# Neural Graph Matching for Pre-training Graph Neural Networks

Yupeng Hou Work done during internship at Ant Group.Gaoling School of
Artificial Intelligence, Renmin University of China.
<EMAIL_ADDRESS><EMAIL_ADDRESS>Key
Laboratory of Big Data Management and Analysis Methods. Binbin Hu Ant Group.
<EMAIL_ADDRESS>Wayne Xin Zhao
22footnotemark: 2 33footnotemark: 3 Corresponding author. Zhiqiang Zhang
44footnotemark: 4 Jun Zhou 44footnotemark: 4 Ji-Rong Wen 22footnotemark: 2
33footnotemark: 3 School of Information, Renmin University of China.

###### Abstract

Recently, graph neural networks (GNNs) have been shown powerful capacity at
modeling structural data. However, when adapted to downstream tasks, it
usually requires abundant task-specific labeled data, which can be extremely
scarce in practice. A promising solution to data scarcity is to pre-train a
transferable and expressive GNN model on large amounts of unlabeled graphs or
coarse-grained labeled graphs. Then the pre-trained GNN is fine-tuned on
downstream datasets with task-specific fine-grained labels.

In this paper, we present a novel Graph Matching based GNN Pre-Training
framework, called GMPT. Focusing on a pair of graphs, we propose to learn
structural correspondences between them via neural graph matching, consisting
of both intra-graph message passing and inter-graph message passing. In this
way, we can learn adaptive representations for a given graph when paired with
different graphs, and both node- and graph-level characteristics are naturally
considered in a single pre-training task. The proposed method can be applied
to fully self-supervised pre-training and coarse-grained supervised pre-
training. We further propose an approximate contrastive training strategy to
significantly reduce time/memory consumption. Extensive experiments on multi-
domain, out-of-distribution benchmarks have demonstrated the effectiveness of
our approach. The code is available at: https://github.com/RUCAIBox/GMPT.

##  1 Introduction

In the past few years, graph neural networks (GNNs) have emerged as a powerful
technical approach for graph representation learning [9, 3]. By leveraging
graph structure as well as node and edge features, GNNs can effectively learn
low-dimensional representation vectors for each node or the entire graph.
However, to apply GNNs to downstream applications, it usually requires
abundant task-specific labeled data, which can be extremely scarce in
practice. To alleviate the data scarcity issue [16], pre-training GNNs [6, 7]
has been proposed as a promising solution. It first learns a transferable and
expressive GNN on a large number of unlabeled graphs or coarse-grained labeled
graphs. Then, the pre-trained GNN is fine-tuned on downstream datasets with
task-specific labels.

For GNN pre-training, existing studies mostly focus on the design of suitable
tasks, such as graph structure reconstruction [3, 6, 7], mutual information
maximization [21, 28, 17] and properties prediction [6]. Generally, these
tasks can be classified into two main categories: (1) _node-level tasks_
utilize node representations to predict the localized properties in the graph
(_e.g.,_ link prediction); (2) _graph-level tasks_ focus on the entire graph
and learn graph representations when designing the globalized optimization
goal (_e.g.,_ graph’s property prediction).

Figure 1: An example of neural graph matching and comparison with existing
studies of static graph representations.

Given the two kinds of GNN pre-training tasks, it is essential to combine
node- and graph-level optimization goals [6], since they capture the graph
characteristics in different views. Existing approaches either adopt a two-
stage approach arranging multi-level pre-training tasks sequentially [6], or
frame them in a multi-task learning manner [15]. In this way, each individual
pre-training task is not aware of all the optimization goals at different
levels, which might result in locally optimal representations _w.r.t._ some
specific level (_e.g.,_ node- or graph-level). Ideally, a good pre-training
task can capture node- and graph-level characteristics simultaneously in order
to derive more comprehensive node (and graph) representations.

For this purpose, we attempt to design new GNN pre-training tasks that are
able to learn node- and graph-level graph semantics _in one single pre-
training task_. Our solution is based on _neural graph matching_ [23, 13, 26],
a neural approach to learning structural correspondence among graphs. We
present an illustrative example of our idea in Figure 1. At each time, a pair
of two associated graphs (_e.g.,_ with the same labels or augmented graphs)
are given, and we evaluate whether the two graphs have similar structural
properties. As a major advantage, neural graph matching naturally combines
node-level correspondence (_e.g.,_ $v_{1}$ to $v_{2}$) and graph-level
properties (_e.g.,_ whether containing shared substructure) when establishing
their correspondence. That is the major reason why we adopt it as the GNN pre-
training task. Another merit of this approach is that a graph will correspond
to different representations when paired with different graphs. As shown in
this example, we will derive different representations for graph $A$ when
paired with graph $A_{1}$ or $A_{2}$, since neural graph matching will enforce
one graph to refer to another graph’s information when learning graph
representations. Therefore, we call the learned representations _adaptive
graph representations_. As a comparison, existing graph-level pre-training
tasks usually adopt static graph representations.

To this end, in this paper, we propose a novel Graph Matching based GNN Pre-
Training method, named as GMPT. The key contribution lies in a neural graph
matching module, where we pair two associated graphs as input at each time. To
learn structural correspondences, we perform intra-graph as well as inter-
graph message passing. In this way, the representations of a given graph are
learned by referring to another paired graph, which derives adaptive graph
representations. Such a method can capture both node- and graph-level
characteristics when learning the graph representations. The proposed method
can be applied to both fully self-supervised and coarse-grained supervised
pre-training settings. In self-supervised setting, GNNs are optimized by a
graph matching-based contrastive loss. To accelerate the learning of graph
pairs during pre-training, we further proposed an _approximate contrastive
training_ strategy to significantly reduce the time/memory consumption,
without loss of accuracy. While in supervised setting, we design different
supervised tasks according to different coarse-grained labels.

In summarization, we design a new GNN pre-training task based on neural graph
matching, devoted to adaptive graph representation learning by modeling both
node- and graph-level characteristics in a single pre-training task. We also
propose an approximate contrastive training strategy to reduce the time/memory
consumption. Extensive experiments on public out-of-distribution benchmarks
from multiple domains on various GNN architectures have demonstrated the
effectiveness of our approach.

Figure 2: Overall framework of our proposed graph matching-based GNN pre-
training methods.

##  2 Related Work

In this section, we review the most related work about pre-training graph
neural networks and graph matching.

###  2.1 Pre-training Graph Neural Networks

Though graph neural networks are powerful tools to characterize graph-
structured data, they heavily rely on fine-grained domain-specific labels
while training, which is usually scarce and difficult to obtain. To alleviate
the above issues, pre-training for graph neural networks has drawn much
attention recently, which empowers GNNs to capture the structural and semantic
information of the input unlabeled graphs (or with few coarse-grained labels),
followed by several fine-tuning steps on the downstream tasks of interest.
Obviously, developing effective supervised (self-supervised) signals to guide
GNNs to exploit structural and semantic properties on original graphs is at
the heart. Generally, existing designed supervised signals can be classified
into two main categories. The first is called _node-level tasks_ , which aims
at predicting localized properties utilizing node representations, such as
graph structure reconstruction [3, 6, 7, 15, 11], localized attribute
prediction [6, 7] and node representation recovery [4]. Another is called
_graph-level tasks_ , which defines globalized optimization goal for the
entire graph, such as graph property prediction [6, 12] and mutual information
maximization [21, 28, 17, 15, 27]. Our proposed framework differs from the
above approaches in the following two aspects: learning node- and graph-level
graph semantics _in one single pre-training task_ and _adaptive graph
representations_.

###  2.2 Graph Matching

Graph matching refers to establishing node correspondences between two (or
among multiple) graphs [1], such that the similarity between the matched
graphs is maximized. Some researches focus on the accuracy of the node
correspondence, and regard graph matching as a quadratic assignment
programming (QAP) problem [14], which is NP-complete. Thus, researchers mainly
employ approximate techniques to seek inexact solutions, such as spectral
approximation [10], double-stochastic approximation [2], and learning-based
approaches [1, 29, 23]. While others care about the similarity calculation
between graphs. Early efforts are mainly based on heuristic rules, such as
minimal graph edit distance [24] and graph kernel methods [8, 22]. With the
development of GNNs, recent work leverage message passing techniques to
explore neural-based graph matching [13, 26]. In this work, we apply neural
graph matching for GNN pre-training, to learn adaptive graph representations
and encourage GNNs to integrate localized and globalized domain-specific
features.

