# Consistency and Discrepancy-Based Contrastive Tripartite Graph Learning for
Recommendations

Linxin Guo<EMAIL_ADDRESS>Chongqing UniversityChongqingChina ,  Yaochen
Zhu<EMAIL_ADDRESS>University of VirginiaCharlottesvilleUSA ,  Min Gao
<EMAIL_ADDRESS>Chongqing UniversityChongqingChina ,  Yinghui Tao
<EMAIL_ADDRESS>Institute of Guizhou Aerospace Measuring and Testing
TechnologyGuiyangChina ,  Junliang Yu<EMAIL_ADDRESS>the University of
QueenslandQueenslandAustralia and  Chen Chen<EMAIL_ADDRESS>University
of VirginiaCharlottesvilleUSA

(2024)

###### Abstract.

Tripartite graph-based recommender systems markedly diverge from traditional
models by recommending unique combinations such as user groups and item
bundles. Despite their effectiveness, these systems exacerbate the long-
standing cold-start problem in traditional recommender systems, because any
number of user groups or item bundles can be formed among users or items. To
address this issue, we introduce a Consistency and Discrepancy-based graph
contrastive learning method for tripartite graph-based Recommendation (CDR).
This approach leverages two novel meta-path-based metrics—consistency and
discrepancy—to capture nuanced, implicit associations between the recommended
objects and the recommendees. These metrics, indicative of high-order
similarities, can be efficiently calculated with infinite graph convolutional
networks (GCN) layers under a multi-objective optimization framework, using
the limit theory of GCN. Additionally, we introduce a novel Contrastive
Divergence (CD) loss, which can seamlessly integrate the consistency and
discrepancy metrics into the contrastive objective as the positive and
contrastive supervision signals to learn node representations, enhancing the
pairwise ranking of recommended objects and proving particularly valuable in
severe cold-start scenarios. Extensive experiments demonstrate the
effectiveness of the proposed CDR. The code is released at
https://github.com/foodfaust/CDR.

Tripartite Graph-Based Recommendations, Extreme Cold-Start, Consistency and
Discrepancy

††ccs: Information systems Recommender systems††journalyear: 2024††copyright:
acmlicensed††conference: Proceedings of the 30th ACM SIGKDD Conference on
Knowledge Discovery and Data Mining ; August 25–29, 2024; Barcelona,
Spain.††booktitle: Proceedings of the 30th ACM SIGKDD Conference on Knowledge
Discovery and Data Mining (KDD ’24), August 25–29, 2024, Barcelona,
Spain††isbn: 979-8-4007-0490-1/24/08††doi: 10.1145/3637528.3672056

##  1\. Introduction

With the rapid development of the Internet, recommender systems have become
crucial in online platforms (Lü et al., 2012; Cheng et al., 2016, 2021).
Traditional systems, modeling the interactions between user and item, have
achieved significant results in both industry and academia (Rendle et al.,
2009; He et al., 2020; Wang et al., 2019; Zhu and Chen, 2022a). Recently,
recommender system research has embraced a trend of formulating users, items,
and their interactions as a bipartite graph, prompting the development of
bipartite graph-based methods like NGCF (Chen et al., 2019c) and LightGCN (He
et al., 2020), which have notably advanced the field. However, tripartite
graph-based recommendation, involving interactions among users, items, and
user groups/item bundles (e.g., group recommendations (Guo et al., 2020;
Sankar et al., 2020; Zhang et al., 2021a) and bundle recommendations (Chang et
al., 2023; Chen et al., 2019a; Ma et al., 2022; Liu et al., 2017)), has
recently emerged and garnered substantial research interest due to its unique
interaction model among three heterogeneous entities/nodes (Cao et al., 2018;
Ghaemmaghami and Salehi-Abari, 2021; Pathak et al., 2017; Bai et al., 2019).

Tripartite graph-based recommendations differ substantially from general
recommender systems. For instance, group recommendations suggest items to
diverse-interest user groups, e.g., recommending a travel route to a group of
tourists, and bundle recommendations propose multiple items to a user
simultaneously. Unlike bipartite graphs that encompass interactions between
two entities (users and items), tripartite graphs engage three types of
entities: users, items, and groups/bundles, allowing arbitrary groupings of
users and items. Despite the relative abundance of user-item interactions,
interactions between recommended objects (user/group) and recommendees
(bundle/item) are markedly sparse (Hao et al., 2023, 2021; Zhu and Chen,
2022b) (demonstrated in Fig. 1). The complexities and sparsity, particularly
noticeable in scenarios like group recommendations where groups engage with
fewer items, inhibit the efficacy of standard pair-wise ranking methods during
positive and negative sampling. This exacerbates the difficulty of accurately
modeling group nodes’ latent semantics in a tripartite graph and,
subsequently, making precise recommendations.

Figure 1. Comparison of bipartite and tripartite graph-based recommendation.

Mitigation of data sparsity has been critically addressed through various
sophisticated mechanisms in group and bundle recommendation models. Attention
mechanisms manage data sparsity and cold-start issues in group or bundle
recommendations by dynamically prioritizing interactions and aggregating item
significance (Guo et al., 2020; Cao et al., 2018; Ma et al., 2022). Multi-task
learning is also an effective way, which enhances robustness in user-group and
user-item interactions by jointly optimizing parallel tasks and effectively
utilizing sparse data through integrating various interaction types (Jia et
al., 2021; Chen et al., 2019a). Additionally, graph-based approaches navigate
data sparsity by traversing user-item, user-group, and user-bundle
interactions within a heterogeneous graph, facilitating information
propagation and ensuring cohesive embedding of item, group, and bundle
semantics (Chang et al., 2023). Thus, attention application, multi-task
learning, and graph-based propagation, skillfully interweave varied
interactions and affiliations to adeptly mitigate challenges in sparse data
scenarios. However, these methods face challenges in extreme cold-start
scenarios due to their reliance on a minimum amount of positive and negative
samples for pair-wise ranking training in the recommendation-related loss.
Details can be seen in Appendix A.

Addressing challenges in tripartite graph-based recommendations involves two
core issues. First, the substantial sparsity in interactions between
recommended objects and recommendees necessitates novel metrics to bridge the
semantic gap between lower user-item interactions and user-group/item-bundle
affiliations and higher relationships between recommended objects and
recommendees. Second, the effective use of metrics as proxy supervision
signals becomes critical to guide the training of the recommendation model,
ensuring refined embeddings for recommended objects and recommendees with rare
interactions, which facilitates accurate future recommendations.

To address these challenges, we propose a Consistency and Discrepancy-based
contrastive learning Recommendation method, i.e., CDR, for tripartite graph-
based recommendation. We initiate with two new metrics, consistency and
discrepancy, between the recommended object and recommendee nodes in the
tripartite graph, defining a hard multi-objective optimization problem to
establish high-order relationships based on abundant user-item interactions
and affiliations. We define reachable and non-reachable meta-path between
recommended objects and recommendees, mediated/blocked by other nodes in the
tripartite graph. We efficiently pre-calculate two coefficients related to the
solution, encouraging similar (consistency) or dissimilar (discrepancy)
embeddings of recommended objects and recommendees, utilizing graph
convolutional networks (GCN) limit theory.

Subsequently, a contrastive inspired CD loss, integrating the designed
consistency and discrepancy metrics as positive and contrastive supervision
signals, is introduced to train the model without requiring direct
interactions between objects and recommendees. Furthermore, when direct
interactions become available, they can fine-tune consistency and discrepancy
metrics, enhancing recommendation accuracy. The contributions of this paper
can be concretely summarized into four folds as follows:

  * •

We propose a new contrastive graph learning-inspired tripartite graph-based
recommender system, i.e., CDR, based on new supervision signals and a new
optimization procedure. CDR is shown to be robust to the extreme cold-start
case where no direct interaction exists between the recommended objects and
the recommendees.

  * •

Two novel metrics, i.e., consistency and discrepancy, are proposed by solving
a multi-objective optimization problem based on reachable/non-reachable meta-
paths between recommended objects and recommendees, with available user-item
interactions and user-group/item-bundle affiliations. The metrics can be
efficiently pre-calculated before optimization with GCN limit theory.

  * •

Furthermore, a contrastive learning-inspired novel CD loss is proposed
accordingly, where the introduced consistency and discrepancy metrics can be
seamlessly integrated into the contrastive objective as the positive and
contrastive supervision signals to learn node representations, without the
requirement of available direct user-bundle/group-item interactions.

  * •

Extensive experiments on real-world datasets demonstrate the superiority of
the proposed CDR model, where the advantages of the proposed consistency and
discrepancy metrics and the contrastive learning-based CD loss are clearly
demonstrated by the ablation study.

##  2\. Preliminaries

In this section, we introduce the symbols used in this paper, define related
concepts, and formally formulate the problem of tripartite graph-based
recommendation as the preliminary knowledge of the proposed CDR model.

Table 1. Summary of notations.

Notation | Descriptions
---|---
$\mathcal{T}$, $\mathcal{O}$, $\mathcal{M}$ | The sets of tuple, object, and member nodes
$|\mathcal{T}|$, $|\mathcal{O}|$, $|\mathcal{M}|$ | The number of tuple, object, and member nodes
$t$, $o$, $m$ | A tuple / object / member node
($v_{1}$, $v_{2}$) | A node pair
$\boldsymbol{Y}$, $\boldsymbol{X}$, $\boldsymbol{Z}$ |  | The matrices of tuple interactions, member
---
interactions, and tuple-member affiliations
$e_{t}$, $e_{o}$, $e_{m}$ | The embedding of tuple, object, and member nodes
$\mathcal{N}(t)$, $\mathcal{N}(o)$, $\mathcal{N}(m)$ |  | The set of neighbors of tuple node $t$, object node $o$,
---
and member node $m$
$deg_{v}$, $\delta_{v_{1},v_{2}}$ |  | The degree of node $v$ and the composite term of the
---
degrees of nodes $v_{1}$ and $v_{2}$
$c_{v_{1},v_{2}}$, $d_{v_{1},v_{2}}$ | The consistency and discrepancy metrics
$\boldsymbol{C}^{\Phi}$, $\boldsymbol{D}^{\Phi}$ |  | The consistency and discrepancy sub-matrices extracted
---
from meta-path $\Phi$
$\boldsymbol{C}_{tuple}$, $\boldsymbol{D}_{tuple}$ |  | The consistency and discrepancy matrices extracted
---
from tuple interactions
$\boldsymbol{C}_{member}$, $\boldsymbol{D}_{member}$ |  | The consistency and discrepancy matrices extracted
---
from member interactions and tuple-member
affiliations

###  2.1. Relevant Concepts

###### Definition 1.

Tripartite Graph (Sun and Han, 2013). Given three node sets $\mathcal{T}$,
$\mathcal{O}$, and $\mathcal{M}$,
$\mathcal{V}=\mathcal{T}\cup\mathcal{O}\cup\mathcal{M}$, a graph
$G=(\mathcal{V},\mathcal{E}$) is called a tripartite graph, if the edge set
$\mathcal{E}\subseteq\mathcal{T}\times\mathcal{O}\cup\mathcal{O}\times\mathcal{M}\cup\mathcal{T}\times\mathcal{M}$,
where $\times$ represents Cartesian product. In tripartite graph-based
recommendation, $G$ is a heterogeneous graph since $\mathcal{T}$,
$\mathcal{O}$, and $\mathcal{M}$ represent three different types of nodes,
i.e., user, item, and user/item groups.

