Self-Supervised Multi-Channel Hypergraph
Convolutional Network for Social Recommendation
Item Type Preprint
Authors Yu, Junliang; Yin, Hongzhi; Li, Jundong; Wang, Qinyong; Hung,
Nguyen Quoc Viet; Zhang, Xiangliang
Eprint version Pre-print
Publisher arXiv
Rights Archived with thanks to arXiv
Download date 10/08/2022 08:39:11
Link to Item http://hdl.handle.net/10754/668790
Self-Supervised Multi-Channel Hypergraph Convolutional
Network for So cial Recommendation
Junliang Yu
The University of Queensland
jl.yu@uq.edu.au
Hongzhi Yin
โ
The University of Queensland
h.yin1@uq.edu.au
Jundong Li
University of Virginia
jundong@virginia.edu
Qinyong Wang
The University of Queensland
qinyong.wang@uq.edu.au
Nguyen Quoc Viet Hung
Gri๎th University
quocviethung1@gmail.com
Xiangliang Zhang
King Abdullah University of Science
and Technology
xiangliang.zhang@kaust.edu.sa
ABSTRACT
Social relations are often used to improve recommendation quality
when user-item interaction data is sparse in recommender systems.
Most existing social recommendation models exploit pairwise re-
lations to mine potential user preferences. However, real-life in-
teractions among users are very complicated and user relations
can be high-order. Hypergraph provides a natural way to model
complex high-order relations, while its potentials for improving
social recommendation are under-explored. In this paper, we ๎ll
this gap and propose a multi-channel hypergraph convolutional net-
work to enhance social recommendation by leveraging high-order
user relations. Technically, each channel in the network encodes
a hypergraph that depicts a common high-order user relation pat-
tern via hypergraph convolution. By aggregating the embeddings
learned through multiple channels, we obtain comprehensive user
representations to generate recommendation results. However, the
aggregation operation might also obscure the inherent characteris-
tics of di๎erent types of high-order connectivity information. To
compensate for the aggregating loss, we innovatively integrate
self-supervised learning into the training of the hypergraph con-
volutional network to regain the connectivity information with
hierarchical mutual information maximization. The experimental
results on multiple real-world datasets show that the proposed
model outperforms the SOTA methods, and the ablation study
veri๎es the e๎ectiveness of the multi-channel setting and the self-
supervised task. The implementation of our model is available via
https://github.com/Coder-Yu/RecQ.
CCS CONCEPTS
โข Information systems โ Recommender systems
;
Social rec-
ommendation.
KEYWORDS
Social Recommendation, Self-supervised Learning, Hypergraph
Learning, Graph Convolutional Network, Recommender System
โ
Corresponding author and having equal contribution with the ๎rst author.
WWW โ21, April 19โ23, 2021, Ljubljana, Slovenia
ยฉ 2021 Association for Computing Machinery.
This is the authorโs version of the work. It is posted here for your personal use. Not for
redistribution. The de๎nitive Version of Record was published in WWW โ21: ACM The
Web Conference, April 19โ23, 2021, Ljubljana, Slovenia, https://doi.org/10.1145/nnnnnnn.
nnnnnnn.
ACM Reference Format:
Junliang Yu, Hongzhi Yin, Jundong Li, Qinyong Wang, Nguyen Quoc Viet
Hung, and Xiangliang Zhang. 2021. Self-Supervised Multi-Channel Hyper-
graph Convolutional Network for Social Recommendation. In WWW โ21:
ACM The Web Conference, April 19โ23, 2021, Ljubljana, Slovenia. ACM, New
York, NY, USA, 12 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn
Friends
FriendsFriends
Buy Buy
Friends
Buy Buy
Figure 1: The common types of high-order user relations in
social recommender systems.
1 INTRODUCTION
Over the past decade, the social media boom has dramatically
changed peopleโs ways of thinking and behaving. It has been re-
vealed that people may alter their attitudes and behaviors in re-
sponse to what they perceive their friends might do or think, which
is known as the social in๎uence [
7
]. Meanwhile, there are also
studies [
25
] showing that people tend to build connections with
others who have similar preferences with them, which is called
the homophily. Based on these ๎ndings, social relations are often
integrated into recommender systems to mitigate the data spar-
sity issue [
13
,
33
]. Generally, in a social recommender system, if
a user has few interactions with items, the system would rely on
her friendsโ interactions to infer her preference and generate better
recommendations. Upon this paradigm, a large number of social
recommendation models have been developed [
12
,
21
,
23
,
55
,
57
,
61
]
and have shown stronger performance compared with general rec-
ommendation models.
