|\cdots\|\mathbf{h}_{l-1,v_{i}}^{*}]$.
This allows the higher layers to utilize not only the features through their
previous layer, but also the low-level features at lower layers. For each item
$v_{i}$, we obtain its representations
$\mathbf{h}_{v_{i}}^{\prime}\in\mathbb{R}^{d}$ by stacking multiple AIL and
RIL.

###  4.4. Session Representation Learning Layer

Through the three layers mentioned above, given session
$s=[v_{1},v_{2},\ldots,v_{n}]$, we can obtain all item representations in it,
i.e.,
$\mathbf{H}=[\mathbf{h}^{{}^{\prime}}_{v_{1}},\mathbf{h}^{{}^{\prime}}_{v_{2}},\ldots,\mathbf{h}^{{}^{\prime}}_{v_{n}}]$.
Then, we can generate session representation with the representations of items
in it.

To reflect the different importance of different positions in the session
sequence to the target item, we utilize a learnable position embedding matrix
$\mathbf{P}=\left[\mathbf{p}_{1},\mathbf{p}_{2},\ldots,\mathbf{p}_{n}\right]$,
where $\mathbf{p}_{i}\in\mathbb{R}^{d}$ is a position vector for specific
position $i$ and $n$ is the length of the session. We combine the position
information with item representations through concatenation and non-linear
transformation:

(7)
$\small\begin{split}\mathrm{z}_{i}=\tanh\left(\mathbf{W}_{2}\left[\mathbf{h}_{v_{i}}^{\prime}\|\mathbf{p}_{n-i+1}\right]+\mathbf{b}_{2}\right),\quad\mathbf{s}^{\prime}&=\frac{1}{n}\sum_{i=1}^{n}\mathbf{h}_{v_{i}}^{\prime},\end{split}$

where parameters $\mathbf{W}_{2}\in\mathbb{R}^{d\times 2d}$ and
$\mathbf{b}_{2}\in\mathbb{R}^{d}$ are trainable parameters,
$\mathbf{s}^{\prime}$ is the session information computed as the average of
representations of items in the session.

Next, we adopt a soft-attention mechanism to learn the contribution of item
$v_{i}$ to the next prediction, and then we can obtain the session
representation by linearly combining the item representations:

(8)
$\small\begin{split}\beta_{i}=\mathbf{q}^{\top}\sigma\left(\mathbf{W}_{3}\mathbf{z}_{i}+\mathbf{W}_{4}\mathbf{s}^{\prime}+\mathbf{b}_{3}\right),\quad\mathbf{s}=\sum_{i=1}^{l}\beta_{i}\mathbf{h}_{v_{i}}^{\prime},\end{split}$

where $\mathbf{W}_{3},\mathbf{W}_{4}\in\mathbb{R}^{d\times d}$ and
$\mathbf{q},\mathbf{b}_{3}\in\mathbb{R}^{d}$ are learnable parameters.

###  4.5. Prediction Layer

We first utilize dot product and then apply softmax function to predict the
click probability $\hat{\mathbf{y}}$ for the item $v_{i}$ :

(9)
$\small\hat{\mathbf{y}}_{i}=\operatorname{Softmax}\left(\mathbf{s}^{\top}\mathbf{h}_{v_{i}}\right),$

where $\hat{\mathbf{y}}_{i}\in\hat{\mathbf{y}}$ denotes the probability of
item $v_{i}$ to be the true next item. The loss function is defined as the
cross-entropy of the prediction results.

##  5\. Experiments and Analysis

###  5.1. Experimental Settings

####  5.1.1. Datasets.

We select two datasets from the Amazon
dataset222https://nijianmo.github.io/amazon/index.html for our experiments:
_Pet Supplies_ and _Movies and TV_. The datasets contain purchase history of
users and users’ reviews for products. Following a common manner, we remove
items appearing less than 5 times. Following (Song et al., 2019), we split
user’s purchase behaviors into week-long sessions. To evaluate our method more
comprehensively, we prepare two versions for each dataset. The first version
(Case 1) keeps all sessions with more than 1 item (Li et al., 2017; Wang et
al., 2020; Wu et al., 2019) while the second version (Case 2) keeps sessions
with more than 5 items (Wu et al., 2019). Obviously, Case 2 is a subset of
Case 1 which keeps long sessions only. We set the sessions of last year as the
test data, and the remaining sessions for training. Then, we adopt sequence
splitting preprocessing method which is commonly adopted in SBR. For an input
session $[v_{1}^{s},v_{2}^{s},\ldots,v_{n}^{s}]$, we generate multiple input
sequence-label pairs, i.e.,
$([v_{1}^{s}],v_{2}^{s}),([v_{1}^{s},v_{2}^{s}],v_{3}^{s})$, …,
$([v_{1}^{s},v_{2}^{s},\ldots,v_{n-1}^{s}],v_{n}^{s})$.

After the preprocessing, the statistic of datasets are summarized in Table 1.

