[table]capposition=top capbtabboxtable[][]

# Permutation-Invariant Variational Autoencoder for Graph-Level Representation
Learning

Robin Winter
Bayer AG
Freie Universität Berlin
<EMAIL_ADDRESS>
&Frank Noé
Freie Universität Berlin
<EMAIL_ADDRESS>&Djork-Arné Clevert
Bayer AG
<EMAIL_ADDRESS>

###### Abstract

Recently, there has been great success in applying deep neural networks on
graph structured data. Most work, however, focuses on either node- or graph-
level supervised learning, such as node, link or graph classification or node-
level unsupervised learning (e.g., node clustering). Despite its wide range of
possible applications, graph-level unsupervised representation learning has
not received much attention yet. This might be mainly attributed to the high
representation complexity of graphs, which can be represented by $n!$
equivalent adjacency matrices, where $n$ is the number of nodes. In this work
we address this issue by proposing a permutation-invariant variational
autoencoder for graph structured data. Our proposed model indirectly learns to
match the node order of input and output graph, without imposing a particular
node order or performing expensive graph matching. We demonstrate the
effectiveness of our proposed model for graph reconstruction, generation and
interpolation and evaluate the expressive power of extracted representations
for downstream graph-level classification and regression.

##  1 Introduction

Graphs are an universal data structure that can be used to describe a vast
variety of systems from social networks to quantum mechanics [1]. Driven by
the success of Deep Learning in fields such as Computer Vision and Natural
Language Processing, there has been an increasing interest in applying deep
neural networks on non-Euclidean, graph structured data as well [2, 3]. Most
notably, generalizing Convolutional Neural Networks and Recurrent Neural
Networks to arbitrarily structured graphs for supervised learning has lead to
significant advances on task such as molecular property prediction [4] or
question-answering [5]. Research on unsupervised learning on graphs mainly
focused on node-level representation learning, which aims at embedding the
local graph structure into latent node representations [6, 7, 8, 9, 10].
Usually, this is achieved by adopting an autoencoder framework where the
encoder utilizes e.g., graph convolutional layers to aggregate local
information at a node level and the decoder is used to reconstruct the graph
structure from the node embeddings. Graph-level representations are usually
extracted by aggregating node-level features into a single vector, which is
common practice in supervised learning on graph-level labels [4].
Unsupervised learning of graph-level representations, however, has not yet
received much attention, despite its wide range of possible applications, such
as feature extraction, pre-training for graph-level classification/regression
tasks, graph matching or similarity ranking. This might be mainly attributed
to the high representation complexity of graphs arising from their inherent
invariance with respect to the order of nodes within the graph. In general, a
graph with $n$ nodes, can be represented by $n!$ equivalent adjacency
matrices, each corresponding to a different node order. Since the general
structure of a graph is invariant to the order of their individual nodes, a
graph-level representation should not depend on the order of the nodes in the
input representation of a graph, i.e. two isomorphic graphs should always be
mapped to the same representation. This poses a problem for most neural
network architectures which are by design not invariant to the order of their
inputs. Even if carefully designed in a permutation invariant way (e.g., Graph
Neural Networks with a final node aggregation step), there is no straight-
forward way to train an autoencoder network, due to the ambiguous
reconstruction objective, requiring the same discrete order of input and
output graphs to compute the reconstruction loss.
How can we learn a permutation-invariant graph-level representation utilizing
a permutation-variant reconstruction objective? In this work we tackle this
question proposing a graph autoencoder architecture that is by design
invariant to the order of nodes in a graph. We address the order ambiguity
issue by training alongside the encoder and decoder model an additional
_permuter_ model that assigns to each input graph a permutation matrix to
align the input graph node order with the node order of the reconstructed
graph.

Figure 1: Network architecture of the proposed model. Input graph is depicted
as fully connected graph (dashed lines for not direct neighbours) with the
additional embedding node $v_{0}$ and edges to it (black color). Different
node and edge types are represented by different colors and real edges by
solid lines between nodes. Transformations parameterized by a neural network
are represented by block arrows.