###### Definition 2.

Tripartite Graph-Based Recommendation. The tripartite graph-based
recommendation contains three types of entities, i.e., user, item, and
user/item group (where item groups are known as bundles in the literature). To
simplify the discussion, we use the term tuples to represent user groups and
item bundles, use the term members to represent users in group recommendation
and items in bundle recommendation, and use the term objects to represent
items in group recommendation and users in bundle recommendation. For
simplicity, we use tuple interactions to represent the interaction between
tuple and object, and member interactions to represent interactions between
member and object.

Tripartite graph-based recommendation unifies group and bundle recommendations
as the interaction prediction problem between tuples and objects, where the
proposed CDR can be viewed as a universal solution.

###### Definition 3.

Meta-Path (Sun et al., 2011). A (reachable) meta-path $\Phi$ defined on the
heterogeneous graph $G=(\mathcal{V},\mathcal{E})$ is a path pattern in the
form of $V_{1}-V_{2}-\cdots-V_{l+1}$, where
$V_{k}\in\\{\mathcal{T},\mathcal{O},\mathcal{M}\\}$. Specifically, $\Phi$
describes the composite relationship $R=R_{V_{1}V_{2}}\circ
R_{V_{2}V_{3}}\circ\cdots\circ R_{V_{l}V_{l+1}}$ between the head node $V_{1}$
and the tail node $V_{l+1}$, where $\circ$ denotes the composition operator
and $R_{V_{k}V_{k+1}}$ denotes the pairwise relationship defined in the edge
set $\mathcal{E}$.

We extend the meta-path schema definition to include non-reachable information
between nodes, allowing explicit consideration of negative pairwise
relationships. Specifically, Tuple$\sim$ Member$-$Tuple represents non-
reachable meta-path schemas, with ”$-$” signifying an edge and ”$\sim$” no
edge between nodes, as depicted in Fig. 2.

(a) Reachable meta-path Tuple-Member-Tuple.

(b) Non-reachable meta-path Tuple$\sim$Member-Tuple.

Figure 2. Two kinds of meta-paths, as examples, for deriving consistency and
discrepancy metrics.

###  2.2. Problem Formulation

Consider the sets of tuple nodes
$\mathcal{T}=\left\\{t_{1},\ldots,t_{|\mathcal{T}|}\right\\}$, object nodes
$\mathcal{O}=\left\\{o_{1},\ldots,o_{|\mathcal{O}|}\right\\}$, and member
nodes $\mathcal{M}=\left\\{m_{1},\ldots,m_{|\mathcal{M}|}\right\\}$, where
$|\mathcal{T}|$, $|\mathcal{O}|$, and $|\mathcal{M}|$ represent the number of
the tuple, object, and member nodes, respectively. Three relations could exist
among $\mathcal{T}$, $\mathcal{O}$, and $\mathcal{M}$, i.e., tuple
interactions, member interactions, and member-tuple affiliations. We use a
sparse matrix $\boldsymbol{Y}\in\\{0,1\\}^{|\mathcal{T}|\times|\mathcal{O}|}$
to represent tuple interactions, where $y_{to}=1$ indicates that the tuple
node $t$ has interacted with the object $o$, and $y_{to}=0$ indicates
otherwise. Since members can be arbitrarily grouped into tuples,
$\boldsymbol{Y}$ can be extremely sparse, and in the extreme cold-start case,
$\boldsymbol{Y}$ can be an empty matrix. Similarly, member interactions are
represented by the sparse matrix
$\boldsymbol{X}\in\\{0,1\\}^{|\mathcal{M}|\times|\mathcal{O}|}$, where
$x_{mo}=1$ indicates that the member $m$ has interacted with the object $o$,
and $x_{mo}$ = 0 indicates otherwise. $\boldsymbol{X}$ is usually much denser
than $\boldsymbol{Y}$ due to the comparatively abundant member interactions.
Finally, we denote the observed affiliations as
$\boldsymbol{Z}\in\\{0,1\\}^{|\mathcal{T}|\times|\mathcal{M}|}$, where the
element $z_{tm}=1$ means the member $m$ is affiliated to the tuple $t$, and
$z_{tm}=0$ otherwise.

The aim of tripartite graph-based recommendation is to generate a top-K list
of objects (bundles/items) for the recommendee (users/user groups) using
member interactions $\boldsymbol{X}$, tuple interactions $\boldsymbol{Y}$, and
tuple-member affiliations $\boldsymbol{Z}$. It also addresses the extreme
cold-start where no tuple interactions (i.e., $\boldsymbol{Y}$ is a zero-
matrix) exist. Table 1 summarizes symbols.

##  3\. Methodology

###  3.1. Tuple-Object Metric Learning

In the first stage of the proposed CDR algorithm, we compute consistency and
discrepancy metrics using defined reachable/non-reachable meta-paths and
abundant member interactions plus member-tuple affiliations. These metrics
gauge the similarity or difference between pairs of recommended object and
recommendee nodes, crucial for the positive and negative relations needed by
the contrastive learning-inspired CD loss, discussed subsequently.

####  3.1.1. Optimization Objective

The derivation of consistency and discrepancy metrics begins with an auxiliary
graph embedding learning task, intending to map all nodes of the tuple-member-
object tripartite graph into an embedding space while maximally preserving
tuple-member affiliation and object-member interaction information in the node
embeddings. Specifically, utilizing the positive and negative pairwise
relationships between tuples/members and members/objects, we aim to optimize
the following objectives:

(1)
$\max\sum_{m\in\mathcal{N}(t)}e_{t}^{\top}e_{m},\quad\min\sum_{\tilde{m}\notin\mathcal{N}(t)}e_{t}^{\top}e_{\tilde{m}},$
(2)
$\max\sum_{m\in\mathcal{N}(o)}e_{o}^{\top}e_{m},\quad\min\sum_{\tilde{m}\notin\mathcal{N}(o)}e_{o}^{\top}e_{\tilde{m}},$

where $e$ is the node embedding, $\mathcal{N}(t)$ and $\mathcal{N}(o)$ are the
neighbor sets of tuple $t$ and object $o$, and $m$ and $\tilde{m}$ are the
first-order member neighbors and non-first-order neighbor nodes (members) of
tuple $t$ and object $o$, respectively. To simplify the discussion, we only
consider nodes $t,m,\tilde{m},o$ in the objective, but the generalization of
Eqs. (1) and (2) to include all nodes in the graph is straightforward.

Prior research primarily optimizes object and member embeddings (user-item
interactions) as per Eq. (2), often using techniques like attention mechanisms
(Vaswani et al., 2017) to aggregate member embeddings into tuple embeddings,
neglecting Eq. (1). Such simple aggregation from member to tuple level lacks
theoretical and explanatory backing. Contrarily, our method effectively
optimizes all four objectives in Eqs. (1) and (2) concurrently.

####  3.1.2. Message Passing Mechanism

Optimizing Eq. (2) involves message passing among heterogeneous nodes.
Utilizing the LightGCN strategy, celebrated for effectively mining bipartite
graph pair-wise relations (He et al., 2020), we employ it as our message-
passing mechanism. The user embedding information aggregation process in the
LightGCN network, incorporating self-connection, is formalized as follows:

(3)
$\begin{split}e_{m}^{(l+1)}=\frac{1}{deg_{m}+1}e_{m}^{(l)}&+\sum_{t\in\mathcal{N}(m)}\frac{1}{\sqrt{deg_{m}+1}\sqrt{deg_{t}+1}}e_{t}^{(l)}\\\
&+\sum_{o\in\mathcal{N}(m)}\frac{1}{\sqrt{deg_{m}+1}\sqrt{deg_{o}+1}}e_{o}^{(l)},\end{split}$

where $e^{(l)}$ represents the node embedding at layer $l$, and $deg_{v}$
represents the degree of node $v$, respectively.

####  3.1.3. Solution Based on the Limit Theory of GCN

The direct generalization of Eq. (3) to solve Eqs. (1) and (2) is
computationally expensive due to multiple aggregation objectives. Moreover, it
is difficult to decide how many aggregation processes are needed to acquire a
good embedding conducive to recommendation. Inspired by UltraGCN(Mao et al.,
2021), we note that a convergent form of $e_{m}$ can be obtained with infinite
propagation layers. Specifically, according to the aggregation formula of
graph convolution, after passing through infinite layers of aggregation, the
embeddings of the model will converge to the following form:

(4)
$e_{m}=\lim_{l\rightarrow\infty}e_{m}^{(l+1)}=\lim_{l\rightarrow\infty}e_{m}^{(l)},$

where the node embeddings are consistent before and after the convolution in
the limit. Based on Eq. (4), the message passing mechanism defined in Eq. (3)
can be simplified as:

(5) $\displaystyle
e_{m}=\sum_{t\in\mathcal{N}\left(m\right)}\frac{1}{deg_{m}}\sqrt{\frac{deg_{m}+1}{deg_{t}+1}}e_{t}+\sum_{o\in\mathcal{N}\left(m\right)}\frac{1}{deg_{m}}\sqrt{\frac{deg_{m}+1}{deg_{o}+1}}e_{o}.$

We denote the composite term of the degree of member $m$ and tuple $t$ (object
$o$) as $\delta_{m,t}$ ($\delta_{m,o}$) defined as follows:

(6)
$\delta_{m,t}=\frac{1}{deg_{m}}\sqrt{\frac{deg_{m}+1}{deg_{t}+1}},\quad\delta_{m,o}=\frac{1}{deg_{m}}\sqrt{\frac{deg_{m}+1}{deg_{o}+1}},$

Eq. (5) can be simplified into the following form:

(7)
$e_{m}=\sum_{t\in\mathcal{N}\left(m\right)}\delta_{m,t}e_{t}+\sum_{o\in\mathcal{N}\left(m\right)}\delta_{m,o}e_{o}.$

Essentially, Eq. (7) uses tuple embeddings and object embeddings to represent
member embeddings. To further utilize tuple-member affiliations, we can
substitute Eq. (7) into Eq. (1) and maximize the two parts simultaneously to
approximate the final result, where the objective can be re-formulated as:

(8)
$\begin{split}\max\sum_{m\in\mathcal{N}(t)}e_{t}^{\top}e_{m}\simeq&\max\sum_{m\in\mathcal{N}(t)}\sum_{t^{\prime}\in\mathcal{N}\left(m\right)}\delta_{m,t^{\prime}}e_{t}^{\top}e_{t^{\prime}}+\\\
&\max\sum_{m\in\mathcal{N}(t)}\sum_{o\in\mathcal{N}\left(m\right)}\delta_{m,o}e_{t}^{\top}e_{o}.\end{split}$

Since the member embedding $e_{m}^{\prime}$ is eliminated from the righ-hand
side of Eq. (8), we can utilize the relationship of member nodes without
training their embeddings.