Table 1. Statistic details of datasets. Dataset | Pet Supplies | Movies and TV
---|---|---
# Interactions | 1416,391 | 2114,752
# Train sessions | 894,646 | 586,696
# Test sessions | 99,024 | 20,823
# All the items | 30,694 | 46,341
# Average session length | 3.35 | 3.48

####  5.1.2. Evaluation Metrics and Baselines.

Following (Wang et al., 2020; Wu et al., 2019), we adopt P@K (Precision) and
MRR@K (Mean Reciprocal Rank) as evaluation metrics. We compare RI-GNN with the
following representative methods, including S-POP (Jannach and Ludewig, 2017),
S-KNN (Jannach and Ludewig, 2017), GRU4Rec (Hidasi et al., 2016), NARM (Li et
al., 2017), STAMP (Liu et al., 2018), BERT4Rec (Sun et al., 2019), SR-GNN (Wu
et al., 2019), GCE-GNN (Wang et al., 2020) and DHCN (Xia et al., 2021). (1)
S-POP recommends the top-_N_ frequent items in the current session. (2) S-KNN
(Jannach and Ludewig, 2017) is a session-based k-nearest-neighbors approach.
(3) GRU4Rec (Hidasi et al., 2016) applies RNN to SBR, which adopts GRU to
model short-term user preferences. (4) NARM (Li et al., 2017) is a RNN-based
model that utilizes attention mechanism to capture user’s main intents and
sequential behavior. (5) STAMP (Liu et al., 2018) employs MLP and attention
mechanism to capture the long-term and short-term interests of users,
respectively. (6) BERT4Rec (Sun et al., 2019) introduces the Bert model to
capture item transitions for the sequential recommendation. (7) SR-GNN (Wu et
al., 2019) converts each of session to a directed graphs and updates item
embeddings by gated GNN. (8) GCE-GNN (Wang et al., 2020) captures item-
transitions from all sessions, not just a single session. (9) DHCN (Xia et
al., 2021) captures the complex high-order information among items by modeling
session-based data as a hypergraph.

####  5.1.3. Hyper-parameters Settings.

Following previous studies (Li et al., 2017; Wang et al., 2020; Wu et al.,
2019), the dimension of node embedding is 100, the size of mini-batch is 100,
and the $L_{2}$ regularization is $10^{-5}$ for all models. For RI-GNN, we use
the Adam optimizer with the initial learning rate 0.001. The dropout ratio of
session graph is 0.2. Moreover, the number of topics are empirically set to 24
and 20 on _Pet Supplies_ and _Movies and TV_ dataset, respectively.

####  5.1.4. Overall Performance.

Table 2. Experimental Results on Sessions with More than 1 Item.

Dataset | Pet Supplies | Movies and TV
---|---|---
Metrics | P@10 | P@20 | MRR@10 | MRR@20 | P@10 | P@20 | MRR@10 | MRR@20
S-POP | 6.52 | 6.52 | 5.25 | 5.26 | 1.83 | 1.84 | 1.59 | 1.59
S-KNN | 21.86 | 24.30 | 12.22 | 12.39 | 20.33 | 23.46 | 8.40 | 8.62
GRU4Rec | 8.79 | 9.84 | 6.26 | 6.34 | 8.32 | 9.47 | 5.64 | 5.72
NARM | 24.48 | 27.68 | 19.48 | 19.70 | 22.40 | 25.40 | 17.05 | 17.26
STAMP | 21.50 | 24.66 | 16.41 | 16.60 | 20.59 | 23.96 | 14.69 | 14.92
BERT4Rec | 24.18 | 27.40 | 19.29 | 19.51 | 23.21 | 26.46 | 17.75 | 17.97
SR-GNN | 24.64 | 27.81 |  19.53 |  19.75 | 23.69 | 26.74 |  18.24 |  18.45
GCE-GNN |  24.73 |  28.28 | 18.16 | 18.41 |  24.25 |  27.64 | 17.02 | 17.25
DHCN | 24.23 | 27.45 | 17.80 | 18.02 | 23.33 | 26.49 | 15.94 | 16.16
RI-GNN | $\mathbf{25.43}^{*}$ | $\mathbf{28.88}^{*}$ | $\mathbf{19.93}^{*}$ | $\mathbf{20.15}^{*}$ | $\mathbf{24.69}^{*}$ | $\mathbf{27.94}^{*}$ | $\mathbf{18.93}^{*}$ | $\mathbf{19.14}^{*}$
Improv.(%) | 2.83 | 2.12 | 2.05 | 2.02 | 1.81 | 1.09 | 3.78 | 3.74

  * 1

1 The best results of each column are highlighted in boldface, the suboptimal
one is underlined, the improvements are calculated by using the difference
between the performance of our proposed RI-GNN and the best baseline, and *
denotes the significant difference for t-test.

Table 3. Experimental Results on Sessions with More than 5 Items.