##  3 The Proposed Method

In this section, we first introduce the notation and problem definition. Then
we present the proposed graph matching based GNN pre-training method GMPT for
both self-supervised setting and coarse-grained supervised setting. Our
approach takes pairs of graphs as input. With a carefully designed cross-graph
message passing mechanism, one graph can be adaptively encoded into different
graph representations when paired with different graphs. Figure 2 presents the
overall architecture of our proposed framework.

###  3.1 Notation and Problem Definition

A graph can be represented as $G=(\mathcal{V},\mathcal{E},\bm{X},\bm{E})$,
where $\mathcal{V}=\\{v_{1},v_{2},\ldots,v_{|\mathcal{V}|}\\}$ denotes the
node set, $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ denotes the edge
set, $\bm{X}\in\mathbb{R}^{|\mathcal{V}|\times d_{v}}$ and
$\bm{E}\in\mathbb{R}^{|\mathcal{E}|\times d_{e}}$ represent the $d_{v}$\- and
$d_{e}$-dimensional attribute matrix for nodes and edges, respectively.
Furthermore, each graph is possibly associated with some label $y$ from a
label set $\mathcal{Y}$. Given a set of graphs with labels
$\\{(G_{1},y_{1}),(G_{2},y_{2}),\ldots,(G_{N},y_{N})\\}$, graph neural
networks (GNNs) leverage graph structure as well as node and edge features to
learn a representation vector for the entire graph $\bm{h}_{G}$, and further
utilize $\bm{h}_{G}$ to predict the corresponding label $y$. A detailed
preliminary of GNNs is provided in Supplementary Material S.1111Please refer
to:
https://github.com/RUCAIBox/GMPT/blob/main/paper/supplementary_material_gmpt.pdf.

In this work, we focus on pre-training GNNs: GNNs that are initialized with
pre-trained parameters are fine-tuned according to various downstream tasks.
Given the defined graph learning task, we consider two kinds of GNN pre-
training paradigms based on whether graph labels are used or not during pre-
training: _self-supervised setting_ (without graph labels) and _supervised
setting_ (with graph labels).

###  3.2 Self-supervised Pre-training

We have no available labeled data for pre-training in self-supervised setting.
The pre-training task is to evaluate whether a pair of augmented graph views
are generated based on the same graph. We adopt contrastive learning to
construct the learning objective and name our approach in this setting as
GMPT-CL.

####  3.2.1 Graph Representation Learning via Graph Matching

We first present how to learn graph representations via graph matching.

Graph augmentation and encoding. Given a list of $n$ graph examples, we first
apply stochastic data augmentation to transform any given graph example into
two correlated views randomly ($2n$ views in total). We consider various
augmentation techniques, including node/edge perturbation [28], subgraph
sampling [28], diffusion [5], and adaptive methods [30]. The selection of
graph augmentation techniques depends on the actual data domain [28]. To pre-
train an expressive GNN encoder, we consider whether a pair of graph views
denoted by $\tilde{G}_{1}$ and $\tilde{G}_{2}$ are matched or not based on
their graph representations. Specially, we first apply the GNN encoder to
obtain node representations in the two graph views. Let $\bm{h}^{(1)}_{s}$ and
$\bm{h}^{(2)}_{t}$ denote the representations of node $s$ from $\tilde{G}_{1}$
and node $t$ from $\tilde{G}_{2}$, respectively.

Neural graph matching. Following recent progress in neural graph matching [13,
23], we incorporate message passing within a graph (called _intra-graph
message_) and between a pair of graphs (called _inter-graph messages_). Given
a intra-graph node pair $\langle s,t\rangle$ and a inter-graph node pair
$\langle s^{\prime},t^{\prime}\rangle$, we define the two kinds of message
passing mechanisms formally as:

(3.1) $\displaystyle\bm{m}_{s\rightarrow t}$
$\displaystyle=\operatorname{MSG}_{\operatorname{intra}}\left(\bm{h}_{s}^{(1)},\bm{h}_{t}^{(1)},\bm{e}_{st}\right),$
(3.2) $\displaystyle\bm{\mu}_{s^{\prime}\rightarrow t^{\prime}}$
$\displaystyle=\operatorname{MSG}_{\operatorname{inter}}\left(\bm{h}_{s^{\prime}}^{(1)},\bm{h}_{t^{\prime}}^{(2)}\right),$

where $\bm{m}_{s\rightarrow t}$ and $\bm{\mu}_{s^{\prime}\rightarrow
t^{\prime}}$ are intra-graph and inter-graph messages, respectively. Intra-
graph message passing can be defined in a similar way following standard GNN
architectures, like GIN [25]. While for
$\operatorname{MSG}_{\operatorname{inter}}$, we adopt a cross-graph attention
mechanism as:

$\displaystyle\bm{\mu}_{s^{\prime}\rightarrow t^{\prime}}=$ $\displaystyle\
a_{s^{\prime}\rightarrow t^{\prime}}\cdot\bm{h}_{s^{\prime}}^{(1)},$

where $a_{s^{\prime}\rightarrow
t^{\prime}}=\frac{\exp(\operatorname{sim}(\bm{h}_{s^{\prime}}^{(1)},\bm{h}_{t^{\prime}}^{(2)}))}{\sum_{k\in\tilde{G}_{2}}\exp(\operatorname{sim}(\bm{h}_{s^{\prime}}^{(1)},\bm{h}_{k}^{(2)}))}$
and $\operatorname{sim}(\cdot)$ is a similarity function, such as dot product
and cosine similarity. The above attention mechanism allows an adaptive
message exchange between the paired graphs. Intuitively, messages passed
between similar substructures of graphs will have higher attention weights.
Here, we normalize the attention weights of messages from the same _source_
node, which means $\sum_{t^{\prime}}a_{s^{\prime}\rightarrow t^{\prime}}=1$.
Besides, it is also optional to normalize the attention weights of messages to
the same _target_ nodes ($\sum_{s^{\prime}}a_{s^{\prime}\rightarrow
t^{\prime}}=1$).

Match enhanced graph representations. After passing intra-graph messages from
nodes’ neighbors (denoted by $\mathcal{N}_{\text{intra}}$) and inter-graph
messages from all the nodes of another graph (denoted by
$\mathcal{N}_{\text{inter}}$), we aggregate the messages together and update
to obtain the contextual node features $\bm{Z}$. For a node $t$, we update its
original representation $\bm{h}_{t}$ as:

(3.3) $\displaystyle\bm{z}_{t}$
$\displaystyle=\operatorname{Update}\left(\bm{h}_{t},\sum_{s\in\mathcal{N}_{\text{intra}}}\bm{m}_{s\rightarrow
t},\sum_{s^{\prime}\in\mathcal{N}_{\text{inter}}}\bm{\mu}_{s^{\prime}\rightarrow
t}\right),$

where we use the sum operation for aggregation. Finally, we obtain the entire
graph’s adaptive representation $\bm{z}_{G}$ by employing a permutation-
invariant function $\operatorname{READOUT}$ to pool contextual node features:

(3.4) $\displaystyle\bm{z}_{G}$
$\displaystyle=\operatorname{READOUT}\left(\\{\bm{z}_{v}|v\in\mathcal{V}\\}\right).$

Note that when involved in different pairs, a given graph will correspond to
different representations in our approach. It is a key merit for subsequent
pre-training tasks, since it can adaptively capture structural correspondences
instead of using static graph representations as in previous studies [6, 28].

####  3.2.2 Contrastive Learning with Adaptive Graph Representations

Contrastive learning is a commonly used technique to learn with augmented
graph views in pairs [28]. It aims to increase similarity scores for positive
pairs and decrease similarity scores for negative pairs. However, existing
graph contrastive learning methods mainly adopt static graph representations
[28, 5, 30], where node-level interaction across graphs is not explicitly
modeled in this process. As a comparison, given a pair of graph views, we
first apply the neural graph matching technique (Section 3.2.1) to
characterize inter-graph interaction, and then construct the contrastive loss
based on the adaptive graph representations.

Formally, given a positive pair $(\tilde{G_{i}},\tilde{G_{j}})$, we firstly
adaptively encode them into contextual graph representations
$\bm{z}_{\tilde{G_{i}}}$ and $\bm{z}_{\tilde{G_{j}}}$ (Eqn. (3.4)), and then
formalize the contrastive loss below:

(3.5) $\ell_{i,j}=-\log\frac{\exp\left(s_{i,j}/\tau\right)}{\sum_{k\neq
i}\exp\left(s_{i,k}/\tau\right)},$

where
$s_{i,j}=\operatorname{sim}(\bm{z}_{\tilde{G_{i}}},\bm{z}_{\tilde{G_{j}}})$
and $\tau$ is a temperature parameter. In practice, we usually have a batch of
graph views, and we enumerate all the possible pairs of graphs in a batch for
this loss in the denominator of Eqn. (3.5).