####  3.1.4. Solution with Reachable/Non-Reachable Meta-Paths

While Eq. (8) is notably simplified, it still demands gradient-based
optimization. The optimization is further streamlined by introducing reachable
and non-reachable meta-paths from Section 2.1, allowing the maximization
portion of Eq. (8) to be redefined as follows:

(9) $\begin{split}\mathcal{L}_{tm\\_max}=&\sum_{\left(t,m,t^{\prime}\right)\in
Q_{tmt^{\prime}}}\delta_{m,t^{\prime}}\left(1-\sigma\left(e_{t}^{\top}e_{t^{\prime}}\right)\right)\\\
+&\sum_{\left(t,m,o\right)\in
Q_{tmo}}\delta_{m,o}\left(1-\sigma\left(e_{t}^{\top}e_{o}\right)\right),\end{split}$

where $\sigma(\cdot)$ is the sigmoid function, $Q_{tmt^{\prime}}$ is the set
of reachable Tuple-Member-Tuple meta-paths ($t,m,t^{\prime}$) composed of
tuple node $t$ and its first-order member neighbors $m$ and second-order tuple
neighbors $t^{\prime}$. Similarly, $Q_{tmo}$ is the set of reachable Tuple-
Member-Object meta-paths ($t,m,o$):

$\displaystyle Q_{tmt^{\prime}}=\left\\{\left(t,m,t^{\prime}\right)\mid
m\in\mathcal{N}(t),t^{\prime}\in\mathcal{N}\left(m\right)\right\\},$ (10)
$\displaystyle Q_{tmo}=\left\\{\left(t,m,o\right)\mid
m\in\mathcal{N}(t),o\in\mathcal{N}\left(m\right)\right\\}.$

The minimization part of Eq. (1) can be reformulated as:

(11)
$\begin{split}\mathcal{L}_{tm\\_min}=\sum_{\left(t,\tilde{m},t^{\prime}\right)\in
Q_{t\tilde{m}t^{\prime}}}\delta_{\tilde{m},t^{\prime}}\sigma\left(e_{t}^{\top}e_{t^{\prime}}\right)+\sum_{\left(t,\tilde{m},o\right)\in
Q_{t\tilde{m}o}}\delta_{\tilde{m},o}\sigma\left(e_{t}^{\top}e_{o}\right).\end{split}$

The set $Q_{t\tilde{m}t^{\prime}}$ diverges from $Q_{tmt^{\prime}}$ by
representing non-reachable Tuple$\sim$Member-Tuple meta-paths. Here, triples
$(t,\tilde{m},t^{\prime})$ in $Q_{t\tilde{m}t^{\prime}}$ denote member
$\tilde{m}$ is not adjacent to tuple $t$, but is to tuple $t^{\prime}$.
Additionally, $Q_{t\tilde{m}o}$ defines non-reachable Tuple$\sim$Member-Object
meta-paths as follows:

$\displaystyle
Q_{t\tilde{m}t^{\prime}}=\left\\{\left(t,\tilde{m},t^{\prime}\right)\mid\tilde{m}\notin\mathcal{N}(t),t^{\prime}\in\mathcal{N}\left(\tilde{m}\right)\right\\},$
(12) $\displaystyle
Q_{t\tilde{m}o}=\left\\{\left(t,\tilde{m},o\right)\mid\tilde{m}\notin\mathcal{N}(t),o\in\mathcal{N}\left(\tilde{m}\right)\right\\}.$

####  3.1.5. Final Objective

Examining the loss functions $\mathcal{L}_{tm\\_max}$ and
$\mathcal{L}_{tm\\_min}$ from Eq. (9) and Eq. (11), the initial terms engage
with $t$ and $t^{\prime}$, while the secondary involve $t$ and $o$. Losses for
pairs $t,t^{\prime}$ and $t,o$ are structured accordingly. Focusing on pair
($t,t^{\prime}$), the losses in $\mathcal{L}_{tm\\_max}$ and
$\mathcal{L}_{tm\\_min}$ integrate as follows:

(13)
$\begin{split}l_{t,t^{\prime}}=&\sum_{m\in\mathcal{N}(t),m\in\mathcal{N}\left(t^{\prime}\right)}\delta_{m,t^{\prime}}\left(1-\sigma\left(e_{t}^{\top}e_{t^{\prime}}\right)\right)\\\
+&\sum_{\tilde{m}\notin\mathcal{N}(t),\tilde{m}\in\mathcal{N}\left(t^{\prime}\right)}\delta_{\tilde{m},t^{\prime}}\sigma\left(e_{t}^{\top}e_{t^{\prime}}\right).\end{split}$

We omit the constant that is independent to $e_{t}$ and $e_{t^{\prime}}$, and
use $c_{t,t^{\prime}}$ and $d_{t,t^{\prime}}$ to denote the constants that
concern only $e_{t}$ and $e_{t^{\prime}}$, where
$c_{t,t^{\prime}}=\sum_{m\in\mathcal{N}(t),m\in\mathcal{N}\left(t^{\prime}\right)}\delta_{m,t^{\prime}},\quad
d_{t,t^{\prime}}=\sum_{\tilde{m}\notin\mathcal{N}(t),\tilde{m}\in\mathcal{N}\left(t^{\prime}\right)}\delta_{\tilde{m},t^{\prime}}$,
the loss function of a node pair ($t,t^{\prime}$) can be rewritten as follows:

(14)
$l_{t,t^{\prime}}=\left(d_{t,t^{\prime}}-c_{t,t^{\prime}}\right)\sigma\left(e_{t}^{\top}e_{t^{\prime}}\right).$

All node pairs of arbitrary type can derive their consistencies $c_{vi,vj}$
and discrepancies $d_{vi,vj}$ by predefined reachable and non-reachable meta-
paths. For an arbitrary node pair ($t,o$), we construct meta-path sets
$Q_{tmo}$ and $Q_{t\tilde{m}o}$ subsequently deriving its loss function with
similar steps applied to other node pair types:

(15) $l_{t,o}=(d_{t,o}-c_{t,o})\sigma(e_{t}^{\top}e_{o}).$

Calculating $c_{t,o}$ and $d_{t,o}$ parallels $c_{t,t^{\prime}}$ and
$d_{t,t^{\prime}}$. Given the asymmetry in calculations like Eq. (14) and Eq.
(15) regarding node pair order, and possible pair types (tuple-tuple, tuple-
object, object-tuple, and object-object), coefficients such as $c_{t,o}$ and
$d_{t,o}$ are also computed. The loss function for arbitrary type node pair
($v_{1},v_{2}$) using meta-paths with user nodes as midpoints is:

(16) $\mathcal{L}=\sum_{v_{1},v_{2}\in\mathcal{T\cup
O}}\left(d_{v_{1},v_{2}}-c_{v_{1},v_{2}}\right)\sigma\left(e_{v_{1}}^{\top}e_{v_{2}}\right),$

where coefficients $c_{v_{1},v_{2}}$ and $d_{v_{1},v_{2}}$ can be pre-
calculated as follows:

(17)
$\begin{split}c_{v_{1},v_{2}}&=\sum_{m\in\mathcal{N}(v_{1}),m\in\mathcal{N}\left(v_{2}\right)}\delta_{m,v_{2}},\\\
d_{v_{1},v_{2}}&=\sum_{\tilde{m}\notin\mathcal{N}(v_{1}),\tilde{m}\in\mathcal{N}\left(v_{2}\right)}\delta_{\tilde{m},v_{2}}.\end{split}$

####  3.1.6. Consistency and Discrepancy Metrics

In general recommendations, coefficients $c_{v_{1},v_{2}}$ and
$d_{v_{1},v_{2}}$ in Eq. (16) determine the embedding distance between
recommended object and recommendee pairs, with $c_{v_{1},v_{2}}$ and
$d_{v_{1},v_{2}}$ respectively encouraging closer and more distant embeddings,
termed as consistency and discrepancy. For
$v_{1},v_{2}\in\mathcal{T}\cup\mathcal{O}$, tuple-object relationships derive
from $c_{v_{1},v_{2}}$ and $d_{v_{1},v_{2}}$ without needing direct
interactions, seamlessly integrating with contrastive learning by providing
positive and contrastive supervision signals.

###  3.2. Contrastive Learning-Inspired CD Loss

Utilizing consistency and discrepancy from subsection 3.1.6, we introduce the
contrastive learning-inspired CD loss here, employing them as positive and
negative supervision signals to formulate the contrastive objective.

####  3.2.1. Motivation of the CD Loss

Given Eq. (16), a direct model training strategy might use a general
recommendation algorithm with $d_{v_{1},v_{2}}-c_{v_{1},v_{2}}$ as supervision
signals. However, in tripartite graph-based recommendations, the subtraction
($d_{v_{1},v_{2}}-c_{v_{1},v_{2}}$) potentially erases unique consistency and
discrepancy information. For example, cases (dv1,v2=1, cv1,v2=0.99) and
(dv1,v2=0.01, cv1,v2=0) both result in a subtraction value of 0.01, despite
significant differences in absolute values. Unlike bipartite recommendations,
where subtractions $d-c$ are clearly 1 or -1, we propose a new contrastive
learning-based objective using consistency and discrepancy independently as
positive and contrastive supervision signals.

####  3.2.2. CD Loss

To further utilize the consistency and discrepancy, inspired by contrastive
learning(Gutmann and Hyvärinen, 2010), we design a CD loss for tripartite
graph-based recommendation, leveraging consistency and discrepancy for any
node pair ($v_{1},v_{2}$):

(18) $\mathcal{L}=\sum_{(v_{1},v_{2})\in
Q}-\log\frac{c_{v_{1},v_{2}}\exp\left(\cos\left(\mathbf{e}_{v_{1}},\mathbf{e}_{v_{2}}\right)/\tau\right)}{\sum_{\tilde{v}_{2}\in\mathcal{T\cup
O}}d_{v_{1},\tilde{v}_{2}}\exp\left(\cos\left(\mathbf{e}_{v_{1}},\mathbf{e}_{\tilde{v}_{2}}\right)/\tau\right)},$

where $Q=\\{(v_{1},v_{2})\mid c_{v_{1},v_{2}}>0\quad and\quad
v_{1},v_{2}\in\mathcal{T\cup O}\\}$ is the set of node pairs whose consistency
is positive, and $\cos\left(\cdot,\cdot\right)$ is the cosine similarity
measure. In Eq. (18), the consistency and discrepancy metrics are
independently used to select positive and negative supervision signals. During
the optimization process, the consistency $c_{v_{1},v_{2}}$ encourages the
embeddings of nodes $v_{1}$ and $v_{2}$ to be close, while the discrepancy
$d_{v_{1},\tilde{v}_{2}}$ constrains the embeddings to be distant with each
other.

The contrastive loss in Eq. (18) introduces a temperature parameter $\tau$,
adjustable based on dataset noise levels; a larger $\tau$ suits higher noise,
focusing model updates on easily distinguishable nodes, while a smaller $\tau$
is used for higher-quality data to strictly differentiate similar nodes.

Figure 3. CDR Model Structure. Upper pre-training with tuple-member
affiliations and member interactions, lower fine-tuning with tuple
interactions, featuring variants CDR-P and CDR-F. CDR-P is for extreme cold-
start scenes, whereas CDR-F is a variant that only uses tuple interactions.

###  3.3. The Overall Procedure of the CDR

Fig. 3 depicts the CDR structure, which computes consistency and discrepancy
metrics between tuples and objects either from member interactions and tuple-
member affiliations (upper part) or solely from tuple interactions (lower
part). The former method, referred to as the pre-training stage, utilizes
member interactions, while the latter can fine-tune metrics using available
tuple interactions, aiding the extraction of node representations.

####  3.3.1. Two Variants of CDR

This subsection details the pre-training and fine-tuning of CDR via two
variants: CDR-P and CDR-F. CDR-P, indicating pre-training, is trained using
metrics from member interactions and tuple-member affiliations, applicable in
extreme cold-start scenarios without tuple interactions. Conversely, CDR-F
(fine-tuning) is trained using tuple interaction metrics, aligning closer with
traditional recommendation models. The full CDR model is formulated by
integrating both CDR-P and CDR-F variants.