##  2 Method

###  2.1 Notations and Problem Definition

An undirected Graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ is defined by the
set of $n$ nodes $\mathcal{V}=\\{v_{1},\dots,v_{n}\\}$ and edges
$\mathcal{E}=\\{(v_{i},v_{j})|v_{i},v_{j}\in\mathcal{V}\\}$. We can represent
a graph in matrix form by its node features
$\mathbf{X}_{\pi}\in\mathbb{R}^{n\times d_{v}}$ and adjacency matrix
$\mathbf{A}_{\pi}\in\\{0,1\\}^{n\times n}$ in the node order $\pi\in\Pi$,
where $\Pi$ is the set of all $n!$ permutations over $\mathcal{V}$. We define
the permutation matrix $\mathbf{P}$ that reorders nodes from order $\pi$ to
order $\pi^{\prime}$ as
$\mathbf{P}_{\pi\rightarrow\pi^{\prime}}=(p_{ij})\in\\{0,1\\}^{n\times n}$,
with $p_{ij}=1$ if $\pi(i)=\pi^{\prime}(j)$ and $p_{ij}=0$ everywhere else.
Since Graphs are invariant to the order of their nodes, note that

$\mathcal{G}_{\pi}=\mathcal{G}(\mathbf{X}_{\pi},\mathbf{A}_{\pi})=\mathcal{G}(\mathbf{P}_{\pi\rightarrow\pi^{\prime}}\mathbf{X}_{\pi},\mathbf{P}_{\pi\rightarrow\pi^{\prime}}\mathbf{A}_{\pi}\mathbf{P}^{\top}_{\pi\rightarrow\pi^{\prime}})=\mathcal{G}(\mathbf{X}_{\pi^{\prime}},\mathbf{A}_{\pi^{\prime}})=\mathcal{G}_{\pi^{\prime}},$
(1)

where $\top$ is the transpose operator. Let us now consider a dataset of
graphs $\mathbf{G}=\\{\mathcal{G}^{(i)}\\}_{i=0}^{N}$ we would like to
represented in a low-dimensional continuous space. We can adopt a latent
variable approach and assume that the data is generate by a process
$p_{\theta}(\mathcal{G}|\mathbf{z})$, involving an unobserved continuous
random variable $\mathbf{z}$. Following the work of Kingma and Welling [11],
we approximate the intractable posterior by
$q_{\phi}(\mathcal{G}|\mathbf{z})\approx p_{\theta}(\mathcal{G}|\mathbf{z})$
and minimize the lower bound on the marginal likelihood of graph $G^{(i)}$:

$\log
p_{\theta}(\mathcal{G}^{(i)})\geq\mathcal{L}(\phi,\theta;\mathcal{G}^{(i)})=-\text{KL}\left[q_{\phi}(\mathbf{z}|\mathcal{G}^{(i)})||p_{\theta}(\mathbf{z})\right]+\mathbb{E}_{q_{\phi}(\mathbf{z}|\mathcal{G}^{(i)})}\left[\log
p_{\theta}(\mathcal{G}^{(i)}|\mathbf{z})\right],$ (2)

where the Kullback–Leibler (KL) divergence term regularizes the encoded latent
codes of graphs $\mathcal{G}^{(i)}$ and the second term enforces high
similarity of decoded graphs to their encoded counterparts. As graphs can be
completely described in matrix form by their node features and adjacency
matrix, we can parameterize $q_{\phi}$ and $p_{\theta}$ in Eq. (2) by neural
networks that encode and decode node features $\mathbf{X}_{\pi}^{(i)}$ and
adjacency matrices $\mathbf{A}_{\pi}^{(i)}$ of graphs
$\mathcal{G}^{(i)}_{\pi}$. However, as graphs are invariant under arbitrary
node re-ordering, the latent code $\mathbf{z}$ should be invariant to the node
order $\pi$:

$q_{\phi}(\mathbf{z}|\mathcal{G}_{\pi})=q_{\phi}(\mathbf{z}|\mathcal{G}_{\pi^{\prime}}),\,\text{for
all $\pi,\pi^{\prime}\in\Pi$}.$ (3)

This can be achieved by parameterizing the encoder model $q_{\phi}$ by a
permutation invariant function. However, if the latent code $\mathbf{z}$ does
not encode the input node order $\pi$, input graph $\mathcal{G}_{\pi}$ and
decoded graph $\hat{\mathcal{G}}_{\pi^{\prime}}$ are no longer necessarily in
the same order, as the decoder model has no information about the node order
of the input graph. Hence, the second term in Eq. (2) cannot be optimized
anymore by minimizing the reconstruction loss between encoded graph
$\mathcal{G}_{\pi}$ and decoded graph $\hat{\mathcal{G}}_{\pi^{\prime}}$ in a
straight-forward way. They need to be brought in the same node order first. We
can rewrite the expectation in Eq. (2) using Eq. (1):

$\mathbb{E}_{q_{\phi}(\mathbf{z}|\mathcal{G}_{\pi}^{(i)})}\left[\log
p_{\theta}(\mathcal{G}_{\pi^{\prime}}^{(i)}|\mathbf{z})\right]=\mathbb{E}_{q_{\phi}(\mathbf{z}|\mathcal{G}_{\pi}^{(i)})}\left[\log
p_{\theta}(\mathbf{\hat{P}}_{\pi\rightarrow\pi^{\prime}}\mathcal{G}_{\pi}^{(i)}|\mathbf{z})\right].$
(4)