Although the above contrastive loss is also defined at the graph level
(whether two views are augmented from the same graph), the derived graph
representations $\bm{z}_{\tilde{G_{i}}}$ and $\bm{z}_{\tilde{G_{j}}}$ are
enhanced with inter-graph node interaction via neural graph matching. As such,
optimizing $\ell_{i,j}$ will encourage GNNs to capture both node- and graph-
level characteristics in graph representations.

####  3.2.3 Approximate Contrastive Training

A major issue with graph matching is it incurs a quadratic time and space cost
in terms of the number of nodes. Here, we propose an approximate contrastive
training strategy to improve algorithm efficiency.

Complexity analysis. We consider the setting with a mini-batch of $n$ graphs.
As mentioned before, we would generate $2n$ augmented graph views for graph
matching and contrastive learning. Typically, for a mini-batch of $n$ graphs,
GMPT-CL considers $2n\times 2n$ times of graph comparisons (two augmented
views each graph) in total. Each comparison performs a node-to-node similarity
calculation (refer to Eqn. (3.2)), taking an additional cost of $O(m^{2}\cdot
d)$ time and space, where $m=\sum^{2n}_{i=1}|\mathcal{V}_{i}|$ denotes the
total number of nodes in $2n$ graph views and $d$ is the dimensionality of
representation vectors.

Approximate calculation. In order to reduce time and memory consumption, the
key idea is to perform an approximate calculation of the proposed contrastive
loss (Eqn. (3.5)): we sample $q$ out of $2n$ graph views to contrast with all
the other views ($q\times 2n$ times comparisons totally). In this way, the
additionally expected time complexity can be reduced to $O(\frac{q}{2n}\cdot
m^{2}\cdot d)$. For a further reduction of space complexity, we adopt the
gradient accumulation technique. For each time of sampling, we perform
$1\times 2n$ times of comparison. After calculating the contrastive loss, the
model backpropagates prediction error and calculates the gradients, but
doesn’t update model parameters immediately. Instead, the gradients are
accumulated until all the $q$ samples are calculated. In this way, the sampled
$q$ graphs only require an additional space complexity of
$O\left(\frac{1}{2n}\cdot m^{2}\cdot d\right)$.

Theoretical analysis. We provide a theoretical analysis to reveal the
connection between GMPT-CL with approximate contrastive training and mutual
information maximization. Firstly, we show that,

###### Lemma 3.1

Minimizing Eqn. (3.5) is equivalent to maximizing a lower bound of the mutual
information between the latent representations of two views of graphs.

Furthermore, we show that,

###### Lemma 3.2

Optimizing Eqn. (3.5) with approximate contrastive training algorithm has the
same optimization lower bound as originally in expectation.

Thus we can see that the proposed approximate contrastive training fits the
formulation of the InfoNCE loss [19, 18] in expectation. The proofs are
provided in Supplementary Material S.3.

Empirically, experiments in Section 4.3 will show that performances of the
fine-tuned GNNs on downstream tasks don’t drop (even improve) with the
proposed approximate contrastive training. The overall pre-training algorithm
of GMPT-CL in one mini-batch is provided in Supplementary Material S.3.

###  3.3 Supervised Pre-training

Besides a large number of unlabeled graphs for self-supervised pre-training,
sometimes we can obtain graphs with coarse-grained labels. Different from
elaborately created fine-grained labels, coarse-grained labels have a
relatively weak correlation with downstream task goals, but can be obtained in
an easier way. For example, in molecular property prediction, we can easily
collect the properties of molecules that have been experimentally measured so
far [6].

In supervised pre-training setting, we pre-train GNNs on graphs with coarse-
grained labels, and then the pre-trained GNNs are fine-tuned according to
fine-grained labels in downstream tasks. Note that coarse-grained labels used
for supervised pre-training are not the real labels of downstream tasks. Based
on whether coarse-grained labels are continuous or discrete, we propose two
variants of GMPT in supervised setting, named as GMPT-Sup and
GMPT-$\text{Sup}_{++}$, respectively.

Continuous labels. Labels with continuous properties can be regarded as real-
value vectors. We assume that similar pairs of graphs also correspond to
similar labels. Based on this consideration, we propose GMPT-Sup to learn the
similarities between graphs via graph matching module, and then minimize the
difference between the learned similarity and the actual label similarity.
Given a pair of graphs $G_{1}$ and $G_{2}$ and their coarse-grained labels
$y_{1}$ and $y_{2}$, we firstly obtain their adaptive graph representations
$\bm{z}_{G_{1}}$ and $\bm{z}_{G_{2}}$ as Eqn. (3.4). Then we define the loss
function as $\ell_{c}=\operatorname{MSE}(s_{p},s_{g})$, where
$s_{p}=\operatorname{sim}(\bm{y}_{1},\bm{y}_{2}),s_{g}=\operatorname{sim}(\bm{z}_{G_{1}},\bm{z}_{G_{2}})$,
and $\operatorname{MSE}(\cdot,\cdot)$ denotes the standard mean squared error
loss. This loss drives the representation similarity of two graphs to be close
to their label similarity.

Discrete labels. For discrete labels, we do not enforce a direct comparison of
the two graphs in a pair. Instead, we construct a classification-based
approach GMPT-$\text{Sup}_{++}$ to associated graph representations with
suitable coarse-grained labels. For a given pair of graphs $G_{1}$ and $G_{2}$
(encoded by graph matching module into $\bm{z}_{G_{1}}$ and $\bm{z}_{G_{2}}$),
we jointly predict the corresponding coarse-grained labels $y_{1}$ and
$y_{2}$. The loss for discrete labels $\ell_{d}$ can be calculated as,

$\displaystyle\ell_{d}=\sum_{k=1,2}\operatorname{BCE}(\bm{y}_{k},\bm{W}_{k}\cdot\bm{z}_{G_{k}}+\bm{b}_{k}),$

where $\operatorname{BCE}(\cdot,\cdot)$ denotes the popular binary cross-
entropy loss, and $\bm{y}_{1}$ and $\bm{y}_{2}$ are vectorized representations
of labels $y_{1}$ and $y_{2}$, respectively. Although the loss of the two
graphs in a pair is calculated separately, their representations are obtained
from the graph matching module (_e.g.,_ cross-graph message passing in Eqn.
(3.2)).

##  4 Experiments

In this section, we conduct extensive experiments to verify the effectiveness
of our proposed methods in both self-supervised and supervised settings.
Moreover, we also give an in-depth analysis of training strategy and
transferability. Additional experiments on key parameters sensitivity analysis
are provided in Supplementary Material S.5.4.

Table 1:  Statistics of the datasets. _PT_ denotes Pre-Training and _FT_
denotes Fine-Tuning. * denotes the total number of tasks for the eight
downstream datasets.

Dataset | Bio | Chem
---|---|---
#(sub)graphs for self-supervised PT |  $307$K |  $2,000$K
#(sub)graphs for supervised PT/FT |  $88$K |  $456$K
#Coarse-grained labels for PT | $5,000$ | $1,310$
#Downstream FT tasks | $40$ |  $678$*

###  4.1 Experimental Setup

####  4.1.1 Datasets

We conduct experiments on two public out-of-distribution (sub)graph
classification benchmarks from different domains, namely Bio and Chem [6]. We
strictly adopt the same way of splitting and pre-processing of these
benchmarks as previous work [6]. Dataset statistics are summarized in Table 1.
Detailed descriptions of the datasets are given in Supplementary Material
S.5.1.