We denote the embedding matrices for the pre-training and fine-tuning stages
as $\mathbf{E}^{p}$ and $\mathbf{E}^{f}$, respectively. The matrix
$\mathbf{E}^{p}$ initializes $\mathbf{E}^{f}$ in the fine-tuning phase,
providing improved initial embedding values and fostering quicker model
convergence. We also retain $\mathbf{E}^{p}$ as a static feature and
concatenate it with $\mathbf{E}^{f}$ to establish the final embedding
$\mathbf{E}$:

(19) $\mathbf{E}=\mathbf{E}^{f}\parallel\mathbf{E}^{p},$

where $\parallel$ is the concatenate operation. $\mathbf{E}$ is used for
subsequent fine-tuning and recommendation. Our model only updates
$\mathbf{E}^{f}$ but not $\mathbf{E}^{p}$ in the fine-tuning stage, which can
avoid the node preferences extracted in the pre-training stage being erased by
downstream tasks, where member interaction and member-tuple affiliation
information is substantially preserved.

####  3.3.2. Generalization of Consistency and Discrepancy

We seek to extend the consistency and discrepancy metrics from Section 3.1.6
for the fine-tuning stage. During pretraining, eight meta-path types are
utilized. Specifically, $TMT$, $TMO$, $OMT$, and $OMO$ stand for node types on
meta-paths, and each item can represent both reachable and non-reachable meta-
paths with the same node type. Matrices, subscripted with member, denote
consistency and discrepancy across these meta-paths:

(20)
$\begin{split}\boldsymbol{C}_{member}&=\left[\begin{array}[]{ll}\boldsymbol{C}^{TMT}&\boldsymbol{C}^{TMO}\\\
\boldsymbol{C}^{OMT}&\boldsymbol{C}^{OMO}\end{array}\right],\\\
\boldsymbol{D}_{member}&=\left[\begin{array}[]{ll}\boldsymbol{D}^{TMT}&\boldsymbol{D}^{TMO}\\\
\boldsymbol{D}^{OMT}&\boldsymbol{D}^{OMO}\end{array}\right],\end{split}$

where
$\boldsymbol{C}^{TMT}=\left[c_{t,t^{\prime}}\right]^{|\mathcal{T}|\times|\mathcal{T}|}$
represents the consistency extracted through the meta-path Tuple-Member-Tuple
($TMT$), as in the other sub-matrices. Elements in these matrices are used in
the pre-training stage.

In the fine-tuning stage, with accessible tuple interactions, additional four
types of meta-paths are considered, namely Tuple-Object-Tuple ($TOT$) and
Object-Tuple-Object ($OTO$) for consistency, Tuple$\sim$Object-Tuple ($TOT$),
and Object$\sim$Tuple-Object ($OTO$) for the discrepancy, alongside direct
$TO$ and $OT$ relationships, given the asymmetry of consistency and
discrepancy. Corresponding matrices are denoted as follows:

(21)
$\begin{split}\boldsymbol{C}_{tuple}&=\left[\begin{array}[]{ll}\boldsymbol{C}^{TOT}&\boldsymbol{C}^{TO}\\\
\boldsymbol{C}^{OT}&\boldsymbol{C}^{OTO}\end{array}\right],\\\
\boldsymbol{D}_{tuple}&=\left[\begin{array}[]{ll}\boldsymbol{D}^{TOT}&\boldsymbol{D}^{TO}\\\
\boldsymbol{D}^{OT}&\boldsymbol{D}^{OTO}\end{array}\right].\end{split}$

Given that one-hop meta-paths $TO$ and $OT$ cannot be utilized directly via
Eq. (17), a redefinition is required. Drawing inspiration from (Mao et al.,
2021), we propose a generalized form for consistency and discrepancy by
modifying $\delta$ in Eq. (17) as follows:

(22)
$\boldsymbol{C}^{TO}=\left[\delta_{to}*Y_{to}\right]^{|\mathcal{T}|\times|\mathcal{O}|},\quad\boldsymbol{D}^{TO}=\left[\delta_{to}\right]^{|\mathcal{T}|\times|\mathcal{O}|},$

where $\boldsymbol{Y}\in\\{0,1\\}^{|\mathcal{T}|\times|\mathcal{O}|}$ is the
tuple interaction matrix. The node relationship on the meta-path $OT$ is
similar to $TO$. With the elements in $\boldsymbol{C}_{tuple}$ and
$\boldsymbol{D}_{tuple}$, the model can be fine-tuned to get a better
recommendation performance. The algorithm procedure of CDR is presented in
Appendix B.

##  4\. Experiments

In this section, we detail extensive experiments on various real-world
datasets to validate CDR’s effectiveness, addressing the following four
research questions:

  * •

RQ1 How does CDR compare to baselines in sparse tuple interactions?

  * •

RQ2 How does CDR compare to other methods in extreme cold-start scenarios with
no tuple interactions?

  * •

RQ3 How do CDR’s components benefit its performance?

  * •

RQ4 How does the hyperparameter setting of temperature impact CDR’s
performance?

Furthermore, we conduct extra analysis to compare the recommendation diversity
between CDR and other models. We also apply the pre-training task of CDR to
other models to demonstrate its effectiveness.

###  4.1. Experimental Setup

####  4.1.1. Datasets

We experiment with public datasets Mafengwo, Youshu, and Last-FM. Mafengwo
evaluates group recommendation models with tourist ratings for travel spots,
while Youshu is used for bundle recommendations in book lists. Although Last-
FM is not directly applicable to the tripartite graph-based recommendation, it
can test CDR’s generalization to other tasks due to its contained social
relations. Experiments related to this dataset are given in Appendix C.
Dataset statistics are given in Table 2, and the meta-path types are given in
Appendix E.

Table 2. The statistics of the datasets.

Dataset |  Tuple |  Member |  Object |  T-O |  M-O |  | Avg.M
---
per.T
Mafengwo | 995 | 5,275 | 1,513 | 3,595 | 39,761 | 7.19
Youshu | 4,771 | 32,770 | 8,039 | 51,377 | 138,515 | 37.03
Last-FM |  User |  Item |  U-I | Social relation
1,892 | 17,632 | 92,834 | 25,434

####  4.1.2. Baselines

To validate the proposed CDR model, we juxtapose it with several top-tier
baseline models across diverse recommendation domains, which are given below:

  * •

BPR (Rendle et al., 2009): A classic user-oriented item ranking algorithm.

  * •

LINE (Tang et al., 2015): Embed nodes into low-dimensional space for
recommendation.

  * •

NGCF (Wang et al., 2019): Aggregate information in the interaction graph for
collaborative filtering.

  * •

SimGCL (Yu et al., 2022): Create contrastive views by adding noises to
user/item embedding.

  * •

XSimGCL (Yu et al., 2023): A method which generate contrastive views by
employing a simple noise-based embedding augmentation.

  * •

ALDI (Huang et al., 2023): A cold-start method which transnfer warm items’
behavioral information to cold items.

  * •

BUIR (Lee et al., 2021): Uses only positive samples and data augmentation to
alleviate data sparsity.

  * •

DAM (Chen et al., 2019a): Utilize attention to aggregate item embeddings for
bundle recommendations.

  * •

BGCN (Chang et al., 2023): Unite user, item, and bundle nodes into a
heterogeneous graph for information propagation.

  * •

AGFN (Li et al., 2023): A bundle recommendation method which refines the
neighborhood aggregation mechanism from different aspects.

  * •

AGREE (Cao et al., 2018): Determine group embedding fusion ratios using an
attention mechanism.

  * •

HCR (Jia et al., 2021): Employ a two-channel hypergraph network for multi-
task, group-user interest modeling.

  * •

GroupIM (Sankar et al., 2020): Maximize mutual information between groups and
corresponding users.

  * •

CrossCBR (Ma et al., 2022): Utilize and combines information from bundle and
item views.

  * •

ConsRec (Wu et al., 2023): Explore consensus behind group behavior data.

  * •

SBPR (Zhao et al., 2014): Integrate social relations to augment training
samples for the model.

  * •

DiffNet (Wu et al., 2019): Merge user representations from user-item bipartite
graphs and social networks.

  * •

SocialLGN (Liao et al., 2022): Embed social relations into interaction graphs
and executes graph convolution.

####  4.1.3. Evaluation Metrics

We utilize several metrics for evaluating recommendation performance,
including F1-Score@$K$, Precision@$K$, Recall@$K$, and NDCG@$K$, which are
prevalent in prior research (He et al., 2020; Chang et al., 2023; Jia et al.,
2021).

####  4.1.4. Implementation Details

Addressing tripartite graph-based recommender systems’ cold-start issue, this
study enforces strict data constraints on models necessitating available tuple
interactions. Models are trained using merely 5% of tuple interactions, while
20% are utilized for final performance assessment. Unlike previous work (Chen
et al., 2019a; Cao et al., 2018; Jia et al., 2021) that employs 100 random
negative samples per test data, we apply Top-$K$ ranking on all dataset
items/bundles, enhancing model evaluation accuracy. All models are set to a 64
embedding dimension to maintain fair comparison. Additionally, the early
stopping method’s patience is set at 10, halting training after 10 consecutive
unimproved epochs. Model parameters are updated via Adam (Kingma and Ba,
2015), with a 0.001 learning rate. Temperature hyperparameters $\tau$ for
Mafengwo and Youshu datasets are configured to 1 and 0.3 for fine-tuning, and
3.8 and 1 for pre-training. For Last-FM dataset, $\tau$ is set to 1.2 (pre-
training) and 3 (fine-tuning).

Table 3. Comparison of CDR and baselines on group and bundle recommendation
datasets. Bold indicates best, and underlined shows suboptimal results.