Recently, graph neural networks (GNNs) [
43
] have achieved
great success in a wide range of areas. Owing to their powerful
capability in modeling relational data, GNNs-based models also
have shown prominent performance in social recommendation
[
9
,
19
,
40
โ
42
,
58
]. However, a key limitation of these GNNs-based
social recommendation models is that they only exploit the sim-
ple pairwise user relations and ignore the ubiquitous high-order
relations among users. Although the long-range dependencies of
arXiv:2101.06448v3 [cs.IR] 21 Jan 2021
WWW โ21, April 19โ23, 2021, Ljubljana, Slovenia Junliang Yu, Hongzhi Yin, Jundong Li, Qinyong Wang, Nguyen ๎oc Viet Hung, and Xiangliang Zhang
relations (i.e. transitivity of friendship), which are also considered
high-order, can be captured by using k graph neural layers to in-
corporate features from k-hop social neighbors, these GNNs-based
models are unable to formulate and capture the complex high-order
user relation patterns (as shown in Fig. 1) beyond pairwise relations.
For example, it is natural to think that two users who are socially
connected and also purchased the same item have a stronger rela-
tionship than those who are only socially connected, whereas the
common purchase information in the former is often neglected in
previous social recommendation models.
Hypergraph [
4
], which generalizes the concept of edge to make
it connect more than two nodes, provides a natural way to model
complex high-order relations among users. Despite the great ad-
vantages over the simple graph in user modeling, the strengths of
hypergraph are under-explored in social recommendation. In this
paper, we ๎ll this gap by investigating the potentials of fusing hy-
pergraph modeling and graph convolutional networks, and propose
a
M
ulti-channel
H
ypergraph
C
onvolutional
N
etwork (
MHCN
) to
enhance social recommendation by exploiting high-order user re-
lations. Technically, we construct hypergraphs by unifying nodes
that form speci๎c triangular relations, which are instances of a set
of carefully designed triangular motifs with underlying semantics
(shown in Fig. 2). As we de๎ne multiple categories of motifs which
concretize di๎erent types of high-order relations such as โhaving a
mutual friendโ, โfriends purchasing the same itemโ, and โstrangers
but purchasing the same itemโ in social recommender systems, each
channel of the proposed hypergraph convolutional network under-
takes the task of encoding a di๎erent motif-induced hypergraph.
By aggregating multiple user embeddings learned through multiple
channels, we can obtain the comprehensive user representations
which are considered to contain multiple types of high-order rela-
tion information and have the great potentials to generate better
recommendation results with the item embeddings.
However, despite the bene๎ts of the multi-channel setting, the ag-
gregation operation might also obscure the inherent characteristics
of di๎erent types of high-order connectivity information [
54
], as dif-
ferent channels would learn embeddings with varying distributions
on di๎erent hypergraphs. To address this issue and fully inherit the
rich information in the hypergraphs, we innovatively integrate a
self-supervised task [
15
,
37
] into the training of the multi-channel
hypergraph convolutional network. Unlike existing studies which
enforce perturbations on graphs to augment the ground-truth [
53
],
we propose to construct self-supervision signals by exploiting the
hypergraph structures, with the intuition that the comprehensive
user representation should re๎ect the user nodeโs local and global
high-order connectivity patterns in di๎erent hypergraphs. Con-
cretely, we leverage the hierarchy in the hypergraph structures
and hierarchically maximizes the mutual information between rep-
resentations of the user, the user-centered sub-hypergraph, and
the global hypergraph. The mutual information here measures the
structural informativeness of the sub- and the whole hypergraph
towards inferring the user features through the reduction in local
and global structure uncertainty. Finally, we unify the recommenda-
tion task and the self-supervised task under a primary & auxiliary
learning framework. By jointly optimizing the two tasks and lever-
aging the interplay of all the components, the performance of the
recommendation task achieves signi๎cant gains.
The major contributions of this paper are summarized as follows:
โข
We investigate the potentials of fusing hypergraph modeling and
graph neural networks in social recommendation by exploiting
multiple types of high-order user relations under a multi-channel
setting.
โข
We innovatively integrate self-supervised learning into the train-
ing of the hypergraph convolutional network and show that a
self-supervised auxiliary task can signi๎cantly improve the social
recommendation task.