Table 2:  Evaluation in self-supervised setting. We test ROC-AUC (%) performance using different pre-training methods. Besides, performances with different GNN architecture on Bio are also presented. The macro-average results over all GNN architectures on Bio, and results of GIN over all subtasks on Chem are also reported. In L2P-GNN, GNN is fine-tuned with a parameterized global pooling layer, while others use average pooling.  Pre-training methods | Bio | Chem
---|---|---
GCN | GraphSAGE | GAT | GIN | Average
w/o pre-training | $63.20\pm 1.00$ | $65.70\pm 1.20$ | $68.20\pm 1.10$ | $64.80\pm 1.00$ | $65.48$ | $67.0$
Infomax | $62.83\pm 1.22$ | $67.21\pm 1.84$ | $66.94\pm 2.61$ | $64.10\pm 1.50$ | $65.27$ | $70.3$
EdgePred | $63.18\pm 1.12$ | $66.05\pm 0.78$ | $65.72\pm 1.17$ | $65.70\pm 1.30$ | $65.16$ | $70.3$
ContextPred | $62.81\pm 1.87$ | $66.47\pm 1.27$ | $67.86\pm 1.19$ | $65.20\pm 1.60$ | $65.59$ | $71.1$
AttrMasking | $62.40\pm 1.35$ | $63.32\pm 1.01$ | $61.72\pm 2.70$ | $64.40\pm 1.30$ | $62.96$ | $70.9$
GraphCL | $67.05\pm 1.16$ |  71.53 $\pm 0.46$ | $65.68\pm 3.98$ | $67.88\pm 0.85$ | $68.04$ | $70.8$
L2P-GNN | $66.48\pm 1.59$ | $69.89\pm 1.63$ | $69.15\pm 1.86$ | $70.13\pm 0.95$ | $68.91$ | $70.4$
GMPT-CL |  70.65 $\pm\ 0.53$ | $70.29\pm 0.21$ |  71.07 $\pm\ 0.14$ |  72.53 $\pm\ 0.42$ | 71.13 | 71.5

####  4.1.2 Baselines

We compare our pre-training methods with the following representative GNN pre-
training methods:
$-$ Infomax [21] maximizes the mutual information between patch
representations and corresponding high-level summaries of graphs.
$-$ EdgePred [3] directly predicts the connectivity of node pairs, _a.k.a.,_
link prediction task.
$-$ ContextPred [6] uses subgraphs to predict their surrounding graph
structures.
$-$ AttrMasking [6] predicts nodes’ or edges’ attributes, which are randomly
masked.
$-$ GraphCL [28] contrast the static representations of augmented views and
judge whether they are generated from the same graph.
$-$ L2P-GNN [15] utilizes meta-learning to alleviate the divergence between
multi-task pre-training and fine-tuning objectives.
$-$ PropPred [6] predicts the coarse-grained labels of graphs in the pre-
training datasets.

Among the baselines, PrepPred is designed for supervised pre-training setting,
while others are targeted at self-supervised pre-training setting. Moreover,
results of the non-pre-trained model are also reported.

####  4.1.3 Parameter Settings

To enhance the reproducibility, we elaborately present the implementation
details as follows. Detailed hyper-parameters for GNN architecture and
training are provided in Supplementary Material S.5.2.

GNN architecture. We mainly experiment on Graph Isomorphism Networks (GINs)
[25], the most expressive GNN architecture for graph-level prediction tasks.
We also experiment with other popular architectures: GCN [9], GraphSAGE [3]
and GAT [20]. We select the same GNN hyper-parameters as previous works [6,
15]. For the proposed graph matching methods, we adopt dot production for the
$\operatorname{sim}(\cdot)$ function and select $\tau=0.07$ in Eqn. (3.5). We
utilize a multiple layer perceptron (MLP) as function of
$\operatorname{Update}$ in Eqn. (3.3).

Pre-training and fine-tuning settings. Results of baselines on different
datasets are directly taken if they have been reported literaturely. For the
other method, we pre-train the models with a learning rate of $0.001$, and
fine-tune the GNNs with a learning rate tuned in $\\{0.01,0.001,0.0001\\}$ for
all the methods. We report the ROC-AUC for both datasets. The downstream
experiments are run with $10$ random seeds, and we report the mean and
standard deviation of the metrics.

###  4.2 Performance Comparison

####  4.2.1 Self-supervised Setting

Table 2 presents the performance comparison in self-supervised pre-training
setting between GMPT-CL and the baselines.

The proposed pre-training method GMPT-CL achieves the best performance
$72.53\%$ over all the compared methods on Bio. with the most expressive GNN
architecture GIN. On Chem dataset, we also notice that GMPT-CL gains the best
results ($71.5\%$) compared to all the baselines.

Applying GMPT-CL on currently popular GNN architectures, as shown in Table 2,
GMPT-CL achieves the best macro-average result ($71.13\%$) over all the
compared baseline methods. In particular, we can see that GMPT-CL is powerful
even with less expressive GNN architectures like GCN or GAT, which brings
$3.60\%$ and $1.92\%$ absolutely gains compared to the best baseline,
respectively.

In sum, we make the following observations.

(1) On average, the proposed GMPT-CL yields the best performance on benchmarks
of different domains ($72.53\%$ on Bio and $71.5\%$ on Chem). As GMPT-CL is a
hybrid-level pre-training task, it encourages GNNs to capture both localized
and globalized domain-specific semantics. Graph matching module of GMPT-CL can
generate adaptive graph representations, in which shared substructures are
enhanced.

(2) Pre-training GNNs with a large amount of unlabeled data is clearly helpful
to downstream tasks, as GMPT-CL brings $6.69\%$ and $4.5\%$ absolutely gains
compared to non-pre-trained models on the macro-average results over datasets
of two domains, respectively.

####  4.2.2 Supervised Setting

Table 3:  Evaluation in supervised setting. We report ROC-AUC (%) performance on Bio and Chem using different supervised pre-training methods with GIN.  Pre-training methods | Bio | Chem
---|---|---
w/o pre-training | $64.8\pm 1.0$ | $67.0$
PropPred | $69.0\pm 2.4$ | $70.0$
GMPT-Sup | 70.84 $\pm$ 0.59 | –
GMPT-$\text{Sup}_{++}$ | $70.73\pm 0.42$ | 70.4

Table 3 presents the performance comparison between GMPT-Sup,
GMPT-$\text{Sup}_{++}$, and the baselines (i.e., non-pre-trained GIN and
PropPred) in supervised pre-training setting. Though coarse-grained labels in
Bio datasets are multi-hot vectors, we still view them as continuous
properties in GMPT-Sup. While coarse-grained labels in Chem datasets contain
plenty of missing values. As similarity over missing values is hard to define,
GMPT-Sup is not a suitable method for labels with missing values, and we only
report the result of GMPT-$\text{Sup}_{++}$ on Chem.

The compared baseline PropPred always outperforms the non-pre-trained method,
reflecting that GNNs pre-trained on coarse-grained labels can characterize
domain-specific semantics. Compared with PropPred, which encodes graphs into
static representations with a graph-level pre-training task, the proposed
hybrid-level methods GMPT-Sup and GMPT-$\text{Sup}_{++}$ generate adaptive
graph representations (Section 3.2.1). Consequently, we can see that the
proposed methods achieve the best performances on Bio and Chem, respectively,
demonstrating the effectiveness of our pre-training methods. Besides, we
notice that GMPT-Sup outperforms GMPT-$\text{Sup}_{++}$ slightly on Bio
dataset. A possible reason is that directly predicting every single property
of coarse-grained labels (as GMPT-$\text{Sup}_{++}$) may cause overfitting and
limit the transferability of the pre-trained model, especially when coarse-
grained labels lack precious domain-specific semantics.

Figure 3: Tuning of approximate contrastive training with different numbers of
sampled views on Bio dataset. (a) Performance with different batch sizes. (b)
Time/Memory Consumption with batch size $32$. Labels on the histogram indicate
memory consumption.

###  4.3 Training Strategy Analysis

As mentioned above, we propose an approximate contrastive training strategy to
reduce the time and space consumption of GMPT-CL and the corresponding
theoretical analysis about time and memory complexity can be found in In
Section 3.2.3. Here we conduct detailed experiments to show how the fine-tuned
GNNs’ performance is affected by the batch size $n$ and number of sampled
views $q$, and what’s the actual running time/memory when the proposed
approximate training strategy is applied.

####  4.3.1 Performance Comparison _w.r.t._ Batch Size and the Number of
Sampled Views

We pre-train GIN models in different batch sizes $n\in\\{4,8,16\\}$ and
numbers of sampled views $q\in\\{1,2,4,8,16,32\\}$, and compare the fine-tuned
model’s performance on downstream dataset of Bio. Note that for a certain
batch size $n$, the maximum number of sampled views is $q=2n$ (totally $2n$
views).