Dataset | Method | R@10 | R@20 | R@30 | N@10 | N@20 | N@30
---|---|---|---|---|---|---|---
Mafengwo | Tuple Interactions
BPR | 0.0604 | 0.1147 | 0.1682 | 0.0317 | 0.0460 | 0.0583
LINE | 0.0243 | 0.0316 | 0.0566 | 0.0143 | 0.0164 | 0.0219
NGCF | 0.0391 | 0.0795 | 0.1135 | 0.0159 | 0.0263 | 0.0334
LightGCN | 0.0766 | 0.1384 | 0.1943 | 0.0385 | 0.0557 | 0.0682
BUIR | 0.0969 | 0.1509 | 0.1928 | 0.0593 | 0.0740 | 0.0839
ColdNAS | 0.0184 | 0.0234 | 0.0282 | 0.0214 | 0.0352 | 0.0501
SGL |  0.1288 | 0.2121 | 0.2626 | 0.0885 |  0.1088 |  0.1212
SimGCL | 0.1263 | 0.2096 | 0.2500 | 0.0522 | 0.0740 | 0.0840
XSimGCL | 0.0400 | 0.0643 | 0.0836 |  0.0988 | 0.0929 | 0.0928
ALDI | 0.1150 |  0.2460 |  0.3203 | 0.0554 | 0.0880 | 0.0999
CDR-F | 0.1954 | 0.2806 | 0.3238 | 0.1268 | 0.1493 | 0.1594
Impro. | 51.71% | 14.07% | 1.09% | 28.34% | 37.22% | 31.52%
Tuple Interaction/Member Interaction/Affiliation
AGREE | 0.0388 | 0.0685 | 0.0994 | 0.0186 | 0.0268 | 0.0338
HCR | 0.0559 | 0.0899 | 0.1219 | 0.0275 | 0.0327 | 0.0388
GroupIM | 0.1453 | 0.2606 | 0.3170 | 0.0796 | 0.1109 | 0.1238
ConsRec |  0.2312 |  0.3085 |  0.3487 |  0.1256 |  0.1449 |  0.1534
CDR | 0.2694 | 0.3833 | 0.4309 | 0.1402 | 0.1715 | 0.1825
Impro. | 16.52% | 24.25% | 23.57% | 11.62% | 18.36% | 18.97%
Youshu | Tuple Interactions
BPR | 0.0188 | 0.0337 | 0.0442 | 0.0122 | 0.0170 | 0.0202
LINE | 0.0234 | 0.0358 | 0.0467 | 0.0170 | 0.0208 | 0.0239
NGCF | 0.0252 | 0.0479 | 0.0676 | 0.0167 | 0.0239 | 0.0298
LightGCN | 0.0362 | 0.0573 | 0.0727 | 0.0269 | 0.0335 | 0.0379
BUIR |  0.0508 |  0.0785 |  0.1034 |  0.0363 |  0.0444 | 0.0510
SGL | 0.0247 | 0.0411 | 0.0530 | 0.0202 | 0.0257 | 0.0299
SimGCL | 0.0313 | 0.0506 | 0.0602 | 0.0263 | 0.0322 | 0.0602
XSimGCL | 0.0349 | 0.0581 | 0.0734 | 0.0331 | 0.0401 | 0.0452
ALDI | 0.0228 | 0.0290 | 0.0612 | 0.0159 | 0.0179 | 0.0199
CDR-F | 0.0648 | 0.1000 | 0.1217 | 0.0398 | 0.0501 |  0.0563
Impro. | 27.41% | 27.32% | 17.78% | 9.59% | 12.74% | -
Tuple Interaction/Member Interaction/Affiliation
DAM | 0.0195 | 0.0316 | 0.0452 | 0.0134 | 0.0169 | 0.0214
CrossCBR | 0.0281 | 0.0462 | 0.0611 | 0.0201 | 0.0262 | 0.0306
BGCN | 0.0543 | 0.0848 | 0.1090 | 0.0358 | 0.0455 | 0.0528
AGFN |  0.0626 |  0.0981 |  0.1247 |  0.0457 |  0.0565 |  0.0652
CDR | 0.0903 | 0.1299 | 0.1520 | 0.0627 | 0.0744 | 0.0809
Impro. | 44.25% | 32.42% | 21.89% | 37.20% | 31.68% | 24.08%

###  4.2. Performance Comparison

To address RQ1, we assess models with highly sparse tuple interactions through
experiments. Table 3 showcases the performance of CDR and baseline models,
yielding the following analyses.

Firstly, CDR significantly surpasses baselines in group and bundle
recommendations, averaging 18.88% and 31.92% improvement on the Mafengwo and
Youshu datasets respectively. CDR-F, using only tuple interactions, remains
highly competitive in general recommendation tasks. Notably, ConsRec and AGFN
perform sub-optimally in extremely sparse settings, sometimes underperforming
models like LightGCN and BUIR that solely utilize tuple interactions. The
issue may stem from the inapplicability of multi-task joint optimization
methods during the cold-start phase due to a severe data imbalance problem:
with 5% of tuple interactions for training amidst abundant member
interactions. Thus, models like AGREE, HCR, and DAM, balancing tuple and
member recommendations, struggle. ConsRec, by exploring consensus behind group
behavior data, evades this imbalance issue, ranking as the best baseline in
group recommendation. AGFN refines the aggregation mechanism to reach the best
result in bundle recommendation. They still trail behind CDR.

Summarily, CDR’s pre-eminence arises chiefly due to: (1) The utility of
consistency and discrepancy metrics, refining model learning of object-
recommendee relationships. (2) The application of a pre-training&fine-tuning
approach, enhancing the generalization of our self-supervised task and adeptly
sidestepping tuple-member interaction data imbalances. (3) The innovative
contrastive CD loss, markedly unlocking the potential of quantified node
relationships. We also conducted experiments about the performance of our CDR
on social recommendation for detailed discussion and clarification on Last-FM
dataset, which are given in Appendix C. Time complexity analysis is given in
Appendix D.

Table 4. Performance comparison of CDR-P in extreme cold-start scenarios and
baselines in warm-start scenarios.

Method | Mafengwo (Group)
---|---
Recall@10 | Recall@20 | NDCG@10 | NDCG@20
BPR | 0.0604 | 0.1147 | 0.0317 | 0.0460
LINE | 0.0243 | 0.0316 | 0.0140 | 0.0164
NGCF | 0.0391 | 0.0795 | 0.0159 | 0.0263
LightGCN | 0.0766 | 0.1384 | 0.0385 | 0.0557
BUIR |  0.0969 |  0.1509 |  0.0593 |  0.0740
CDR-P | 0.1275 | 0.1633 | 0.0653 | 0.0745
| Youshu (Bundle)
| Recall@10 | Recall@20 | NDCG@10 | NDCG@20
BPR | 0.0188 | 0.0337 | 0.0122 | 0.0170
LINE | 0.0234 | 0.0350 | 0.0170 | 0.0208
NGCF | 0.0252 | 0.0479 | 0.0167 | 0.0239
LightGCN |  0.0362 |  0.0573 |  0.0269 |  0.0335
BUIR | 0.0508 | 0.0785 | 0.0363 | 0.0444
CDR-P | 0.0280 | 0.0476 | 0.0190 | 0.0251

Table 5. Performance of variants of CDR.

Method | Mafengwo (Group)
---|---
F1-Score | Precision@20 | Recall@20 | NDCG@20
CDR-P | 0.0195 | 0.0104 | 0.1633 | 0.0745
CDR-F | 0.0336 | 0.0179 | 0.2806 | 0.1493
CDR-R | 0.0425 | 0.0226 | 0.3531 | 0.1747
w/o $c$ | 0.0296 | 0.0157 | 0.2411 | 0.1160
w/o $d$ | 0.0349 | 0.0186 | 0.2790 | 0.1193
w/o $c$&$d$ | 0.0298 | 0.0158 | 0.2466 | 0.1177
CDR | 0.0474 | 0.0253 | 0.3833 | 0.1715
| Youshu (Bundle)
| F1-Score | Precision@20 | Recall@20 | NDCG@20
CDR-P | 0.0135 | 0.0079 | 0.0476 | 0.0251
CDR-F | 0.0207 | 0.0116 | 0.1000 | 0.0501
CDR-R | 0.0171 | 0.0099 | 0.0621 | 0.0348
w/o $c$ | 0.0277 | 0.0157 | 0.1166 | 0.0648
w/o $d$ | 0.0179 | 0.0114 | 0.0423 | 0.0280
w/o $c$&$d$ | 0.0180 | 0.0114 | 0.0427 | 0.0281
CDR | 0.0334 | 0.0191 | 0.1299 | 0.0744

Figure 4. Performance comparison of different loss functions of CDR on the
Mafengwo dataset.

###  4.3. Performance in Extreme Cold-Start

Addressing RQ2, we test CDR-P in extreme cold-start situations, with no tuple
interactions available during training, contrasting with baselines using
available tuple interactions. It is an unfair comparison since other models
have more information from tuple interactions. Table 4 reveals that our model,
even without utilizing tuple interactions, outperforms all baselines on the
Mafengwo dataset, showcasing CDR’s robustness in extreme scenarios. The
average performance of CDR-P in Youshu is the third for the lack of
information.

CDR-P excels in extreme cold-start scenarios because its supervision signals,
consistency and discrepancy metrics, derive from accessible and non-accessible
meta-path views, built upon existing member interactions and affiliations.
These member interactions and affiliations, being relatively abundant, furnish
ample information for recommendations. While data noise could potentially
derail model training, elevating the temperature hyperparameter concentrates
on easily differentiated nodes, thereby boosting robustness against noisy
datasets. Consequently, CDR-P discerns approximate node preferences,
mitigating the adverse effects of data noise.

###  4.4. Ablation Study

Addressing RQ3, ablation tests on CDR components confirm each part’s
significance. CDR-P and CDR-F denote models using only pre-training and solely
tuple interactions, respectively, while CDR-R swaps pre-training and fine-
tuning data—using tuple interactions for pre-training and member ones for
fine-tuning. Table 5 reveals omitting pre-training or fine-tuning stages
reduces performance, affirming CDR’s pre-training and fine-tuning efficacy.
Swapping training data stages also hampers performance, with a notable
degradation on the Youshu dataset after fine-tuning via member interactions
and affiliations, indicating the pivotal role and ordering of tuple
interactions in tripartite graph-based recommendations.

Figure 5. The effect of different temperatures $\tau$ on the pre-training and
fine-tuning phases. Figure 6. The performance comparison of the two baselines
on the Mafengwo dataset before and after CDR-P pre-training.

####  4.4.1. Ablation on Consistency and Discrepancy Metrics

Our consistency and discrepancy metrics, reflecting complex tuple-object node
relationships beyond binary labels, embody real-life varied user preferences
for numerous purchased items. Validating these metrics’ effectiveness, we
binarize calculated consistency $c$ and discrepancy $d$. Table 5 reveals
substantial performance drops when omitting either metric, affirming their
complementarity in contrastive learning-based loss and thus their collective
necessity.

####  4.4.2. Ablation on the CD Loss

The CDR model’s loss function was substituted with three alternatives to
validate the proposed contrastive learning-inspired CD loss. Specifically,
CDR-Origin employs the original loss (Eq. (16)), while CDR-MSE and CDR-CE
utilize mean squared error (Hu et al., 2008) and cross-entropy loss (He et
al., 2017) respectively. Results in Fig. 4 reveal CDR-Origin and CDR-MSE
underperform compared to CDR, potentially due to information loss from the
subtraction between consistency and discrepancy, as discussed in Section
3.2.1.

The performance of CDR-CE and CDR, which use these node relationships
separately, is better than CDR-Origin and CDR-MSE. The CD loss of CDR
performed best due to its characteristics to distinguish relatively easy or
difficult nodes by the temperature $\tau$. CD loss can adapt to data
containing noise of different degrees.

###  4.5. Hyperparameter Sensitivity Analysis

To further answer RQ4, we conducted hyperparameter sensitivity experiments on
CDR, where Fig. 5 shows the effect of different temperature values $\tau$ in
the pre-training and fine-tuning phases of Mafengwo and Youshu datasets. We
can find that CDR is sensitive to the temperature $\tau$, and the optimal
temperature value in the pre-training stage is higher than that in the fine-
tuning stage on these two datasets. This occurs because member interactions
are noisier than tuple interactions.

Therefore, a lower temperature is conducive to distinguishing similar nodes,
enabling the learning of fine-grained node representations for tuple
recommendations. Furthermore, an analysis concerns the correlation between
consistency & discrepancy and loss is given in Appendix F.

###  4.6. Generality of the Pre-Training Task

Finally, we use the consistency and discrepancy derived in the pre-training
phase of CDR as the supervision signal to train two traditional recommender
baselines to verify the generalization ability of the two metrics. In Fig. 6,
Base and Pretrain represent the performance of the baselines with/without
metrics obtained from CDR-P pre-training as supervision signals, respectively.
From Fig. 6, we can find that, with consistency and discrepancy as the
supervision signal, the performance of NGCF and LightGCN improves
substantially compared to the models that use only binary tuple interactions.
The results further demonstrate the importance of member interactions and
member affiliations and the effectiveness of the derived two metrics.