โข
We conduct extensive experiments on multiple real-world datasets
to demonstrate the superiority of the proposed model and thor-
oughly ablate the model to investigate the e๎ectiveness of each
component with an ablation study.
The rest of this paper is organized as follows. Section 2 introduces
the related work. Section 3 details the multi-channel hypergraph
convolutional network and elaborates on how self-supervised learn-
ing further improves the performance. The experimental results
and analysis are illustrated in Section 4. Finally, Section 5 concludes
this paper.
2 RELATED WORK
2.1 Social Recommendation
As suggested by the social science theories [
7
,
25
], usersโ prefer-
ences and decisions are often in๎uenced by their friends. Based on
this fact, social relations are integrated into recommender systems
to alleviate the issue of data sparsity. Early exploration of social
recommender systems mostly focuses on matrix factorization (MF),
which has a nice probabilistic interpretation with Gaussian prior
and is the most used technique in social recommendation regime.
The extensive use of MF marks a new phase in the research of
recommender systems. A multitude of studies employ MF as their
basic model to exploit social relations since it is very ๎exible for MF
to incorporate prior knowledge. The common ideas of MF-based
social recommendation algorithms can be categorized into three
groups: co-factorization methods [
22
,
46
], ensemble methods [
20
],
and regularization methods [
23
]. Besides, there are also studies
using socially-aware MF to model point-of-interest [
48
,
51
,
52
],
preference evolution [
39
], item ranking [
55
,
61
], and relation gen-
eration [11, 57].
Over the recent years, the boom of deep learning has broadened
the ways to explore social recommendation. Many research e๎orts
demonstrate that deep neural models are more capable of capturing
high-level latent preferences [
49
,
50
]. Speci๎cally, graph neural net-
works (GNNs) [
63
] have achieved great success in this area, owing
to their strong capability to model graph data. GraphRec [
9
] is the
๎rst to introduce GNNs to social recommendation by modeling the
user-item and user-user interactions as graph data. Di๎Net [
41
] and
its extension Di๎Net++ [
40
] model the recursive dynamic social
di๎usion in social recommendation with a layer-wise propagation
structure. Wu et al. [
42
] propose a dual graph attention network
to collaboratively learn representations for two-fold social e๎ects.
Song et al. develop DGRec [
34
] to model both usersโ session-based
interests as well as dynamic social in๎uences. Yu et al. [
58
] propose
a deep adversarial framework based on GCNs to address the com-
mon issues in social recommendation. In summary, the common
Self-Supervised Multi-Channel Hypergraph Convolutional Network for Social Recommendation WWW โ21, April 19โ23, 2021, Ljubljana, Slovenia
idea of these works is to model the user-user and user-item inter-
actions as simple graphs with pairwise connections and then use
multiple graph neural layers to capture the node dependencies.
2.2 Hypergraph in Recommender Systems
Hypergraph [
4
] provides a natural way to model complex high-
order relations and has been extensively employed to tackle various
problems. With the development of deep learning, some studies
combine GNNs and hypergraphs to enhance representation learn-
ing. HGNN [
10
] is the ๎rst work that designs a hyperedge convolu-
tion operation to handle complex data correlation in representation
learning from a spectral perspective. Bai et al. [
2
] introduce hyper-
graph attention to hypergraph convolutional networks to improve
their capacity. However, despite the great capacity in modeling com-
plex data, the potentials of hypergraph for improving recommender
systems have been rarely explored. There are only several studies
focusing on the combination of these two topics. Bu et al. [
5
] intro-
duce hypergraph learning to music recommender systems, which
is the earliest attempt. The most recent combinations are HyperRec
[
38
] and DHCF [
16
], which borrow the strengths of hypergraph
neural networks to model the short-term user preference for next-
item recommendation and the high-order correlations among users
and items for general collaborative ๎ltering, respectively. As for the
applications in social recommendation, HMF [
62
] uses hypergraph
topology to describe and analyze the interior relation of social
network in recommender systems, but it does not fully exploit
high-order social relations since HMF is a hybrid recommenda-
tion model. LBSN2Vec [
47
] is a social-aware POI recommendation
model that builds hyperedges by jointly sampling friendships and
check-ins with random walk, but it focuses on connecting di๎erent
types of entities instead of exploiting the high-order social network
structures.