As Figure 3 shows, with different batch sizes and number of sampled views, our
method can reach no less than $69\%$ ROCAUC performance. We notice that
applying approximate contrastive training usually improves the performance of
GMPT-CL. Generally, GMPT-CL reaches the best performance when the number of
sampled views $q=4$. We speculate that the performance gain comes from the
randomness introduced by contrastive approximate training. Besides, we find
that even with a small sampling number $q=1$, our method can still have
competitive performances.

####  4.3.2 Time and Memory Consumption _w.r.t._ the Number of Sampled Views

Here, we take an intuitive look at the actual time and space consumption of
the proposed approximate contrastive training strategy. As Figure 3 shows, we
find that as $q$ decreases, the training time of GMPT-CL per epoch also
decreases, which verifies that choosing a relatively small $q$ can
dramatically reduce the training time of GMPT-CL. Besides, we find that the
memory consumption doesn’t change a lot with different choices of $q$,
verifying that the space complexity doesn’t affect much by $q$.

In summary, we suggest adopting the proposed approximate contrastive training
strategy when pre-training GNNs with GMPT-CL, which has been shown to reduce
the time/space consumption and gain a slight performance increasing. For the
choice of the number of sampled views $q$ with the batch size $n$, we suggest
to select a relative small $q$ (i.e., $4\leq q\leq n$).

Figure 4: Analysis of transferability for (a) L2P-GNN and (b) GMPT-CL over
$40$ out-of-distribution subtasks of Bio. “PT” denotes as “pre-training”. The
green line indicates the borderline. The purple line indicates the worst
negative transfer across the 40 subtasks.

###  4.4 Transferability Analysis

Out-of-distribution (OOD) problem widely exists in real-world applications,
meaning that graphs in the training set are structurally very different from
graphs in the test set [6]. Existing study shows that improperly designed GNN
pre-training tasks may cause serious negative transfer. Thus, we analyze the
proposed GMPT-CL and the best baseline L2P-GNN to analyze the transfer status
over the individual subtasks of out-of-distribution datasets. The left and up
area indicates positive transfer, and the right and bottom area indicates
negative transfer. As shown in Figure 4, we can see that compared to the best
baseline L2P-GNN, the proposed GMPT-CL has less negative transfer cases ($12$
_v.s._ $17$), as well as a slighter negative transfer extent ($-0.07$ _v.s._
$-0.13$). Besides we can see that GNN pre-trained by GMPT-CL gets AUC result
$>0.5$ on all downstream subtasks of Bio. All the observations above show the
good transferability of the proposed GMPT-CL method.

##  5 Conclusion

In this work, we propose GMPT, a general graph matching-based GNN pre-training
framework for both self-supervised pre-training and coarse-grained supervised
pre-training. By structuralized neural graph matching module, we generate
adaptive representations for the matched graphs, encouraging GNNs to learn
both globalized and localized domain-specific semantics in a single pre-
training task. We also propose approximate contrastive training strategy,
which significantly reduces the time/memory consumption brought by the graph
matching module. Extensive experiments on multi-domain out-of-distribution
benchmarks show the effectiveness and transferability of our method.

Besides GMPT, more hybrid-level GNN pre-training tasks can be explored in the
future. In addition, we will also consider generalizing our framework to more
complicated graph structures (e.g. dynamic graphs, knowledge graphs, and
heterogeneous graphs).

## Acknowledgement

This work was partially supported by the National Natural Science Foundation
of China under Grant No. 61872369 and 61832017, Beijing Outstanding Young
Scientist Program under Grant No. BJJWZYJH012019100020098, and CCF-Ant Group
Research Fund. Xin Zhao is the corresponding author.

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# Entanglement Polygon Inequalities for A Class of Mixed States

Xian Shi<EMAIL_ADDRESS>College of Information Science and Technology,
Beijing University of Chemical Technology, Beijing 100029, China

###### Abstract

The study on the entanglement polygon inequality of multipartite systems has
attracted much attention. However, most of the results are on pure states.
Here we consider the property for a class of mixed states, which are the
reduced density matrices of generalized W-class states in multipartite higher
dimensional systems. First we show the class of mixed states satisfies the
entanglement polygon inequalities in terms of Tsallis-q entanglement, then we
propose a class of tighter inequalities for mixed states in terms of Tsallis-q
entanglement. At last, we get an inequality for the mixed states, which can be
regarded as a relation for bipartite entanglement.

###### pacs:

03.65.Ud, 03.67.Mn

##  I introduction

Entanglement is an essential feature in quantum mechanics compared with the
classical horodecki2009quantum ; plenio2014introduction . It serves a key role
in many information tasks, such as quantum teleportation
bennett1993teleporting ; bouwmeester1997experimental , quantum dense coding
harrow2004superdense , and quantum cryptography ekert1991quantum ;
mirhosseini2015high .

One of the most meaningful properties of entanglement in multipartite systems
is that entanglement cannot be freely shared. Monogamy of entanglement (MoE)
coffman2000distributed ; zong2022monogamy  is one manifestation of the
property. For a tripartite entangled state $\rho_{ABC}$, the mathematical
representation of MoE in terms of an entanglement measure $E$ is that

$\displaystyle E_{A|B}+E_{A|C}\leq E_{A|BC}.$ (1)

The above inequality means that the sum of entanglement in $AB$ and $AC$ is
bounded by entanglement between subsystem $A$ and the remaining $BC$. This
relation is valid in terms of concurrence coffman2000distributed , the
squashed entanglement measure christandl2004squashed , and Tsallis-$q$
entanglement kim2010tsallis ; luo2016general  for $n$-qubit systems. Except
(1), other representations of MoE have been proposed gour2018monogamy ;
guo2020multipartite ; shi2021multilinear ; jin2022new ; guo2023complete .

Recently, the other entanglement distribution property, entanglement polygon
inequality (EPI), was proposed for pure states in multiqubit systems in terms
of some entanglement measure $E$ qian2018entanglement ,

$\displaystyle E_{j|\overline{j}}\leq\sum_{k\neq j}E_{k|\overline{k}},$

here $j$ denotes a subsystem $\mathcal{H}_{j}$, $\overline{j}$ is the
remaining subsystems. Recently, EPI was proved valid for pure states in
multiqudit systems in terms of some entanglement measures yang2022entanglement
; shi2023entanglement . Based on the property, a method to build the genuine
entanglement measure for multipartite systems was proposed xie2021triangle .
However, there are few results on EPI for mixed states of multipartite systems
in terms of some entanglement measure.

The generalized $W$ class (GW) states of multipartite higher dimensional
systems own fine properties on their entanglement distribution. In 2008, Kim
and Sanders showed that the GW states in multipartite higher dimensional
systems satisfy the monogamy relations in terms of the squared concurrence
san2008generalized . Moreover, there the authors showed that inequality (1) is
saturated for the GW states in terms of the squared concurrence. In recent
years, there have been many results on monogamy of entanglement for the GW
states in terms of convex roof extended negativity choi2015negativity ,
Tsallis-$q$ entanglement shi2020monogamy , Renyi-$\alpha$ entanglement
lai2021tighter , and the unified-$(q,s)$ entanglement li2024monogamy .

This article is organized as follows. In section II, we present the
preliminary knowledge of this article. In section III, we present our main
results. First, we present a class of tripartite mixed states satisfy the EPI
in terms of Tsallis-$q$ entanglement, which are generated by the partial trace
map over a generalized $W$-class state in terms of Tsallis-$q$ entanglement,
and then we show a tighter inequality for the mixed states. At last, we
present another type of inequalities for the mixed states. In section IV, we
end with a summary.

##  II Preliminary Knowledge

In this section, we present the preliminary knowledge needed here. First, we
present the definitions and properties of entanglement measures for bipartite
systems, and then we recall the generalized $W$-class states in multipartite
higher dimensional systems.