##  5\. Conclusions

We introduce CDR, a novel tripartite graph-based recommendation model to
improve its performance in cold-start scenarios. The model utilizes two new
metrics, consistency and discrepancy, representing tuple-object relationships
via reachable/non-reachable meta-paths, based on ample user-item interactions
and group-user/bundle-item affiliations. Leveraging the GCN’s limit theory,
these metrics are efficiently pre-calculated before optimization. Furthermore,
a novel contrastive learning-based objective is proposed, independently
exploiting consistency and discrepancy as positive and contrastive supervision
signals. Extensive experiments across various recommendation tasks and
multiple real-world datasets confirm the proposed metrics and contrastive
learning-inspired CD loss’s generalization and robustness to tripartite graph-
based recommendation’s cold-start issues.

## Acknowledgement

We thank the anonymous reviewers for their insightful feedback and
constructive comments. This work was partially supported by the National
Natural Science Foundation of China (Grant No. 6217602), the Australian
Research Council through the Industrial Transformation Training Centre (Grant
No. IC200100022), and the Science and Technology Research Program of Chongqing
Municipal Education Commission (Grant No. KJZD-K202204402 and
KJZD-K202304401). Co-authors Yaochen Zhu and Chen Chen consulted on this
project on unpaid weekends for personal interests, and appreciated
collaborators and family for their understanding.

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##  Appendix A Related Work

###  A.1. Tripartite Graph-Based Recommendation

In conventional recommendation systems, user-item interactions are typically
represented as a bipartite graph, with various Graph Convolutional Network
(GCN)-based methods modeling these interactions. This paper introduces a novel
concept: tripartite graph-based recommendations, which entail recommendation
tasks involving interactions among three distinct entity types. The third
entity often comprises user and item groups, introducing a novel affiliation
between items and bundles or users and groups. In this model, if an item
(recommended object) is recommended to a user group (recommendee), it is
termed a “group recommendation task” (Yin et al., 2022; Du et al., 2020b, a);
if an item group (recommended object) is recommended to a user (recommendee),
it is known as a “bundle recommendation” (Chang et al., 2023; Chen et al.,
2019a; Ma et al., 2022). As the first to consolidate these into one
overarching problem, we delineate group and bundle recommendations separately
in the ensuing section.

####  A.1.1. Group Recommendation

Group recommendation targets recommending a single item to a user group,
accounting for varied interests. Often formed temporarily, user groups usually
yield sparse group-item interactions(Yin et al., 2022; Sankar et al., 2020;
Zhang et al., 2021a). Guo et al. (Guo et al., 2020) introduced a model
addressing sparsity by leveraging a self-attention mechanism to learn users’
social influence in temporary group recommendations. Similarly, Cao et al.
(Cao et al., 2018) proposed AGREE, merging neural collaborative filtering and
attention mechanisms to dynamically determine group embedding fusion weights.
Jia et al. (Jia et al., 2021) advanced this by fusing member and group
networks into a dual-channel hypergraph convolutional model, HCR.

####  A.1.2. Bundle Recommendation

Bundle recommendation seeks to suggest multiple items collectively to a single
user and, akin to group recommendation, faces a pervasive data sparsity issue
due to the arbitrary bundling of items. Attention networks are often utilized
to ascertain the weight of items in a bundle (Ma et al., 2022). Furthermore,
multi-task learning is commonly employed to meld item and bundle interaction
data, mitigating sparsity. For instance, Chen et al. (Chen et al., 2019a)
devised DAM, a bundle recommendation model that interlinks user-bundle and
user-item interactions within a multi-task joint optimization framework. Gao
et al. (Chang et al., 2023) amalgamated user-item interactions, user-bundle
interactions, and bundle-item affiliations into a heterogeneous graph to
ensure bundle embeddings encapsulate item semantics.

###  A.2. Pre-Training and Fine-Tuning in Recommendation

Pre-training, aiming to optimize model parameters utilizing label ground truth
or auxiliary data, facilitates faster convergence or reduces label needs
during subsequent fine-tuning stages (Liu et al., 2021; Wang et al., 2021; Liu
et al., 2023; Hao et al., 2023). In recommendations, pre-training and fine-
tuning, viewed as self-supervised learning, entails initial training on
augmented data, followed by fine-tuning on original data, adapting to the
designated recommendation task.

Key aspects of pre-training involve creating augmented data and devising a
pretext task. Chen et al. (Chen et al., 2019b) and Zhang et al. (Zhang et al.,
2021b), inspired by the NLP model BERT (Devlin et al., 2018), designed similar
models for recommendations where side information on users/items is available.
Wang et al. (Wang et al., 2023) leveraged this by viewing interaction data
through heterogeneous graphs and building specific subgraphs. Similarly,
models utilizing side information in pre-training have been proposed (Zhou et
al., 2020). Qiu et al. (Qiu et al., 2021) masked sections of user and item
reviews, using bidirectional context to reconstruct them.

##  Appendix B Algorithm

The overall process of CDR is presented in Algorithm 1.

Input: Tuple interaction matrix $Y$ Member interaction matrix $X$ Tuple-member
affiliation matrix $Z$

Output: Final node embeddings $\mathbf{E}$

1 Extract $\boldsymbol{C}_{member}$ and $\boldsymbol{D}_{member}$ from member
interaction $X$ and affiliation $Z$

2 Randomly initialize embedding $\mathbf{E}^{p}$

3 while  _not converge_ do

4 for  _each epoch_ do

5 Evaluate $\mathcal{L}$ through $\boldsymbol{C}_{member}$,
$\boldsymbol{D}_{member}$, and $\mathbf{E}^{p}$ (Eq. (18))

6 Backpropagation and update embeddings $\mathbf{E}^{p}$

7

8 end for

9

10 end while

11Extract $\boldsymbol{C}_{tuple}$ and $\boldsymbol{D}_{tuple}$ from tuple
interactions $Y$

12 Stop gradient of $\mathbf{E}^{p}$

13 Initialize node embedding $\mathbf{E}^{f}$ with $\mathbf{E}^{p}$

14 Concatenate final embeddings
$\mathbf{E}\leftarrow\mathbf{E}^{f}\parallel\mathbf{E}^{p}$

15 while  _not converge_ do

16 for  _each epoch_ do

17 Evaluate $\mathcal{L}$ through $\boldsymbol{C}_{tuple}$,
$\boldsymbol{D}_{tuple}$, and $\mathbf{E}$ (Eq. (18))

18 Back propagation and update node embeddings $\mathbf{E}^{f}$

19

20 end for

21

22 end while

23return Final node embeddings $\mathbf{E}$

Algorithm 1 The overall process of CDR.

##  Appendix C Supplementary Experiments

To further explore if CDR can generalize to other recommendation types with
existing consistency and discrepancy metrics. we validate CDR’s adaptability
to other recommendation tasks, testing on the Last-FM social recommendation
dataset. The baseline methods we choose are SBPR (Zhao et al., 2014), DiffNet
(Wu et al., 2019), and SocialLGN (Liao et al., 2022). As Table 6 indicates,
models utilizing social relations outperform those relying solely on
interaction data. CDR-F remains competitive using only user-item interactions.
In social recommendation, which only involves user social relationships, CDR-P
outperforms CDR-F by leveraging both social and interaction data. Post-fine-
tuning, CDR, amalgamating the benefits of CDR-P and CDR-F, enhances all
metrics, substantiating its generalizability across various recommendation
domains where a tripartite graph can be constructed.

Table 6. Performance comparison of CDR model and baselines under social
recommendation scenarios.

Dataset | Method | F1-Score | Precision@20 | Recall@20 | NDCG@20
---|---|---|---|---|---
Last-FM | User-item interaction
BPR | 0.0277 | 0.0207 | 0.0417 | 0.0353
BUIR | 0.0484 | 0.0361 | 0.0731 | 0.0685
NGCF | 0.0505 | 0.0379 | 0.0756 | 0.0721
LightGCN |  0.0568 |  0.0425 |  0.0857 | 0.0844
CDR-F | 0.0576 | 0.0428 | 0.0878 |  0.0798
User-item interaction/Social relation
SBPR | 0.0544 | 0.0405 | 0.0829 | 0.0725
DiffNet | 0.0570 | 0.0427 | 0.0859 | 0.0805
SocialLGN | 0.0665 | 0.0495 | 0.1011 | 0.0894
CDR-P |  0.0756 |  0.0563 |  0.1151 |  0.1108
CDR | 0.0836 | 0.0622 | 0.1274 | 0.1195

##  Appendix D Time Complexity Analysis

Our analysis covers each critical stage of our method, i.e., pre-processing,
pre-training, and fine-tuning. In the pre-processing stage, for each member
$m$, we need to calculate two kinds of terms, i.e., $\delta_{m,t}$ and
$\delta_{m,o}$, and the time complexity is
$O(|\mathcal{M}||\mathcal{T}|)+O(|\mathcal{M}||\mathcal{O}|)$, where
$\mathcal{T,O,M}$ are sets of the three types of nodes in a tripartite graph-
based recommendation, and $||$ is the number of elements in a set. Similarly,
we need to calculate two other kinds of terms $\delta_{o,t}$ and
$\delta_{t,o}$, and the time complexity is $O(\mathcal{|O||T|})$. Then, with
the four kinds of terms, i.e., $\delta_{m,t}$, $\delta_{m,o}$, $\delta_{o,t}$,
and $\delta_{t,o}$, we can directly get the values in $C_{member}$,
$D_{member}$, $C_{tuple}$, and $D_{tuple}$ for further training. All
calculations in pre-processing do not include matrix multiplication, so this
stage does not take a lot of time.

In our model, the time complexity for both the pre-training and fine-tuning
stages is similar. Here, $Q$ represents the set of node pairs that exhibit
positive consistency and whose types belong to tuples or objects. According to
the loss function, we can derive the time complexities for every batch of the
two stages are both $O((\frac{|Q|}{B}+|Q|)d)$, where $d$ is the dimension of
embedding, and $B$ is the batch size. Note that a pair of nodes is included in
the set $Q$ only if they share common neighborhood nodes. Given the sparsity
characteristic of recommendation datasets, $Q$ is significantly smaller than
$\mathcal{O\times T}$. This inherent sparsity greatly reduces the actual
computational time, thereby ensuring efficiency in processing.

We conduct a comparison of model runtime, with the results shown in Table 7.
Our CDR is the fastest because it does not require node aggregation. ALDI
(Huang et al., 2023) and XSimGCL (Yu et al., 2023), which only utilize T-O
interactions, also offer relatively fast speeds. In contrast, AGFN (Li et al.,
2023) processes more slowly for using graph aggregation methods on tripartite
graph.

Table 7. The runtime of different graph-based method.

Method | ALDI | XSimGCL | AGFN | CDR
---|---|---|---|---
Runtime | 0.0820s | 0.0901s | 0.2263s | 0.0738s

##  Appendix E Types of Meta-paths

We summarize all the meta-path types in Mafengwo and Youshu datasets, and the
result is shown in Table 8, where G indicates a group, B indicates a bundle,
and U and I are a user and an item separately.