2.3 Self-Supervised Learning
Self-supervised learning [
15
] is an emerging paradigm to learn
with the ground-truth samples obtained from the raw data. It was
๎rstly used in the image domain [
1
,
59
] by rotating, cropping and
colorizing the image to create auxiliary supervision signals. The
latest advances in this area extend self-supervised learning to graph
representation learning [
28
,
29
,
35
,
37
]. These studies mainly de-
velop self-supervision tasks from the perspective of investigating
graph structure. Node properties such as degree, proximity, and
attributes, which are seen as local structure information, are often
used as the ground truth to fully exploit the unlabeled data [
17
].
For example, InfoMotif [
31
] models attribute correlations in mo-
tif structures with mutual information maximization to regularize
graph neural networks. Meanwhile, global structure information
like node pair distance is also harnessed to facilitate representa-
tion learning [
35
]. Besides, contrasting congruent and incongruent
views of graphs with mutual information maximization [
29
,
37
] is
another way to set up a self-supervised task, which has also shown
promising results.
As the research of self-supervised learning is still in its infancy,
there are only several works combining it with recommender sys-
tems [
24
,
44
,
45
,
64
]. These e๎orts either mine self-supervision
signals from future/surrounding sequential data [
24
,
45
], or mask
attributes of items/users to learn correlations of the raw data [64].
However, these thoughts cannot be easily adopted to social rec-
ommendation where temporal factors and attributes may not be
available. The most relevant work to ours is GroupIM [
32
], which
maximizes mutual information between representations of groups
and group members to overcome the sparsity problem of group
interactions. As the group can be seen as a special social clique,
this work can be a corroboration of the e๎ectiveness of social self-
supervision signals.
3 PROPOSED MODEL
3.1 Preliminaries
Let
๐ = {๐ข
1
, ๐ข
2
, ..., ๐ข
๐
}
denote the user set (
|๐ | = ๐
), and
๐ผ =
{๐
1
, ๐
2
, ..., ๐
๐
}
denote the item (
|๐ผ | = ๐
).
I(๐ข)
is the set of user
consumption in which items consumed by user
๐ข
are included.
๐น โ R
๐ร๐
is a binary matrix that stores user-item interactions. For
each pair
(๐ข, ๐)
,
๐
๐ข๐
= 1
indicates that user
๐ข
consumed item
๐
while
๐
๐ข๐
= 0
means that item
๐
is unexposed to user
๐ข
, or user
๐ข
is not
interested in item
๐
. In this paper, we focus on top-K recommenda-
tion, and
^
๐
๐ข๐
denotes the probability of item
๐
to be recommended
to user
๐ข
. As for the social relations, we use
๐บ โ R
๐ร๐
to de-
note the relation matrix which is asymmetric because we work
on directed social networks. In our model, we have multiple con-
volutional layers, and we use
{๐ท
(1)
, ๐ท
(2)
, ยท ยท ยท , ๐ท
(๐)
} โ R
๐ร๐
and
{๐ธ
(1)
, ๐ธ
(2)
, ยท ยท ยท , ๐ธ
(๐)
} โ R
๐ร๐
to denote the user and item embed-
dings of size
๐
learned at each layer, respectively. In this paper,
we use bold capital letters to denote matrices and bold lowercase
letters to denote vectors.
De๎nition 1
: Let
๐บ = (๐, ๐ธ)
denote a hypergraph, where
๐
is
the vertex set containing
๐
unique vertices and
๐ธ
is the edge set
containing
๐
hyperedges. Each hyperedge
๐ โ ๐ธ
can contain any
number of vertices and is assigned a positive weight
๐
๐๐
, and all the
weights formulate a diagonal matrix
๐พ โ R
๐ร๐
. The hypergraph
can be represented by an incidence matrix
๐ฏ โ R
๐ ร๐
where
๐ป
๐๐
=
1 if the hyperedge
๐ โ ๐ธ
contains a vertex
๐ฃ
๐
โ ๐
, otherwise 0. The
vertex and edge degree matrices are diagonal matrices denoted by
๐ซ
and
๐ณ
, respectively, where
๐ท
๐๐
=
ร
๐
๐=1
๐
๐๐
๐ป
๐๐
; ๐ฟ
๐๐
=
ร
๐
๐=1
๐ป
๐๐
.
It should be noted that, in this paper,
๐
๐๐
is uniformly assigned 1
and hence ๐พ is an identity matrix.