###  II.1 Entanglement measures

Given a bipartite pure state
$|\psi\rangle_{AB}=\sum_{i}\sqrt{\lambda_{i}}|ii\rangle,$ the concurrence is
defined as

$\displaystyle C(|\psi\rangle_{AB})=\sqrt{2(1-\mathrm{tr}\rho_{A}^{2})},$ (2)

where $\rho_{A}=\mathrm{tr}_{B}\rho_{AB}$. When $\rho_{AB}$ is a mixed state,
its concurrence is defined as

$\displaystyle
C(\rho_{AB})=\min_{\\{p_{i},|\psi_{i}\rangle_{AB}\\}}\sum_{i}p_{i}C(|\psi_{i}\rangle_{AB}),$
(3)

where the minimum takes over all the decompositions of
$\rho_{AB}=\sum_{i}p_{i}|\psi_{i}\rangle_{AB}\langle\psi_{i}|.$

For a quantum state $\rho$, the Tsallis-$q$ entropy is

$\displaystyle T_{q}(\rho)=\frac{1}{q-1}[1-\mathrm{tr}\rho^{q}],$

for $q\geq 1.$ When $q\rightarrow 1$, $T_{q}(\cdot)$ converges to the von
Neumann entropy $S(\cdot)$

$\displaystyle\lim\limits_{q\rightarrow
1}T_{q}(\rho)=-\mathrm{tr}\rho\log\rho=S(\rho).$ (4)

For a bipartite pure state $|\psi\rangle_{AB}$, when $q\geq 1$, its
Tsallis-$q$ entanglement is

$\displaystyle\mathbf{T}_{q}(|\psi\rangle_{AB})=T_{q}(\rho_{A}),$

where $\rho_{A}=\mathrm{tr}_{B}|\psi\rangle_{AB}\langle\psi|$. For a mixed
state $\rho_{AB}$, its Tsallis-$q$ entanglement is defined as

$\displaystyle\mathbf{T}_{q}(\rho_{AB})=\min\sum_{i}p_{i}\mathbf{T}_{q}(|\psi_{i}\rangle_{AB}),$

where the minimum takes over all possible pure state decompositions of
$\rho_{AB}=\sum_{i}p_{i}|\psi_{i}\rangle_{AB}\langle\psi_{i}|.$

By combing (4), we have

$\displaystyle\lim\limits_{q\rightarrow
1}\mathbf{T}_{q}(\rho_{AB})=E(\rho_{AB}),$

where $E(\rho_{AB})$ is entanglement of formation for the mixed state
$\rho_{AB}$ Bennett1996Concentrating ; Bennett1996mixed .

Next assume $|\phi\rangle$ is a pure state in
$\mathcal{H}_{2}\otimes\mathcal{H}_{d}$ with its Schmidt decomposition
$|\phi\rangle_{AB}=\sqrt{\lambda}|00\rangle+\sqrt{1-\lambda}|11\rangle,$ its
Tsallis-$q$ entanglement is

$\displaystyle\mathbf{T}_{q}(|\phi\rangle_{AB})=\frac{1}{(1-q)}[\lambda^{q}+(1-\lambda)^{q}-1].$

Combing the definition of concurrence for pure states, we have

$\displaystyle\mathbf{T}_{q}(|\phi\rangle_{AB})=f_{q}(C^{2}(|\phi_{AB}\rangle)),$

where $f_{q}(x)$ is defined in $x\in[0,1]$,

$\displaystyle
f_{q}(x)=\frac{((1+\sqrt{1-x})^{q}+(1-\sqrt{1-x})^{q})-2^{q}}{(1-q)2^{q}}.$

Next we recall several lemmas on the properties of $f_{q}(x)$.

###### Lemma 1

kim2010tsallis  The function $f_{q}(x)$ is a monotonously increasing and
concave function when
$q\in[\frac{5-\sqrt{13}}{2},2]\cup[3,\frac{5+\sqrt{13}}{2}]$.

###### Lemma 2

luo2016general  The function $f_{q}(x^{2})$ is a monotonically increasing
function of the variable $x$ for $q\in(0,\infty)$ and $x\in(0,1)$, it is a
convex function of $x$ when
$q\in[\frac{5-\sqrt{13}}{2},\frac{5+\sqrt{13}}{2}].$

###  II.2 Generalized W-class State in Multipartite Higher Dimensional Systems

Here we first recall the generalized W-class (GW) state
$|W_{n}^{d}\rangle_{A_{1}A_{2}\cdots A_{n}}$ in multi-qudit systems
san2008generalized ,

$\displaystyle|W_{n}^{d}\rangle_{A_{1}A_{2}\cdots A_{n}}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{d-1}(a_{1j}|j0\cdots 0\rangle+a_{2j}|0j\cdots
0\rangle+\cdots+a_{nj}|00\cdots 0j\rangle),$ (5)

here $\\{|j\rangle_{A_{i}}\\}_{j=0}^{d-1}$ is an orthonormal basis of qudit
subsystems $A_{i}$ with $i=1,2,\cdots,n$ and
$\sum_{i=1}^{n}\sum_{j=1}^{d-1}|a_{ij}|^{2}=1$.

###### Lemma 3

san2008generalized  For any n-qudit generalized W-class state
$|\psi\rangle_{AB_{1}\cdots B_{n-1}}$ in (5) and a partition
$P=\\{P_{1},\cdots,P_{m}\\}$ for the subset $S=\\{A,B_{1},\cdots,B_{l-1}\\},$
$m\leq l\leq n,$

$\displaystyle C^{2}_{P_{s}|\overline{P_{s}}}=\sum_{k\neq
s}C^{2}_{P_{s}|P_{k}}=\sum_{k\neq s}(C^{a}_{P_{s}|P_{k}})^{2},$ (6)

for all $k\neq s$, here we denote $\\{P_{1},\cdots,\overline{P_{s}},\cdots
P_{m}\\}=\\{P_{1},\cdots,{P_{s}},\cdots,P_{m}\\}-\\{P_{s}\\}$

Based on Lemma 1 and the relations between $\mathbf{T}_{q}(\cdot)$ and
$C(\cdot)$, the authors in shi2020monogamy  showed that

###### Lemma 4

Assume $\rho_{A_{j_{1}}A_{j_{2}}\cdots A_{j_{m}}}$ is a reduced density matrix
of an $n$-qudit GW state in (5), here $m\leq n$, then we have

$\displaystyle\mathbf{T}_{q}(\rho_{A_{j_{1}}|A_{j_{2}}\cdots
A_{j_{m}}})=f_{q}(C^{2}(\rho_{A_{j_{1}}|A_{j_{2}}\cdots A_{j_{m}}})),$

for $q\in[\frac{5-\sqrt{13}}{2},\frac{5+\sqrt{13}}{2}]$.

##  III Main Results

This section mainly considers classes of entanglement polygon inequalities in
terms of the Tsallis-$q$ entanglement for a class of mixed states, which are
the reduced density matrices of a GW state. First, we show that the EPI is
satisfied by the class of mixed states. Then a tighter EPI is presented for
the class of mixed states. At last, we present a generalized EPI for the mixed
states, this inequality can be regarded as a relationship of a bipartite
entanglement.

###### Theorem 1

Assume $\rho_{A_{j_{1}}A_{j_{2}}A_{j_{3}}}$ is a tripartite reduced density
matrix of an $n$-qudit GW state in
$\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}\otimes\cdots\mathcal{H}_{A_{n}}$,
then

$\displaystyle\mathbf{T}_{q}(\rho_{A_{j_{1}}|A_{j_{2}}A_{j_{3}}})\leq\mathbf{T}_{q}(\rho_{A_{j_{2}}|A_{j_{1}}A_{j_{3}}})+\mathbf{T}(\rho_{A_{j_{3}}|A_{j_{1}}A_{j_{2}}}),$

when $q\in[\frac{5-\sqrt{13}}{2},2]\cup[3,\frac{5+\sqrt{13}}{2}].$

###### Proof.

As $\rho_{A_{j_{1}}A_{j_{2}}A_{j_{3}}}$ is a tripartite reduced density matrix
of a GW state, then

$\displaystyle\mathbf{T}_{q}(\rho_{A_{j_{2}}|A_{j_{1}}A_{j_{3}}})+\mathbf{T}_{q}(\rho_{A_{j_{3}}|A_{j_{1}}A_{j_{2}}})$
$\displaystyle=$ $\displaystyle
f_{q}(C^{2}_{A_{j_{2}}|A_{j_{1}}A_{j_{3}}})+f_{q}(C^{2}_{A_{j_{3}}|A_{j_{2}}A_{j_{1}}})$
$\displaystyle=$ $\displaystyle
f_{q}(C^{2}_{A_{j_{2}}A_{j_{1}}}+C^{2}_{A_{j_{2}}A_{j_{3}}})+f_{q}(C^{2}_{A_{j_{3}}A_{j_{2}}}+C^{2}_{A_{j_{3}}A_{j_{1}}})$
$\displaystyle\geq$ $\displaystyle
f_{q}(C^{2}_{A_{j_{1}}A_{j_{2}}}+C^{2}_{A_{j_{1}}A_{j_{3}}})$ $\displaystyle=$
$\displaystyle f_{q}(C^{2}_{A_{j_{1}}|A_{j_{2}}A_{j_{3}}})$ $\displaystyle=$
$\displaystyle\mathbf{T}_{q}(\rho_{A_{j_{1}}|A_{j_{2}}A_{j_{3}}}),$

here the first equality is due to the Lemma 4, the second equality is due to
Lemma 3, the first inequality is due to the Lemma 1, and the last equality is
due to Lemma 3.  $\sqcap$$\sqcup$