Table 8. The metapaths for different scenarios. Scenarios | Dataset | Meta-path
---|---|---
Group | Mafengwo | Pre-train | GUG/GUI/IUG/IUI
Fine-tune | GIG/GI/IG/IGI
Bundle | Youshu | Pre-train | UIU/UIB/BIU/BIB
Fine-tune | UBU/UB/BU/BUB
General | Mafengwo | GIG/GI/IG/IGI
Youshu | UBU/UB/BU/BUB

##  Appendix F Correlation between Consistence & Discrepancy and the Training
Loss

We conduct an experiment by calculating the Pearson correlation coefficients
between the metrics consistency ($c$) and discrepancy ($d$), and the loss for
each node pair, as presented in Table 9. The experimental results indicate
that there is a negative correlation between consistency and loss, and a
positive correlation between discrepancy and loss. This means node pairs with
high consistency tend to have lower loss, while those with greater discrepancy
exhibit higher loss.

Table 9. The Pearson correlation coefficients between the metrics consistency ($c$) and discrepancy ($d$), and the loss for each node pair. Dataset | Corr($c$,loss) | Corr($d$,loss)
---|---|---
Youshu | -0.60 | 0.52
Mafengwo | -0.46 | 0.43

This pattern of correlations allows us to leverage the metrics $c$ and $d$
strategically during the optimization process. To effectively minimize loss,
an increase in $c_{v_{i},v_{j}}$ should correspond with an increase in
$cos(e_{v_{i}},e_{v_{j}})$, enhancing similarity between embeddings.
Conversely, an increase in $d_{v_{i},v_{j}}$ should result in a decrease in
$cos(e_{v_{i}},e_{v_{j}})$, reducing similarity.


#  Extension of Kodama vector and quasilocal quantities in three-dimensional
axisymmetric spacetimes

Shunichiro Kinoshita<EMAIL_ADDRESS>Department of Physics, Chuo
University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

###### Abstract

Spherically symmetric spacetimes admit the so-called Kodama vector, which
provides a locally conserved current and a preferred time even for dynamical
spacetime without any time translation symmetry. A charge associated with this
conserved current leads to a quasilocal mass which agrees with the Misner-
Sharp mass. In three dimensions, spherically symmetric spacetimes correspond
to axisymmetric ones, while axisymmetry allows spacetimes to be rotating with
angular momentum. We extend the notion of the Kodama vector to axisymmetric
rotating spacetimes in three dimensions. We also define a quasilocal mass
taking into account angular momentum in three-dimensional axisymmetric
spacetimes.

##  I Introduction

Conservation laws of energy and momentum have played significant roles in
physics. In gravitational theories, however, diffeomorphism invariance makes
local notion of energy and momentum for gravitating system ambiguous. Thus,
gravitational energy-momentum should be defined nonlocally in finite spacetime
domain. Such various definitions of quasilocal mass and angular momentum have
been proposed (see Szabados:2009eka ; Jaramillo:2010ay  and references
therein).

On the other hand, it is well-known that if a spacetime admits a Killing
vector field generating some symmetry, a locally conserved current vector can
be obtained by contracting divergence-free, symmetric tensor such as energy-
momentum tensor, or Einstein tensor, with the Killing vector. In particular,
if a Killing vector is timelike, one can obtain a locally conserved energy
current in terms of the Killing vector. In this case the spacetime is static
or stationary with time translation symmetry, so that this conserved current
cannot be applied to dynamics of gravitating systems.

Remarkably, in spherically symmetric spacetimes we can construct a local
conserved energy current by using the so-called Kodama vector, even though the
spacetime is dynamical without any time translation Killing vector
Kodama:1979vn . This conserved current gives us quasi-local mass as Misner-
Sharp mass Misner:1964je ; Hayward:1994bu . Generalizations to higher-
dimensional spacetime or other gravitational theories such as Gauss-Bonnet and
$f(R)$ gravity were studied in Refs. Maeda:2006pm ; Maeda:2007uu ; Cai:2009qf
; Zhang:2014goa .

In three dimensions, the counterparts of spherically symmetric spacetimes are
axisymmetric (or circularly symmetric) ones. However, since axisymmetry does
not rule out rotating systems, which are of physical interest significantly,
we can take account of angular momentum unlike spherical symmetry of higher-
dimensional spacetimes. Such a rotating spacetime, in general, cannot be
described by warped product metric. It turns out that in spacetimes with
angular momentum the conventional Kodama vector does not always yield a
conserved current. (The Kodama vector in warped product spacetimes was
discussed in Ref. An:2017wti .)

In this paper we will extend the notion of the Kodama vector to spacetimes
with nonzero angular momentum, which are not described by warped product
metric. The organization of the paper is as follows. In Sec. II we consider
general axisymmetric spacetimes in three dimensions, and explore a new vector
that satisfies similar conditions to the Kodama vector in spherically
symmetric cases. This new vector provides a locally conserved current. In Sec.
III we define quasilocal quantities as charges associated with locally
conserved currents.

While this paper was in preparation, Ref. Gundlach:2021six  appeared where the
authors have proposed the same quasilocal mass.

##  II Extension of Kodama vector

Let us consider a three-dimensional axisymmetric spacetime with an axial
Killing vector field given by

$\psi^{\mu}=\left(\frac{\partial}{\partial\phi}\right)^{\mu}.$ (1)

The spacetime metric can be generally written as

$g_{\mu\nu}dx^{\mu}dx^{\nu}=h_{ij}(y)dy^{i}dy^{j}+r^{2}(y)[d\phi+a_{i}(y)dy^{i}]^{2},$
(2)

where $h_{ij}$ is a two-dimensional metric on the orbit space of the Killing
vector. Note that $a_{i}(y)$ has a gauge freedom as $a_{i}(y)\to
a_{i}(y)+\partial_{i}\lambda(y)$, where $\lambda(y)$ is an arbitrary function
on the orbit space. If we assume that the axial Killing vector field has
closed orbits with $\phi\sim\phi+2\pi$, the norm of the Killing vector
provides the circumferential radius $r(y)=(\psi^{\mu}\psi_{\mu})^{1/2}$. Greek
indices indicate components of total three-dimensional spacetime, and Latin
indices indicate two-dimensional orbit space components.

When $\psi^{\mu}$ is a hypersurface orthogonal Killing vector
$\psi_{[\mu}\nabla_{\nu}\psi_{\lambda]}=0$, that is, nontrivial $a_{i}(y)$
vanishes, the spacetime becomes simply a warped product. In this case we
obtain the Kodama vector by conventional definition Kodama:1979vn  as
$\widetilde{K}^{i}=-e^{ij}\nabla_{j}r$, where $e^{ij}$ is the two-dimensional
volume form associated with $h_{ij}$. This definition can be naturally adapted
to a vector on the full three-dimensional spacetime as

$\widetilde{K}^{\mu}=-\frac{1}{r}\epsilon^{\mu\nu\alpha}\psi_{\alpha}\nabla_{\nu}r,$
(3)

where $\epsilon_{\mu\nu\alpha}$ is the totally antisymmetric $3$-tensor
corresponding to a three-dimensional volume form. The metric of the two-
dimensional orbit space is written as

$h_{\mu\nu}=g_{\mu\nu}-\frac{1}{r^{2}}\psi_{\mu}\psi_{\nu}.$ (4)

Even though the spacetime is not a warped product, $\widetilde{K}^{\mu}$ can
still satisfy the following properties: $\widetilde{K}^{\mu}\nabla_{\mu}r=0$
and $\nabla_{\mu}\widetilde{K}^{\mu}=0$ in a similar manner as the Kodama
vector. However, it turns out that
$G^{\mu\nu}\nabla_{\mu}\widetilde{K}_{\nu}\neq 0$ for the three-dimensional
Einstein tensor $G_{\mu\nu}$ of generic axisymmetric spacetimes.111 For a
specific spacetime, it is possible that
$G^{\mu\nu}\nabla_{\mu}\widetilde{K}_{\nu}=0$ also holds while the spacetime
is not warped product. We can see an example of such spacetimes in Ref.
Flory:2013ssa , which is a solution of a three dimensional higher derivative
gravity.  Thus, $G^{\mu\nu}\widetilde{K}_{\nu}$ is not always a locally
conserved current unless the axisymmetric spacetime becomes a warped product
such as nonrotating spacetimes.

Now, we define a new vector associated with the axial Killing vector as

$K^{\mu}\equiv-\frac{1}{2}\epsilon^{\mu\alpha\beta}\nabla_{\alpha}\psi_{\beta}.$
(5)

It is worth noting that this vector can be decomposed into normal and
tangential components to $\psi^{\mu}$ as

$\displaystyle
K^{\mu}=-\frac{1}{r}\epsilon^{\mu\nu\alpha}\psi_{\alpha}\nabla_{\nu}r-\frac{\beta}{2r^{2}}\psi^{\mu},\quad\beta=\epsilon^{\lambda\alpha\beta}\psi_{\lambda}\nabla_{\alpha}\psi_{\beta}.$
(6)

Thus, two vectors $K^{\mu}$ and $\widetilde{K}^{\mu}$ are related as
$K^{\mu}=\widetilde{K}^{\mu}-\beta\psi^{\mu}/(2r^{2})$. If $\beta=0$, which
means that the axial Killing vector $\psi^{\mu}$ is hypersurface orthogonal
and the spacetime becomes warped product, then both vectors are identical to
just the Kodama vector.

In what follows we will examine some properties of $K^{\mu}$. By definition,
$K^{\mu}$ itself is (i) divergence-free: $\nabla_{\mu}K^{\mu}=0$. We can see
that (ii) $K^{\mu}$ is tangent to $r=\text{const.}$ surfaces, because

$\displaystyle K^{\mu}\nabla_{\mu}r$
$\displaystyle=K^{\mu}\nabla_{\mu}(\psi^{\nu}\psi_{\nu})^{1/2}=\frac{1}{r}\psi^{\nu}K^{\mu}\nabla_{\mu}\psi_{\nu}=\frac{1}{r}\psi^{\nu}K^{\mu}\epsilon_{\alpha\mu\nu}K^{\alpha}=0.$
(7)

The Killing equation for the Killing vector $\psi^{\mu}$ gives us

$\nabla_{\mu}\nabla_{\alpha}\psi_{\beta}=-R_{\alpha\beta\mu}{}^{\nu}\psi_{\nu},$
(8)

and, in three dimensions the Riemann tensor $R_{\alpha\beta\mu\nu}$ can be
expressed as222 In three-dimensional spacetime, we can directly show
$\displaystyle\epsilon_{\alpha\beta\gamma}\epsilon^{\mu\nu\lambda}G^{\gamma}{}_{\lambda}$
$\displaystyle=-3!g_{[\alpha}{}^{\mu}g_{\beta}{}^{\nu}g_{\gamma]}{}^{\lambda}G^{\gamma}{}_{\lambda}$
$\displaystyle=-\left(g_{\alpha}{}^{\mu}g_{\beta}{}^{\nu}g_{\gamma}{}^{\lambda}+g_{\beta}{}^{\mu}g_{\gamma}{}^{\nu}g_{\alpha}{}^{\lambda}+g_{\gamma}{}^{\mu}g_{\alpha}{}^{\nu}g_{\beta}{}^{\lambda}-g_{\alpha}{}^{\mu}g_{\gamma}{}^{\nu}g_{\beta}{}^{\lambda}-g_{\gamma}{}^{\mu}g_{\beta}{}^{\nu}g_{\alpha}{}^{\lambda}-g_{\beta}{}^{\mu}g_{\alpha}{}^{\nu}g_{\gamma}{}^{\lambda}\right)G^{\gamma}{}_{\lambda}$
$\displaystyle=g_{\alpha}{}^{\mu}R_{\beta}{}^{\nu}-g_{\beta}{}^{\mu}R_{\alpha}{}^{\nu}+R_{\alpha}{}^{\mu}g_{\beta}{}^{\nu}-R_{\beta}{}^{\mu}g_{\alpha}{}^{\nu}-\frac{R}{2}\left(g_{\alpha}{}^{\mu}g_{\beta}{}^{\nu}-g_{\beta}{}^{\mu}g_{\alpha}{}^{\nu}\right)$
$\displaystyle=R_{\alpha\beta}{}^{\mu\nu}.$