Since the ordering of the decoded graph $\pi^{\prime}$ is subject to the
learning process of the decoder and thus unknown in advance, finding
$\mathbf{P}_{\pi\rightarrow\pi^{\prime}}$ is not trivial. In [12], the authors
propose to use approximate graph matching to find the permutation matrix
$\mathbf{P}_{\pi\rightarrow\pi^{\prime}}$ that maximizes the similarity
$s(X_{\pi^{\prime}},\mathbf{P}_{\pi\rightarrow\pi^{\prime}}\hat{X}_{\pi};A_{\pi^{\prime}},\mathbf{P}_{\pi\rightarrow\pi^{\prime}}\hat{A}_{\pi}\mathbf{P}^{\top}_{\pi\rightarrow\pi^{\prime}})$,
which involves up to $O(n^{4})$ re-ordering operations at each training step
in the worst case [13].

###  2.2 Permutation-Invariant Variational Graph Autoencoder

In this work we propose to solve the reordering problem in Eq. (4) implicitly
by inferring the permutation matrix $\mathbf{P}_{\pi^{\prime}\rightarrow\pi}$
from the input graph $\mathcal{G}_{\pi}$ by a model
$g_{\psi}(\mathcal{G}_{\pi})$ that is trained to bring input and output graph
in the same node order and is used by the decoder model to permute the output
graph. We train this _permuter_ model jointly with the _encoder_ model
$q_{\phi}(\mathbf{z}|\mathcal{G}_{\pi})$ and _decoder_ model
$p_{\theta}(\mathcal{G}_{\pi}|\mathbf{z},\mathbf{P}_{\pi^{\prime}\rightarrow\pi})$,
optimizing:

$\mathcal{L}(\phi,\theta,\psi;\mathcal{G}^{(i)}_{\pi})=-\text{KL}\left[q_{\phi}(\mathbf{z}|\mathcal{G}^{(i)}_{\pi})||p_{\theta}(\mathbf{z})\right]+\mathbb{E}_{q_{\phi}(\mathbf{z}|\mathcal{G}^{(i)})}\left[\log
p_{\theta}(\mathcal{G}^{(i)}_{\pi}|\mathbf{z},g_{\psi}(\mathcal{G}^{(i)}_{\pi}))\right].$
(5)

Intuitively, the permuter model has to learn how the ordering of nodes in the
graph generated by the decoder model will differ from a specific node order
present in the input graph. During the learning process, the decoder will
learn its own canonical ordering that, given a latent code $z$, it will always
reconstruct a graph in. The permuter learns to transform/permute this
canonical order to a given input node order. For this, the permuter predicts
for each node $i$ of the input graph a score $s_{i}$ corresponding to its
probability to have a low node index in the decoded graph. By sorting the
input nodes indices by their assigned scores we can infer the output node
order and construct the respective permutation matrix
$\mathbf{P}_{\pi\rightarrow\pi^{\prime}}=(p_{ij})\in\\{0,1\\}^{n\times n}$,
with

$p_{ij}=\begin{cases}1,&\text{if $j=$ argsort$(s)_{i}$}\\\
0,&\text{else}\end{cases}$ (6)

to align input and output node order. Since the argsort operation is not
differentiable, we utilizes the continuous relaxation of the argsort operator
proposed in [14, 15]:

$\mathbf{P}\approx\hat{\mathbf{P}}=\text{softmax}(\frac{-d(\text{sort}(s)\mathbb{1}^{\top},\mathbb{1}s^{\top})}{\tau}),$
(7)

where the softmax operator is applied row-wise, $d(x,y)$ is the $L_{1}$-norm
and $\tau\in\mathbb{R}_{+}$ a temperature-parameter. By utilizing this
continuous relaxation of the argsort operator, we can train the permuter model
$g_{\psi}$ in Eq. (5) alongside the encoder and decoder model with stochastic
gradient descent. In order to push the relaxed permutation matrix towards a
real permutation matrix (only one 1 in every row and column), we add to Eq.
(5) a row- and column-wise entropy term as additional penalty term:

$C(\mathbf{P})=\sum_{i}H(\overline{\mathbf{p}}_{i,\cdot})+\sum_{j}H(\overline{\mathbf{p}}_{\cdot,j}),$
(8)

with Shannon entropy $H(x)=-\sum_{i}x_{i}\log(x_{i})$ and normalized
probabilities
$\overline{\mathbf{p}}_{i,\cdot}=\frac{\mathbf{p}_{i,\cdot}}{\sum_{j}\mathbf{p}_{i,j}}$.
Propositions 1. A square matrix $\mathbf{P}$ is a real permutation matrix if
and only if $C(\mathbf{P})=0$ and the doubly stochastic constraint $p_{ij}\geq
0\;\forall(i,j),\;\sum_{i}p_{ij}=1\;\forall j,\;\sum_{j}p_{ij}=1\;\forall i$
holds.
Proof. See Appendix A.
By enforcing $\hat{\mathbf{P}}\rightarrow\mathbf{P}$, we ensure that no
information about the graph structure is encoded in $\hat{\mathbf{P}}$ and
decoder model $p_{\theta}(\mathcal{G}_{\pi}|\mathbf{z},\mathbf{P})$ can
generate valid graphs during inference, without providing a specific
permutation matrix $\mathbf{P}$ (e.g., one can set $\mathbf{P=I}$ and decode
the learned canonical node order). At this point it should also be noted, that
our proposed framework can easily be generalized to arbitrary sets of
elements, although we focus this work primarily on sets of nodes and edges
defining a graph.

#### Graph Isomorphism Problem.

Equation (5) gives us means to train an autoencoder framework with a
permutation invariant encoder that maps a graph
$f:\mathcal{G}\rightarrow\mathcal{Z}$ in an efficient manner. Such an encoder
will always map two topologically identical graphs (even with different node
order) to the same representation $z$. Consequently, the question arises, if
we can decide for a pair of graphs whether they are topologically identical.
This is the well-studied _graph isomorphism problem_ for which no polynomial-
time algorithm is known yet [16, 17]. As mentioned above, in our framework,
two isomorphic graphs will always be encoded to the same representation.
Still, it might be that two non-isomorphic graphs will be mapped to the same
point (non-injective). However, if the decoder is able to perfectly
reconstruct both graphs (which is easy to check since the permuter can be used
to bring the decoded graph in the input node order), two non-isomorphic graphs
must have a different representation $z$. If two graphs have the same
representation and the reconstruction fails, the graphs might still be
isomorphic but with no guarantees. Hence, our proposed model can solve the
graph isomorphism problem at least for all graphs it can reconstruct.

###  2.3 Details of the Model Architecture

In this work we parameterize the encoder, decoder and permuter model in Eq.
(5) by neural networks utilizing the self-attention framework proposed by
Vaswani et al. [18] on directed messages representing a graph. Figure 1,
visualizes the architecture of the proposed permutation-invariant variational
autoencoder. In the following, we describe the different parts of the model in
detail111Code available at https://github.com/jrwnter/pigvae.

#### Graph Representation by Directional Messages.

In general, most graph neural networks can be thought of as so called Message
Passing Neural Networks (MPNN) [19]. The key idea of MPNNs is the aggregation
of neighbourhood information by passing and receiving messages of each node to
and from neighbouring nodes in a graph. We adopt this view and represent
graphs by its messages between nodes. We represent a graph
$\mathcal{G}(\mathbf{X},\mathbf{E})$, with node features
$\mathbf{X}\in\mathbb{R}^{n\times d_{v}}$ and edge features
$\mathbf{E}\in\mathbb{R}^{n\times n\times d_{e}}$, by its message matrix
$\mathbf{M}=(\mathbf{m}_{ij})\in\mathbb{R}^{n\times n\times d_{m}}$:

$\mathbf{m}_{ij}=\sigma\left(\left[\mathbf{x}_{i}||\mathbf{x}_{j}||\mathbf{e}_{ij}\right]\mathbf{W}+\mathbf{b}\right),$
(9)

with non-linearity $\sigma$, concatenation operator $||$ and trainable
parameters $\mathbf{W}$ and $\mathbf{b}$. Note, that nodes in this view are
represented by _self-messages_ $\text{diag}(\mathbf{M})$, messages between
non-connected nodes exists, although the presence or absence of a connection
might be encoded in $\mathbf{e}_{ij}$, and if $\mathbf{M}$ is not symmetric,
edges have an inherent direction.

#### Self-Attention on Directed Messages.

We follow the idea of aggregating messages from neighbours in MPNNs, but
utilize the self-attention framework proposed by Vaswani et al. [18] for
sequential data. Our proposed model comprises multiple layers of multi-headed
scaled-dot product attention. One attention head is defined by:

$\text{Attention}\left(\mathbf{Q},\mathbf{K},\mathbf{V}\right)=\text{softmax}\left(\frac{\mathbf{Q}\mathbf{K}^{\top}}{\sqrt{d_{k}}}\right)\mathbf{V}$
(10)

with queries $\mathbf{Q}=\mathbf{MW}^{Q}$, keys $\mathbf{K}=\mathbf{MW}^{K}$,
and values $\mathbf{V}=\mathbf{MW}^{V}$ and trainable weights
$\mathbf{W}^{Q}\in\mathbb{R}^{d_{m}\times d_{q}}$,
$\mathbf{W}^{K}\in\mathbb{R}^{d_{m}\times d_{k}}$ and
$\mathbf{W}^{V}\in\mathbb{R}^{d_{m}\times d_{v}}$. For multi-headed self-
attention we concatenate multiple attention heads together and feed them to a
linear layer with $d_{m}$ output features. Since the message matrix
$\mathbf{M}$ of a graph with $n$ nodes comprises $n^{2}$ messages, attention
of all messages to all messages would lead to a $O(n^{4})$ complexity. We
address this problem by letting messages $\mathbf{m}_{ij}$ only attend on
incoming messages $\mathbf{m}_{ki}$, reducing the complexity to $O(n^{3})$. We
achieve this by representing $\mathbf{Q}$ as a $(m\times n\times d)$ tensor
and $\mathbf{K}$ and $\mathbf{V}$ by a transposed $(n\times m\times d)$
tensor, resulting into a $(m\times n\times m)$ attention tensor, with number
of nodes $m=n$ and number of features $d$. That way, we can efficiently
utilize batched matrix multiplications in Eq. (10), in contrast to computing
the whole $(n^{2}\times n^{2})$ attention matrix and masking attention on not
incoming messages out.

#### Encoder

To encode a graph into a fixed-sized, permutation-invariant, continuous latent
representation, we add to input graphs a dummy node $v_{0}$, acting as an
embedding node. To distinguish the embedding node from other nodes, we add an
additional node and edge type to represent this node and edges to and from
this node. After encoding this graph into a message matrix
$\mathbf{M}^{(\text{enc},~{}0)}$ as defined in Eq. (9), we apply $L$
iterations of self-attention to update $\mathbf{M}^{(\text{enc},~{}L)}$,
accumulating the graph structure in the embedding node, represented by the
self-message $\mathbf{m}^{(\text{enc},~{}L)}_{0,0}$. Following [11], we
utilize the reparameterization trick and sample the latent representation
$\mathbf{z}$ of a graph by sampling from a multivariate normal distribution:

$\mathbf{z}\sim\mathcal{N}(f_{\mu}(\mathbf{m}^{(\text{enc},~{}L)}_{0,0}),f_{\sigma}(\mathbf{m}^{(\text{enc},~{}L)}_{0,0})\mathbf{I}),$
(11)

with $f_{\mu}:\mathbf{m}_{0,0}\rightarrow\mathbf{\mu}\in\mathbb{R}^{d_{z}}$
and
$f_{\sigma}:\mathbf{m}_{0,0}\rightarrow\mathbf{\sigma}\in\mathbb{R}^{d_{z}}$,
parameterized by a linear layer.

#### Permuter

To predict how to re-order the nodes in the output graph to match the order of
nodes in the input graph, we first extract node embeddings represented by
self-messages on the main diagonal of the encoded message matrix
$\mathbf{m}^{(\text{enc},~{}L)}_{i,i}=\text{diag}(\mathbf{M}^{(\text{enc},~{}L)})$
for $i>0$. We score these messages by a function
$f_{s}:\mathbf{m}_{i,i}\rightarrow s\in\mathbb{R}$, parameterized by a linear
layer and apply the soft-sort operator (see Eq. (7)) to retrieve the
permutation matrix $\hat{\mathbf{P}}$.

#### Decoder

We initialize the message matrix for the decoder models input with the latent
representation $\mathbf{z}$ at each entry. To break symmetry and inject
information about the relative position/order of nodes to each other, we
follow [18] and define position embeddings in dimension $k$

$\text{PE}(i)_{k}=\begin{cases}&\sin(i/10000^{2k/d_{z}}),\;\;\text{for even
$k$}\\\ &\cos(i/10000^{2k/d_{z}}),\;\;\text{for odd $k$}\\\ \end{cases}$ (12)

It follows for the initial decoder message matrix
$\mathbf{M}^{(\text{dec},~{}0)}$:

$\mathbf{m}^{(\text{dec},~{}0)}_{ij}=\sigma\left(\left[\mathbf{z}+[\text{PE}(i)||\text{PE}(j)]\right]\mathbf{W}+\mathbf{b}\right),$
(13)

Since the self-attention based decoder model is permutation equivariant, we
can move the permutation operation in Eq. (5) in front of the decoder model
and directly apply it to the position embedding sequence (see Figure 1). After
$L$ iterations of self-attention on the message matrix $\mathbf{M}$, we
extract node features $\mathbf{x}_{i}\in\mathbf{X}$ and edge features
$\mathbf{e}_{i,j}\in\mathbf{E}$ by a final linear layer:

$\mathbf{x}_{i}=\mathbf{m}_{i,i}\mathbf{W}^{v}+\mathbf{b}^{v}\hskip
30.0pt\mathbf{e}_{i,j}=0.5\cdot(\mathbf{m}_{i,j}+\mathbf{m}_{j,i})\mathbf{W}^{e}+\mathbf{b}^{e},$
(14)

with learnable parameters $\mathbf{W}^{v}\in\mathbb{R}^{d_{m}\times d_{v}}$,
$\mathbf{W}^{e}\in\mathbb{R}^{d_{m}\times d_{e}}$,
$\mathbf{b}^{v}\in\mathbb{R}^{d_{v}}$ and
$\mathbf{b}^{e}\in\mathbb{R}^{d_{e}}$.

#### Overall Architecture

We now describe the full structure of our proposed method using the
ingredients above (see Figure 1). Initially, the input graph is represented by
the directed message matrix $\mathbf{M}^{(\text{enc},~{}0)}$, including an
additional graph embedding node $v_{0}$. The encoder model performs $L$
iterations of self-attention on incoming messages. Next, diagonal entries of
the resulting message matrix $\mathbf{M}^{(\text{enc},~{}L)}$ are extracted.
Message $m^{(\text{enc},~{}L)}_{0,0}$, representing embedding node $v_{0}$, is
used to condition the normal distribution, graph representation $\mathbf{z}$
is sampled from. The other diagonal entries $m^{(\text{enc},~{}L)}_{i,i}$ are
transformed into scores and sorted by the Soft-Sort operator to retrieve the
permutation matrix $\hat{\mathbf{P}}$. Next, position embeddings (in Figure 1
represented by single digits) are re-ordered by applying $\hat{\mathbf{P}}$
and added by the sampled graph embedding $\mathbf{z}$. The resulting node
embeddings are used to initialize message matrix
$\mathbf{M}^{(\text{dec},~{}0)}$ and fed into the decoding model. After $L$
iterations of self-attention, diagonal entries are transformed to node
features $\mathbf{X}$ and off-diagonal entries to edge features $\mathbf{E}$
to generate the output graph. In order to train and infer on graphs of
different size, we pad all graphs in a batch with empty nodes to match the
number of nodes of the largest graph. Attention on empty nodes is masked out
at all time. To generate graphs of variable size, we train alongside the
variational autoencoder an additional multi-layer perceptron to predict the
number of atoms of graph from its latent representation $\mathbf{z}$. During
inference, this model informs the decoder on how many nodes to attend to.

#### Key Architectural Properties

Since no position embeddings are added to the input of the encoders self-
attention layers, accumulated information in the single embedding node $v_{0}$
($\mathbf{m}_{0,0}$) is invariant to permutations of the input node order.
Hence, the resulting graph embedding $\mathbf{z}$ is permutation invariant as
well. This is in stark contrast to classical graph autoencoder frameworks [8,
20, 21], that encode whole graphs effectively by concatenating all node
embeddings, resulting in a graph-level representation that is different for
isomorphic graphs, as the sequence of node embeddings permutes equivalently
with the input node order. As no information about the node order is encoded
in the graph embedding $\mathbf{z}$, the decoder learns its own (canonical)
node order, distinct graphs are deterministically decoded in. The input node
order does not influence this decoded node order. As the decoder is based on
permutation equivariant self-attention layers, this canonical order is solely
defined with respect to the sequence of position embeddings used to initialize
the decoders input. If the sequence of position embeddings is permuted, the
decoded node order permutes equivalently. Thus, by predicting the right
permutation matrix, input and output order can be aligned to correctly
calculate the reconstruction loss. Input to the permuter model
$[\mathbf{m}_{1,1},\dots,\mathbf{m}_{n+1,n+1}]$ is equivariant to permutations
in the input node order (due to the equivariant self-attention layers in the
encoder). Since the permuter model itself (i.e., the scoring function) is also
permutation equivariant (node-wise linear layer), resulting permutation
matrices $\mathbf{P}$ are equivariant to permutations in the input node order.
Consequently, if the model can correctly reconstruct a graph in a certain node
order, it can do it for all $n!$ input node orders, and the learning process
of the whole model is independent to the node order of graphs in the training
set.