Dataset | Pet Supplies | Movies and TV
---|---|---
Metrics | P@10 | P@20 | MRR@10 | MRR@20 | P@10 | P@20 | MRR@10 | MRR@20
NARM | 26.07 | 30.34 | 19.33 | 19.62 | 20.98 | 24.73 | 14.27 | 14.53
STAMP | 24.44 | 28.06 | 18.28 | 18.53 | 21.27 | 25.40 | 14.49 | 14.77
BERT4Rec | 25.85 | 30.04 |  19.49 |  19.77 | 22.24 | 26.43 | 15.87 | 16.15
SR-GNN |  26.29 |  30.49 | 19.31 | 19.60 | 22.73 | 26.57 |  16.11 |  16.37
GCE-GNN | 25.95 | 30.36 | 17.51 | 17.81 |  23.18 |  27.51 | 14.39 | 14.71
DHCN | 24.96 | 29.20 | 17.25 | 17.54 | 21.83 | 25.71 | 13.01 | 13.27
RI-GNN | $\mathbf{27.72}^{*}$ | $\mathbf{32.11}^{*}$ | $\mathbf{20.26}^{*}$ | $\mathbf{20.57}^{*}$ | $\mathbf{24.15}^{*}$ | $\mathbf{28.30}^{*}$ | $\mathbf{16.99}^{*}$ | $\mathbf{17.27}^{*}$
Improv.(%) | 5.44 | 5.31 | 3.95 | 4.05 | 4.19 | 2.87 | 5.46 | 5.50

The experimental results of overall performance on sessions of different
lengths are reported in Table 2 and Table 3 respectively, according to the
tables, we can draw the following conclusions:

Case 1 (Performance on sessions with more than 1 item): (1) Traditional
methods (i.e., S-POP, S-KNN) show a significant inferiority to neural methods,
except for GRU4Rec. This demonstrates that neural networks can learn more
sophisticated features than the traditional methods. (2) Neural network based
methods (i.e., GRU4Rec, NARM, STAMP, BERT4Rec) usually have better performance
for SBR. GRU4Rec shows worse performance, which is probably because it
strictly defines a session as a sequence, and does not consider the diversity
of user preferences. The other neural methods (i.e., NARM, STAMP, BERT4Rec)
outperform GRU4Rec significantly. Among them, NARM combines RNN and attention
mechanism, STAMP and BERT4Rec are completely based on attention mechanism.
This result demonstrates that attention-based methods are also an effective
way besides the RNN-based methods. (3) Among all the baseline methods, the
GNN-based methods (i.e. SR-GNN, GCE-GNN and DHCN) demonstrate the superiority
over the others. They convert each session into a graph and model item
dependencies with GNN. This indicates that GNN-based models would be more
effective than RNN-based and attention-based models when capturing the complex
dependencies between items in SBR. (4) It is obvious that our proposed RI-GNN
model outperforms all the baselines on all datasets. Compared with the GNN-
based methods, the superiority of RI-GNN may due to that it is equipped with
RIL and review information, which can model item dependencies more accurately.

Case 2 (Performance on sessions with more than 5 items): Table 3 shows that
our proposed RI-GNN model can outperform the state-of-the-art SBR methods by
2.87% to 5.50% on different metrics. Compared with the result in Table 2, RI-
GNN outperforms state-of-the-art SBR methods with a larger margin on long
sessions. This demonstrates the superiority of RI-GNN is more outstanding for
the long sessions. The reason may be that there exist more dependencies
between non-adjacent items within the long sessions. This further verifies our
hypothesis.

Figure 4. Comparison of Ablation Variants.

####  5.1.5. Ablation Study.

To investigate the effectiveness of RIL component and review information for
RI-GNN, we implement three variants of RI-GNN, denoted as RI-GNN-RIL, RI-GNN-
topic, and RI-GNN-review, by removing RIL, topic and review respectively. The
comparison of them is shown in Fig. 4. It is clear that all variants are
inferior to the standard RI-GNN, which demonstrates that RIL, topics and
review information are critical and necessary for the success of RI-GNN model.

##  6\. Conclusion

In this paper, we rethink adjacent dependencies in SBR, argue that adjacent
items in a session are not always dependent and non-adjacent items are not
necessarily independent. Accordingly, we propose a novel review-refined inter-
item graph neural network (RI-GNN), which leverages reviews to reduce the
false dependencies between adjacent but actually independent items and capture
true dependencies between non-adjacent but dependent items. Empirical
evaluations on two real-world datasets demonstrate the superiority of RI-GNN
over the state-of-the-art methods. In the future, we will explore more
powerful methods to refine the dependencies within sessions.

Acknowledgment. Wenpeng Lu is the corresponding author. The research work is
partly supported by National Natural Science Foundation of China under Grant
No.11901325 and No.61502259, National Key R&D Program of China under Grant
No.2018YFC0831700, Key Program of Science and Technology of Shandong Province
under Grant No.2020CXGC010901 and No.2019JZZY020124, and Natural Science
Foundation of Shandong Province under Grant ZR2021MF079.

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