3.2 Multi-Channel Hypergraph Convolutional
Network for Social Recommendation
In this section, we present our model
MHCN
, which stands for
M
ulti-channel
H
ypergraph
C
onvolutional
N
etwork. In Fig. 3, the
schematic overview of our model is illustrated.
3.2.1 Hypergraph Construction. To formulate the high-order infor-
mation among users, we ๎rst align the social network and user-item
interaction graph in social recommender systems and then build
hypergraphs over this heterogeneous network. Unlike prior models
which construct hyperedges by unifying given types of entities
[
5
,
47
], our model constructs hyperedges according to the graph
structure. As the relations in social networks are often directed, the
connectivity of social networks can be of various types. In this pa-
per, we use a set of carefully designed motifs to depict the common
WWW โ21, April 19โ23, 2021, Ljubljana, Slovenia Junliang Yu, Hongzhi Yin, Jundong Li, Qinyong Wang, Nguyen ๎oc Viet Hung, and Xiangliang Zhang
M
1
M
5
M
3
M
6
M
7
M
4
M
8
M
9
M
10
M
2
Social Motifs
Joint Motifs Purchase Motif
Follow Purchase
Figure 2: Triangle motifs used in our work. The green circles denote users and the yellow circles denote items.
types of triangular structures in social networks, which guide the
hypergraph construction.
Motif, as the speci๎c local structure involving multiple nodes, is
๎rst introduced in [
26
]. It has been widely used to describe com-
plex structures in a wide range of networks. In this paper, we only
focus on triangular motifs because of the ubiquitous triadic closure
in social networks, but our model can be seamlessly extended to
handle on more complex motifs. Fig. 2 shows all the used trian-
gular motifs. It has been revealed that
M
1
โ M
7
are crucial for
social computing [
3
], and we further design
M
8
โ M
10
to involve
user-item interactions to complement. Given motifs
M
1
โ M
10
,
we categorize them into three groups according to the underlying
semantics.
M
1
โ M
7
summarize all the possible triangular rela-
tions in explicit social networks and describe the high-order social
connectivity like โhaving a mutual friendโ. We name this group
โSocial Motifsโ.
M
8
โ M
9
represent the compound relation, that
is, โfriends purchasing the same itemโ. This type of relation can
be seen as a signal of strengthened tie, and we name
M
8
โ M
9
โJoint Motifsโ. Finally, we should also consider users who have no
explicit social connections. So, M
10
is non-closed and de๎nes the
implicit high-order social relation that users who are not socially
connected but purchased the same item. We name
M
10
โPurchase
Motif โ. Under the regulation of these three types of motifs, we can
construct three hypergraphs that contain di๎erent high-order user
relation patterns. We use the incidence matrices
๐ฏ
๐
,
๐ฏ
๐
and
๐ฏ
๐
to represent these three motif-induced hypergraphs, respectively,
where each column of these matrices denotes a hyperedge. For
example, in Fig. 3,
{๐ข
1
, ๐ข
2
, ๐ข
3
}
is an instance of
M
4
, and we use
๐
1
to denote this hyperedge. Then, according to de๎nition 1, we have
๐ป
๐
๐ข
1
,๐
1
= ๐ป
๐
๐ข
2
,๐
1
= ๐ป
๐
๐ข
3
,๐
1
= 1.
3.2.2 Multi-Channel Hypergraph Convolution. In this paper, we
use a three-channel setting, including โSocial Channel (s)โ, โJoint
Channel (j)โ, and โPurchase Channel (p)โ, in response to the three
types of triangular motifs, but the number of channels can be ad-
justed to adapt to more sophisticated situations. Each channel is
responsible for encoding one type of high-order user relation pat-
tern. As di๎erent patterns may show di๎erent importances to the
๎nal recommendation performance, directly feeding the full base
user embeddings
๐ท
(0)
to all the channels is unwise. To control
the information ๎ow from the base user embeddings
๐ท
(0)
to each
channel, we design a pre-๎lter with self-gating units (SGUs), which
is de๎ned as:
๐ท
(0)
๐
= ๐
๐
gate
(๐ท
(0)
) = ๐ท
(0)
โ ๐ (๐ท
(0)
๐พ
๐
๐
+ ๐
๐
๐
), (1)
where
๐พ
๐
๐
โ R
๐ร๐
, ๐
๐
๐
โ R
๐
are parameters to be learned,
๐ โ
{๐ , ๐, ๐}
represents the channel,
โ
denotes the element-wise prod-
uct and
๐
is the sigmoid nonlinearity. The self-gating mechanism
Table 1: Computation of motif-induced adjacency matrices.