Table 1: Negative log likelihood (NLL) and area under the receiver operating characteristics curve (ROC-AUC) for reconstruction of the adjacency matrix of graphs from different families. We compare our proposed method (PIGAE) with Graph Autoencoder (GAE) [8] and results of Graphite and Graph Autoencoder (GAE*) reported in [20]. PIGAE∗ utilize topological distances of nodes in a graph as edge feature. Models | Erdos-Renyi | Barabasi-Albert | Ego
---|---|---|---
| NLL | ROC-AUC | NLL | ROC-AUC | NLL | ROC-AUC
PIGAE | $20.5\pm 0.9$ | $98.3\pm 0.1$ | $27.2\pm 0.9$ | $96.7\pm 0.2$ | $23.4\pm 0.5$ | $97.8\pm 0.3$
PIGAE∗ | $\mathbf{19.5\pm 0.8}$ | $\mathbf{99.4\pm 0.1}$ | $\mathbf{15.2\pm 0.8}$ | $\mathbf{99.5\pm 0.1}$ | $\mathbf{22.4\pm 0.5}$ | $\mathbf{98.8\pm 0.3}$
GAE | $186\pm 3$ | $57.9\pm 0.1$ | $199\pm 3$ | $57.4\pm 0.1$ | $191\pm 4$ | $59.1\pm 0.1$
GAE∗ | $222\pm 8$ | - | $236\pm 15$ | - | $197\pm 2$ | -
Graphite | $196\pm 1$ | - | $192\pm 2$ | - | $183\pm 1$ | -

##  3 Related Works

Most existing research on unsupervised graph representation learning focuses
on _node-level_ representation learning and can be broadly categorized in
either shallow methods based on matrix factorization [22, 23, 24, 25] or
random walks [26, 27], and deep methods based on Graph Neural Networks (GNN)
[2, 3]. Kipf and Welling [8] proposed a graph autoencoder (GAE),
reconstructing the adjacency matrix by taking the dot product between the
latent node embeddings encoded by a GNN. Grover et al. [20] build on top of
the GAE framework by parameterizing the decoder with additional GNNs, further
refining the decoding process. Although _graph-level_ representations in GAE-
like approaches can be constructed by concatenating all node-level
representations, note, that as a consequence they are only permutation
equivariant and not permutation invariant. Permutation invariant
representations could be extracted only after training by aggregating node
embeddings into a single-vector representation. However, such a representation
might miss important global graph structure. Samanta et al. [21] proposed a
GAE-like approach, which parameters are trained in a permutation invariant
way, following [28] utilizing breadth-first-traversals with randomized tie
breaking during the child-selection step. However, as graph-level
representations are still constructed by concatenation of node-level
embeddings, this method still encodes graphs only in a permutation-equivariant
way. A different line of work utilized Normalizing Flows [29] to address
variational inference on graph structured data based on node-level latent
variables [30, 31, 32].
Research on _graph-level_ representations has mainly focused on supervised
learning, e.g., graph-level classification by applying a GNN followed by a
global node feature aggregation step (e.g., mean or max pooling) [4] or a
jointly learned aggregation scheme [33]. Research on graph-level unsupervised
representation learning has not yet received much attention and existing work
is mainly based on contrastive learning approaches. Narayanan et al. [34]
adapted the _doc2vec_ method from the field of natural language processing to
represent whole graphs by a fixed size embedding, training a _skipgram_ method
on rooted subgraphs (_graph2vec_). Bai et al. [35] proposed a _Siamese_
network architecture, trained on minimizing the difference between the
Euclidean distance of two encoded graphs and their graph editing distance.
Recently, Sun et al. [36], adapted the _Deep InfoMax_ architecture [37] to
graphs, training on maximizing the mutual information between graph-level
representations and representations of sub-graphs of different granularity
(_InfoGraph_). Although those contrastive learning approaches can be designed
to encode graphs in a permutation invariant way, they cannot be used to
reconstruct or generate graphs from such representations.
Another line of related work concerns itself with _generative models_ for
graphs. Besides methods based on variational autoencoders [8, 20, 12] and
Normalizing Flows [30, 31, 32], graph generative models have also been
recently proposed based on generative adversarial neural networks [38, 39] and
deep auto-regressive models [40, 28, 41]. Moreover, due to its high practical
value for drug discovery, many graph generating methods have been proposed for
molecular graphs [42, 43, 44, 45, 21]. Although graph generative models can be
trained in a permutation invariant way [42, 21], those models can not be used
to extract permutation invariant graph-level representations.
Recently, Yang et al. [46] proposed a GAE-like architecture with a node-
feature aggregation step to extract permutation invariant graph-level
representations that can also be used for graph generation. They tackle the
discussed ordering issue of GAEs in the reconstruction by training alongside
the GAE a Generative Adversarial Neural Network, which’s permutation invariant
discriminator network is used to embed input and output graph into a latent
space. That way, a permutation invariant reconstruction loss can be defined as
a distance in this space. However, as this procedure involves adversarial
training of the reconstruction metric, this only approximates the exact
reconstruction loss used in our work and might lead to undesirable graph-level
representations.