Due to the proof of Theorem 1, we can prove the following similarly,

$\displaystyle\mathbf{T}_{q}(\rho_{A_{j_{2}}|A_{j_{1}}A_{j_{3}}})\leq\mathbf{T}_{q}(\rho_{A_{j_{1}}|A_{j_{2}}A_{j_{3}}})+\mathbf{T}(\rho_{A_{j_{3}}|A_{j_{1}}A_{j_{2}}}),$

that is,

###### Corollary 1

Assume $\rho_{A_{j_{1}}A_{j_{2}}A_{j_{3}}}$ is a tripartite reduced density
matrix of an $n$-qudit GW state in
$\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}\otimes\cdots\mathcal{H}_{A_{n}}$,
then

$\displaystyle|\mathbf{T}_{q}(\rho_{A_{j_{2}}|A_{j_{1}}A_{j_{3}}})-\mathbf{T}_{q}(\rho_{A_{j_{3}}|A_{j_{1}}A_{j_{2}}})|\leq$
$\displaystyle\mathbf{T}_{q}(\rho_{A_{j_{1}}|A_{j_{2}}A_{j_{3}}})\leq\mathbf{T}_{q}(\rho_{A_{j_{2}}|A_{j_{1}}A_{j_{3}}})+$
$\displaystyle\mathbf{T}(\rho_{A_{j_{3}}|A_{j_{1}}A_{j_{2}}}),$

when $q\in[\frac{5-\sqrt{13}}{2},2]\cup[3,\frac{5+\sqrt{13}}{2}].$

Figure 1: Characterization of Corollary 1. In this figure, the length of
segment $i-j$ indicates the value of
$\mathbf{T}_{q}(\rho_{A_{k}|A_{i}A_{j}})$, here $k$ is another party different
from parties $i$ and $j$.

In Fig. 1, we plot a triangle to present the meaning of Corollary 1, the
length of segment $i-j$ denotes the entanglement of
$\mathbf{T}_{q}(\rho_{A_{k}|A_{i}A_{j}})$, here $k$ is another party.

###### Corollary 2

Assume $\rho_{A_{j_{1}}A_{j_{2}}\cdots A_{j_{m}}}$ is a multipartite reduced
density matrix of an $n$-qudit $GW$ state in
$\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}\otimes\cdots\otimes\mathcal{H}_{A_{n}}$,
and $\\{P_{1},P_{2},\cdots,P_{h}\\}$ is a partition of the set
$S=\\{A_{j_{1}},A_{j_{2}},\cdots,A_{j_{m}}\\}$, that is,
$\cup_{g=1}^{h}P_{g}=S$,

$\displaystyle\mathbf{T}_{q}(\rho_{P_{j_{k}}|\overline{P_{j_{k}}}})\leq\sum_{l\neq
k}\mathbf{T}_{q}(\rho_{P_{j_{l}}|\overline{P_{j_{l}}}}),$

here $\overline{P_{j_{i}}}=P_{j_{0}}P_{j_{1}}\cdots
P_{j_{i-1}}P_{j_{i+1}}\cdots P_{j_{m}}$, $i=0,1,\cdots,m$, and $m\leq n$.

Assume $j\in\mathbb{N}^{+}$, then $j$ can be written as

$\displaystyle j=\sum_{i=0}^{n-1}j_{i}2^{i},$ (7)

here we assume $\log_{2}j\leq n,j_{i}\in\\{0,1\\}.$ According to the equality
(7), we have the following bijection:

$\displaystyle j\rightarrow$ $\displaystyle\vec{j}$ $\displaystyle
j\rightarrow$ $\displaystyle(j_{0},j_{1},\cdots,j_{n-1}),$

and we denote its Hamming distance $w_{H}(\vec{j})$ as the number 1 of the set
$\\{j_{0},j_{1},\cdots,j_{n-1}\\}$. Next we present a tighter inequality of
the GW states in terms of the Tsallis-$q$ entanglement.

###### Theorem 2

Let $\beta\in[0,1],$ assume $\rho_{A_{i_{1}}A_{i_{2}}\cdots A_{i_{m}}}$ is a
reduced density matrix of a GW state $|\psi\rangle_{A_{0}A_{1}\cdots
A_{n-1}}$. Let $P=\\{P_{j_{0}},P_{j_{1}},\cdots,P_{j_{m-1}},P_{j_{m}}\\}$ be a
partition of the set $\\{A_{i_{1}},A_{i_{2}},\cdots,A_{i_{m}}\\}$, when
$q\in[\frac{5-\sqrt{13}}{2},2]\cup[3,\frac{5+\sqrt{13}}{2}],$ there exists an
appropriate order of $P_{j_{0}},P_{j_{1}},\cdots,P_{j_{m-1}},P_{j_{m}}$ such
that

$\displaystyle[\mathbf{T}_{q}(\rho_{P_{j_{0}}|\overline{P_{j_{0}}}})]^{\beta}\leq\sum_{i=1}^{m}(2^{\beta}-1)^{w_{H}(\vec{j_{i}})}[\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta},$
(8)

here $\overline{P_{j_{i}}}=P_{j_{0}}P_{j_{1}}\cdots
P_{j_{i-1}}P_{j_{i+1}}\cdots P_{j_{m}}$, $i=0,1,\cdots,m$.

Here we place the proof of Theorem 2 in Sec. VI.

At last, we present the other class of entanglement polygon inequalities for
bipartite entanglement. Assume $|\phi\rangle_{\mathbf{A}\mathbf{B}C}$ is a GW
state in
$\mathcal{H}_{\mathbf{A}}\otimes\mathcal{H}_{\mathbf{B}}\otimes\mathcal{H}_{C},$
$\rho_{\mathbf{A}\mathbf{B}}=\mathrm{tr}_{C}|\phi\rangle_{\mathbf{A}\mathbf{B}C}\langle\phi|$
is a mixed state of the Hilbert space $\mathcal{H}_{\mathbf{A}\mathbf{B}}$.
Here we denote $\mathbf{A}=A_{1}A_{2}\cdots A_{m}$ and
$\mathbf{B}=B_{1}B_{2}\cdots B_{m}.$

###### Theorem 3

Assume $|\phi\rangle_{\mathbf{A}\mathbf{B}C}$ is a GW state in
$\mathcal{H}_{\mathbf{A}\mathbf{B}C},$
$\rho_{\mathbf{A}\mathbf{B}}=\mathrm{tr}_{C}|\phi\rangle_{\mathbf{A}\mathbf{B}C}\langle\phi|$,
here $\mathbf{A}=A_{1}A_{2}\cdots A_{m}$ and $\mathbf{B}=B_{1}B_{2}\cdots
B_{m},$ then

$\displaystyle\mathbf{T}_{q}^{\mathbf{A}|\mathbf{B}}\leq\sum_{i,j=1}^{m}\mathbf{T}_{q}^{A_{i}|B_{j}},$
(9)

when $q\in[\frac{5-\sqrt{13}}{2},2]\cup[3,\frac{5+\sqrt{13}}{2}].$

###### Proof.

As $\rho_{\mathbf{A}\mathbf{B}}$ is reduced density matrix of a GW state, then

$\displaystyle\mathbf{T}_{q}(\rho_{\mathbf{A}\mathbf{B}})=$ $\displaystyle
f_{q}(C^{2}_{\mathbf{A}|\mathbf{B}})$ $\displaystyle=$ $\displaystyle
f_{q}(\sum_{i,j=1}^{m}C_{A_{i}B_{j}}^{2})$ $\displaystyle\leq$
$\displaystyle\sum_{i,j=1}^{m}f_{q}(C^{2}_{A_{i}B_{j}})$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{m}T_{q}(\rho_{A_{i}B_{j}}).$

$\sqcap$$\sqcup$

##  IV Summary

In this manuscript, we have mainly considered the distribution property of
multipartite entanglement for a class of mixed states in terms of the
Tsallis-$q$ entanglement. First, we have proved the EPI in terms of
Tsallis-$q$ entanglement for the reduced density matrices of the GW states.
With Hamming distance, we have presented a tighter EPI in terms of Tsallis-$q$
entanglement for the mixed states. At last, we have shown inequalities for the
mixed states in terms of Tsallis-$q$ entanglement. We hope our results can
provide a reference for future work on the study of multiparty quantum
entanglement.