Motif Matrix Computation ๐จ
๐
๐
=
M
1
๐ช = (๐ผ ๐ผ ) โ ๐ผ
๐
๐ช + ๐ช
โค
M
2
๐ช = (๐ฉ๐ผ ) โ ๐ผ
๐
+ (๐ผ ๐ฉ ) โ ๐ผ
๐
+ (๐ผ ๐ผ ) โ ๐ฉ ๐ช + ๐ช
โค
M
3
๐ช = (๐ฉ๐ฉ) โ ๐ผ + (๐ฉ๐ผ ) โ ๐ฉ + (๐ผ ยท ๐ฉ) โ ๐ฉ ๐ช + ๐ช
โค
M
4
๐ช = (๐ฉ๐ฉ) โ ๐ฉ ๐ช
M
5
๐ช = (๐ผ ๐ผ ) โ ๐ผ + (๐ผ ๐ผ
๐
) โ ๐ผ + (๐ผ
๐
๐ผ ) โ ๐ผ ๐ช + ๐ช
โค
M
6
๐ช = (๐ผ ๐ฉ) โ ๐ผ + (๐ฉ๐ผ
๐
) โ ๐ผ
๐
+ (๐ผ
๐
๐ผ ) โ ๐ฉ ๐ช
M
7
๐ช = (๐ผ
๐
๐ฉ) โ ๐ผ
๐
+ (๐ฉ๐ผ ) โ ๐ผ + (๐ผ ๐ผ
๐
) โ ๐ฉ ๐ช
M
8
๐ช = (๐น๐น
๐
) โ ๐ฉ ๐ช
M
9
๐ช = (๐น๐น
๐
) โ ๐ผ ๐ช + ๐ช
โค
M
10
๐ช = ๐น ๐น
๐
๐ช
e๎ectively serves as a multiplicative skip-connection [
8
] that learns
a nonlinear gate to modulate the base user embeddings at a feature-
wise granularity through dimension re-weighting, then we obtain
the channel-speci๎c user embeddings ๐ท
(0)
๐
.
Referring to the spectral hypergraph convolution proposed in
[10], we de๎ne our hypergraph convolution as:
๐ท
(๐+1)
๐
= ๐ซ
โ1
๐
๐ฏ
๐
๐ณ
โ1
๐
๐ฏ
โค
๐
๐ท
(๐)
๐
. (2)
The di๎erence is that we follow the suggestion in [
6
,
14
] to remove
the learnable matrix for linear transformation and the nonlinear
activation function (e.g. leaky ReLU). By replacing
๐ฏ
๐
with any of
๐ฏ
๐
,
๐ฏ
๐
and
๐ฏ
๐
, we can borrow the strengths of hypergraph convo-
lutional networks to learn user representations encoded high-order
information in the corresponding channel. As
๐ซ
๐
and
๐ณ
๐
are diago-
nal matrices which only re-scale embeddings, we skip them in the
following discussion. The hypergraph convolution can be viewed
as a two-stage re๎nement performing โnode-hyperedge-nodeโ fea-
ture transformation upon hypergraph structure. The multiplication
operation
๐ฏ
โค
๐
๐ท
(๐)
๐
de๎nes the message passing from nodes to hy-
peredges and then premultiplying
๐ฏ
๐
is viewed to aggregate infor-
mation from hyperedges to nodes. However, despite the bene๎ts of
hypergraph convolution, there are a huge number of motif-induced
hyperedges (e.g. there are 19,385 social triangles in the used dataset,
LastFM), which would cause a high cost to build the incidence ma-
trix
๐ฏ
๐
. But as we only exploit triangular motifs, we show that this
problem can be solved in a ๎exible and e๎cient way by leveraging
the associative property of matrix multiplication.
Following [
60
], we let
๐ฉ = ๐บ โ ๐บ
๐
and
๐ผ = ๐บ โ ๐ฉ
be the adjacency
matrices of the bidirectional and unidirectional social networks
respectively. We use
๐จ
๐
๐
to represent the motif-induced adjacency
matrix and
(๐จ
๐
๐
)
๐,๐
= 1
means that vertex
๐
and vertex
๐
appear
in one instance of
M
๐
. As two vertices can appear in multiple