$R_{\alpha\beta\mu\nu}=\epsilon_{\alpha\beta\gamma}\epsilon_{\mu\nu\lambda}G^{\gamma\lambda}.$
(9)

Using the above equations, we find (iii) $G^{\mu\nu}\nabla_{\mu}K_{\nu}=0$ as
follows:

$\displaystyle G^{\mu\nu}\nabla_{\mu}K_{\nu}$
$\displaystyle=-\frac{1}{2}\epsilon_{\nu}{}^{\alpha\beta}G^{\mu\nu}\nabla_{\mu}\nabla_{\alpha}\psi_{\beta}$
(10)
$\displaystyle=\frac{1}{2}\epsilon_{\nu}{}^{\alpha\beta}G^{\mu\nu}R_{\alpha\beta\mu\lambda}\psi^{\lambda}$
$\displaystyle=G^{\mu\nu}\left(\epsilon_{\nu\alpha\lambda}G_{\mu}{}^{\alpha}-\epsilon_{\nu\alpha\mu}R_{\lambda}{}^{\alpha}\right)\psi^{\lambda}$
$\displaystyle=0.$

As a result, we confirm that $K^{\mu}$ can always satisfy the same conditions
(i) $\nabla_{\mu}K^{\mu}=0$, (ii) $K^{\mu}\nabla_{\mu}r=0$, and (iii)
$G^{\mu\nu}\nabla_{\mu}K_{\nu}=0$ as the Kodama vector in spherically
symmetric spacetimes. This implies that the vector $K^{\mu}$ defined by (5) is
natural extension of the Kodama vector to axisymmetric spacetime with angular
momentum in three dimensions.

##  III quasilocal quantities

In spherically symmetric cases, a locally conserved current constructed from
the Kodama vector yields a quasilocal mass as an associated charge
Kodama:1979vn ; Hayward:1994bu . In this section, we will construct quasilocal
quantities related to the extended Kodama vector.

In general, if a conserved current $\mathcal{J}^{\mu}$ satisfying
$\nabla_{\mu}\mathcal{J}^{\mu}=0$ is axisymmetric, we obtain

$0=\pounds_{\psi}\mathcal{J}^{\mu}=\psi^{\nu}\nabla_{\nu}\mathcal{J}^{\mu}-\mathcal{J}^{\nu}\nabla_{\nu}\psi^{\mu}=2\nabla_{\nu}(\psi^{[\nu}\mathcal{J}^{\mu]}),$
(11)

where we have used the fact that $\psi^{\mu}$ and $\mathcal{J}^{\mu}$ are
divergence-free. This implies that there exists a scalar function
$\mathcal{Q}$ such that

$\nabla_{\mu}\mathcal{Q}=2\pi\epsilon_{\mu\alpha\beta}\psi^{\alpha}\mathcal{J}^{\beta}.$
(12)

By the Stokes theorem this scalar function is written in the integral form

$\mathcal{Q}[C]\equiv\frac{1}{2\pi}\oint_{C}\mathcal{Q}d\phi=\int_{S}\mathcal{J}^{\mu}dS_{\mu},$
(13)

where $C$ is a closed Killing orbit of the axial Killing vector and $S$ is an
arbitrary spacelike surface whose boundary is $C$. It turns out that scalar
function $\mathcal{Q}$ is a charge associated with each closed Killing orbit.

Since $K^{\mu}$ and $G^{\mu\nu}K_{\nu}$ are local conserved currents as we
mentioned in the previous section, we obtain scalar functions
$\psi^{\mu}\psi_{\mu}$ and $K^{\mu}K_{\mu}$ associated with each current,
respectively. Indeed, we can explicitly confirm that

$\displaystyle\nabla_{\mu}(\psi^{\nu}\psi_{\nu})$
$\displaystyle=2\psi^{\nu}\nabla_{\mu}\psi_{\nu}$ (14)
$\displaystyle=2K^{\alpha}\epsilon_{\alpha\mu\nu}\psi^{\nu},$

and

$\displaystyle\nabla_{\mu}(K^{\nu}K_{\nu})$
$\displaystyle=-\nabla^{\alpha}\psi^{\beta}\nabla_{\mu}\nabla_{\alpha}\psi_{\beta}$
(15)
$\displaystyle=\nabla^{\alpha}\psi^{\beta}R_{\alpha\beta\mu\nu}\psi^{\nu}$
$\displaystyle=\nabla^{\alpha}\psi^{\beta}\epsilon_{\alpha\beta\gamma}\epsilon_{\mu\nu\lambda}G^{\gamma\lambda}\psi^{\nu}$
$\displaystyle=-2K_{\gamma}G^{\gamma\lambda}\epsilon_{\lambda\mu\nu}\psi^{\nu}.$

Because any linear combination of local conserved currents is also locally
conserved, we should adopt a charge associated with energy current as
quasilocal mass. Now, we define a mass function $m$ as

$m\equiv\frac{1}{8G_{3}}(-\Lambda\psi^{\nu}\psi_{\nu}+K^{\nu}K_{\nu}),$ (16)

where $G_{3}$ and $\Lambda$ denote a three-dimensional gravitational constant
and a cosmological constant, respectively. By using Eqs. (14) and (15), we
obtain

$\displaystyle\nabla_{\mu}m$
$\displaystyle=\frac{1}{4G_{3}}\epsilon_{\mu\nu\alpha}\psi^{\alpha}(G^{\nu\beta}+\Lambda
g^{\nu\beta})K_{\beta}$ (17)
$\displaystyle=2\pi\epsilon_{\mu\nu\alpha}\psi^{\alpha}T^{\nu\beta}K_{\beta},$

where we have assumed the Einstein equation $G_{\mu\nu}+\Lambda
g_{\mu\nu}=8\pi G_{3}T_{\mu\nu}$ in the last line. Thus, $m$ is a charge
associated with the conserved energy current, $-T^{\mu\nu}K_{\nu}$, in
Einstein gravity.

In addition, we have an angular-momentum function as

$j\equiv\frac{1}{8G_{3}}\epsilon^{\alpha\mu\nu}\psi_{\alpha}\nabla_{\mu}\psi_{\nu}=-\frac{1}{4G_{3}}\psi_{\mu}K^{\mu},$
(18)

which satisfies

$\displaystyle\nabla_{\mu}j$
$\displaystyle=-\frac{1}{4G_{3}}\epsilon_{\mu\nu\alpha}\psi^{\alpha}G^{\nu\beta}\psi_{\beta}$
(19)
$\displaystyle=-2\pi\epsilon_{\mu\nu\alpha}\psi^{\alpha}T^{\nu\beta}\psi_{\beta}.$

This also implies that $j$ is a charge associated with the conserved current,
$T^{\mu\nu}\psi_{\nu}$, which is nothing but a well-known conserved current
with respect to the axial Killing vector $\psi^{\mu}$. Note that formula (18)
of angular-momentum function agrees with that of the Komar angular momentum.

The above mass and angular-momentum functions are rewritten as

$8G_{3}m=-\Lambda
r^{2}-\nabla_{\mu}r\nabla^{\mu}r+\frac{(4G_{3}j)^{2}}{r^{2}}.$ (20)

It turns out that the mass function consists of the conventional Misner-Sharp
mass in spherically symmetric cases and the angular-momentum term.333
Precisely speaking, we can add an arbitrary constant term to definition of
quasilocal mass (16). This constant is related to a value of mass in vacuum.
Here, we have set $m=-1/8G_{3}$ in pure AdS3.  In the Bañados-Teitelboim-
Zanelli (BTZ) solution Banados:1992wn , which is stationary, axisymmetric
vacuum solution with a negative cosmological constant ($\Lambda<0$), the
spacetime metric is given by

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\left(d\phi-\frac{4G_{3}J}{r^{2}}dt\right)^{2},\quad
f(r)=-\Lambda r^{2}-8G_{3}M+\frac{16G_{3}^{2}J^{2}}{r^{2}},$ (21)

where $M$ and $J$ denote mass and angular-momentum parameters, respectively.
In this case quasilocal mass $m$ and quasilocal angular momentum $j$ defined
by Eqs. (16) and (18) become constants equal to the mass and angular-momentum
parameters everywhere as $m=M$ and $j=J$. Moreover, the extended Kodama vector
coincides with usual time-translational Killing vector,
$K^{\mu}=(\partial/\partial t)^{\mu}$.

##  IV Conclusion

In this paper we have proposed extending the notion of the Kodama vector to
axisymmetric, three-dimensional spacetimes with nonzero angular momentum. The
extended Kodama vector
$K^{\mu}=-\frac{1}{2}\epsilon^{\mu\alpha\beta}\nabla_{\alpha}\psi_{\beta}$,
which is associated with an axial Killing vector $\psi^{\upsilon}$, satisfies
the same conditions as the conventional Kodama vector in spherically symmetric
cases. In particular, satisfying condition $G^{\mu\nu}\nabla_{\mu}K_{\nu}=0$
for the Einstein tensor makes $G^{\mu\nu}K_{\nu}$ divergence free for generic
axisymmetric spacetimes. This provides a locally conserved energy current
assuming the Einstein equation. We have shown that quasilocal mass like the
Misner-Sharp mass can be defined as a charge associated with the locally
conserved energy current. This quasilocal mass contains angular momentum of
rotating spacetime.

These quasilocal quantities and conservation laws were also pointed out in
Ref. Gundlach:2021six , where they were useful for reducing the Einstein
equations in numerical simulations of axisymmetric Einstein-perfect fluid
systems. What we should emphasis in the present paper is that the extended
Kodama vector guarantees the conserved currents and provides the quasilocal
quantities as conserved charges. The above properties originate from purely
geometrical identities for three-dimensional axisymmetric spacetimes
independent of field equations of a specific gravitational theory. This means
that any axisymmetric, three-dimensional spacetime can admit a conserved
current associated with the extended Kodama vector even in gravitational
theories other than the Einstein gravity. The field equations will give
physical meaning of the conserved current by connecting with energy-momentum
tensor of matter in each gravitational theory.

It is fascinating to generalize the extended Kodama vector to higher-
dimensional axisymmetric spacetimes, or spacetimes different from warped
product, other than three dimensions. It does not seem to be so easy, because
geometrical properties in three-dimensional spacetimes are necessary in order
to explore the extended Kodama vector.444 It was shown that, if a three-
dimensional spacetime admits a non-null Killing vector field, the spacetime
metric and the Ricci tensors are determined by two scalars characterizing the
Killing vector in Ref. Gurses:2010sm . However, once one can reduce to a
three-dimensional axisymmetric spacetime, it is expected that there are a
certain extended Kodama vector and locally conserved current associated with
it. For example, if a cylindrical spacetime in four dimensions are rotating,
there is expected to be a similar conserved current.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant No. JP16K17704.

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