##  V Acknowledgement

This work was supported by the National Natural Science Foundation of China
(Grant No. 12301580).

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##  VI Appendix

###### Lemma 5

Let $\beta\in[0,1],x\in(0,1],$ then we have

$\displaystyle(1+x)^{\beta}\leq 1+(2^{\beta}-1)x^{\beta}.$ (10)

###### Proof.

Let $t=\frac{1}{x},$ then the lemma is equivalent to get the maximum of $f(t)$
when t$\in[1,\infty),$

$\displaystyle f(t)=(1+t)^{\beta}-t^{\beta}.$ (11)

As $t\in[1,\infty),$ and $f^{{}^{\prime}}(t)\leq 0,$ that is, when $t=1,$
$f(t)$ get the maximum $2^{t}-1.$ At last, When we replace t with
$\frac{1}{x},$ we finish the proof.  $\sqcap$$\sqcup$

Theorem 2: _Let $\beta\in[0,1],$ assume $\rho_{A_{i_{1}}A_{i_{2}}\cdots
A_{i_{m}}}$ is a reduced density matrix of a GW state
$|\psi\rangle_{A_{0}A_{1}\cdots A_{n-1}}$, let
$P=\\{P_{j_{0}},P_{j_{1}},\cdots,P_{j_{m-1}},P_{j_{m}}\\}$ be a partition of
the set $\\{A_{i_{1}},A_{i_{2}},\cdots,A_{i_{m}}\\}$, then there exists an
appropriate order of $P_{j_{0}},P_{j_{1}},\cdots,P_{j_{m-1}},P_{j_{m}}$ such
that_

$\displaystyle[\mathbf{T}_{q}(\rho_{P_{j_{0}}|\overline{P_{j_{0}}}})]^{\beta}\leq\sum_{i=1}^{m}(2^{\beta}-1)^{w_{H}(\vec{j_{i}})}[\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta},$
(12)

_here $\overline{P_{j_{i}}}=P_{j_{0}}P_{j_{1}}\cdots
P_{j_{i-1}}P_{j_{i+1}}\cdots P_{j_{m}}$, $i=0,1,\cdots,m$._

###### Proof.

In the process of the proof, we always can order the partite
$P_{j_{0}},P_{j_{1}}$ $\cdots$ $P_{j_{m-1}}$ such that

$\displaystyle\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})\geq\mathbf{T}_{q}(\rho_{P_{j_{i+1}}|\overline{P_{j_{i+1}}}}),i=0,1,\cdots,m-1.$
(13)

Through computation, we have that all the reduced density matrices of a GW
state is can be written as

$\displaystyle\rho_{P_{j_{0}}\cdots P_{j_{m-1}}}=\gamma_{P_{j_{0}}\cdots
P_{j_{k-1}}}\otimes|0_{m-k}\rangle\langle 0_{m-k}|$

Here we will use the mathematical induction method to prove the theorem. When
$\rho_{P_{j_{0}}P_{j_{1}}P_{j_{2}}}$ is a tripartite reduced density matrix of
a GW state $|\psi\rangle_{AA_{1}\cdots A_{n-1}}$,

$\displaystyle[\mathbf{T}_{q}(\rho_{P_{j_{0}}|P_{j_{1}}P_{j_{2}}})]^{\beta}$
$\displaystyle\leq$
$\displaystyle[\mathbf{T}_{q}(\rho_{P_{j_{1}}|P_{j_{0}}P_{j_{2}}})+\mathbf{T}_{q}(\rho_{P_{j_{2}}|P_{j_{0}}P_{j_{1}}})]^{\beta}$
$\displaystyle=$
$\displaystyle[\mathbf{T}_{q}(\rho_{P_{j_{1}}|P_{j_{0}}P_{j_{2}}})]^{\beta}\left[1+[\frac{\mathbf{T}_{q}(\rho_{P_{j_{2}}|P_{j_{0}}P_{j_{1}}})}{\mathbf{T}_{q}(\rho_{P_{j_{1}}|P_{j_{0}}P_{j_{2}}})}]\right]^{\beta}$
$\displaystyle\leq$
$\displaystyle(\mathbf{T}_{q}(\rho_{P_{j_{1}}|P_{j_{0}}P_{j_{2}}}))^{\beta}+(2^{\beta}-1)(\mathbf{T}_{q}(\rho_{P_{j_{2}}|P_{j_{0}}P_{j_{1}}}))^{\beta},$
(14)

here the first inequality is due to the Theorem 1, and when $a>c>0,b>0,$
$a^{b}>c^{b},$ and the second inequality is due to the Lemma 5.
Next assume $m<2^{n},$ the theorem is correct. Then when $m=2^{n}$, from the
inequality (14), we have

$\displaystyle[\mathbf{T}_{q}(\rho_{P_{j_{0}}|\overline{P_{j_{0}}}})]^{\beta}$
$\displaystyle\leq$
$\displaystyle[\sum_{i=1}^{m/2-1}\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta}[1+\frac{\sum_{i=\frac{m}{2}}^{m-1}\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})}{\sum_{i=0}^{\frac{m}{2}-1}\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})}]^{\beta}$
$\displaystyle\leq$
$\displaystyle[\sum_{i=1}^{\frac{m}{2}-1}\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta}+(2^{\beta}-1)[\sum_{i=\frac{m}{2}}^{m-1}\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta}$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{\frac{m}{2}-1}(2^{\beta}-1)^{w_{H}(\vec{j_{i}})}(\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}}))^{\beta}$
$\displaystyle+$
$\displaystyle\sum_{i=\frac{m}{2}}^{m-1}(2^{\beta}-1)\times(2^{\beta}-1)^{w_{H}(\vec{j_{i}})-1}[\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta}$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{m-1}(2^{\beta}-1)^{w_{H}(\vec{j_{i}})}[\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta}.$
(15)

When $m$ is an arbitrary number, we always can choose an $n\in\mathbb{N}^{+}$
such that $2^{n-1}\leq m\leq 2^{n}.$ Let

$\displaystyle\gamma_{P_{j_{0}}\cdots
P_{j_{2^{n}-1}}}=\rho_{P_{j_{0}}P_{j_{1}}\cdots
P_{j_{m}-1}}\otimes\delta_{P_{j_{m}}P_{j_{m+1}}\cdots P_{j_{2^{n}-1}}},$

here

$\delta_{P_{j_{m}}P_{j_{m+1}}\cdots P_{j_{2^{n}-1}}}=|00\cdots
0\rangle_{P_{j_{m}}P_{j_{m+1}}\cdots P_{j_{2^{n}-1}}}\langle 00\cdots 0|.$

Then due to the inequality (15), we have

$\displaystyle[\sum_{i=0}^{2^{n}-1}\mathbf{T}_{q}(\gamma_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta}\leq\sum_{i=0}^{2^{n}-1}(2^{\beta}-1)^{w_{H}(\vec{j_{i}})}[\mathbf{T}_{q}(\gamma_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta}.$
(16)

From the definition of the state $\gamma_{P_{j_{0}}\cdots P_{j_{2^{n}-1}}}$,
we have

$\displaystyle\mathbf{T}_{q}(\gamma_{P_{j_{i}}|\overline{P_{j_{i}}}})=$
$\displaystyle\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}}),\hskip
14.22636pti=1,\cdots,m-1,$ (17)
$\displaystyle\mathbf{T}_{q}(\gamma_{P_{j_{i}}|\overline{P_{j_{i}}}})=$
$\displaystyle 0,\hskip 14.22636pti=m,m+1,\cdots,2^{n},$ (18)

hence,

$\displaystyle[\mathbf{T}_{q}(\gamma_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta}$
$\displaystyle=$
$\displaystyle[\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta}$
$\displaystyle\leq$
$\displaystyle\sum_{i=0}^{2^{n}-1}(2^{\beta}-1)^{w_{H}(\vec{j_{i}})}[\mathbf{T}_{q}(\gamma_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{m-1}(2^{\beta}-1)^{w_{H}(\vec{j_{i}})}[\mathbf{T}_{q}(\rho_{P_{j_{i}}|\overline{P_{j_{i}}}})]^{\beta},$
(19)

here the first equality is due to (17), the second equality is due to (18)
$\sqcap$$\sqcup$