Figure 2: a) Euclidean distance over graph editing distance, averaged over
1000 Barabasi-Albert graphs with $m=2$. b) t-SNE projection of representations
from ten different graph families with different parameters. Example graphs
are shown for some of the clusters. c) Legend of t-SNE plot explaining colours
and symbols. Graph family abbreviations: Binomial (B), Binomial Ego (BE),
Barabasi-Albert (BA), Geometric (G), Regular (R), Powerlaw Tree (PT), Watts-
Strogatz (WA), Extended Barabasi-Albert (EBA), Newman-Watts-Strogatz (NWA),
Dual-Barabasi-Albert (DBA).

##  4 Experimental Evaluation

We perform experiments on synthetically generated graphs and molecular graphs
from the public datasets QM9 and PubChem. At evaluation time, predicted
permutation matrices are always discretized to ensure their validity. For more
details on training, see Appendix C.

###  4.1 Synthetic Data

#### Graph Reconstruction

In the first experiment we evaluate our proposed method on the reconstruction
performance of graphs from graph families with a well-defined generation
process. Namely, Erdos-Renyi graphs [47], with an edge probability of $p=0.5$,
Barabasi-Albert graphs [48], with $m=4$ edges preferentially attached to nodes
with high degree and Ego graphs. For each family we uniformly sample graphs
with 12-20 nodes. The graph generation parameters match the ones reported in
[20], enabling us to directly compare to Graphite. As additional baseline we
compare against the Graph Autoencoder (GAE) proposed by Kipf and Welling [8].
As Grover et al. [20] only report negative log-likelihood estimates for their
method Graphite and baseline GAE we also reevaluate GAE and report both
negative log-likelihood (NLL) estimates for GAE to make a better comparison to
Graphite possible (accounting for differences in implantation or the graph
generation process). In Table 1 we show the evaluation metrics on a fixed test
set of 25000 graphs for each graph family. On all four graph datasets our
proposed model significantly outperforms the baseline methods, reducing the
NLL error in three of the four datasets by approximately one magnitude.
Utilizing the topological distance instead of just the connectivity as edge
feature (compare [49]) further improves the reconstruction performance.

Table 2: Classification Accuracy of our method (PIGAE), classical GAE, InfoGraph (IG), Shortest Path Kernel (SP) and Weisfeiler-Lehman Sub-tree Kernel (WL) on graph class prediction. PIGAE | GAE | IG | SP | WL
---|---|---|---|---
$\mathbf{0.83\pm 0.01}$ | $0.65\pm 0.01$ | $0.75\pm 0.02$ | $0.50\pm 0.02$ | $0.73\pm 0.01$

#### Qualitative Evaluation

To evaluate the representations learned by our proposed model, we trained a
model on a dataset of randomly generated graphs with variable number of nodes
from ten different graph families with different ranges of parameters (see
Appendix B for details). Next, we generated a test set of graphs with a fixed
number of nodes from these ten different families and with different fixed
parameters. In total we generated graphs in 29 distinct settings. In Figure 2,
we visualized the t-SNE projection [50] of the graph embeddings, representing
different families by colour and different parameters within each family by
different symbols. In this 2-D projection, we can make out distinct clusters
for the different graph sets. Moreover, clusters of similar graph sets tend to
cluster closer together. For example, Erdos-Renyi graphs form for each edge
probability setting ($0.25$, $0.35$, $0.5$) a distinct cluster, while
clustering in close proximity. As some graph families with certain parameters
result in similar graphs, some clusters are less separated or tend to overlap.
For example, the Dual-Barabasi-Albert graph family, which attaches nodes with
either $m_{1}$ or $m_{2}$ other nodes, naturally clusters in between the two
Barabasi-Albert graph clusters with $m=m_{1}$ and $m=m_{2}$.

#### Graph Editing Distance

A classical way of measuring graph similarity is the so called _graph editing
distance_ (GED) [51]. The GED between two graphs measures the minimum number
of graph editing operations to transform one graph into the other. The set of
operations typically includes inclusion, deletion and substitution of nodes or
edges. To evaluate the correlation between similarity in graph representation
and graph editing distance, we generated a set of 1000 Barabasi-Albert ($m=3$)
graphs with 20 nodes. For each graph we successively substitute randomly an
existing edge by a new one, creating a set of graphs with increasing GED with
respect to the original graph. In Figure 2, we plot the mean Euclidean
distance between the root graphs and their 20 derived graphs with increasing
GED. We see a strong correlation between GED and Euclidean distance of the
learned representations. In contrast to classical GAEs, random permutations of
the edited graphs have no effect on this correlation (see Appendix D for
comparison).

#### Graph Isomorphism and Permutation Matrix

To empirical analyse if our proposed method detects isomorphic graphs, we
generated for 10000 Barabasi-Albert graphs with up to 28 nodes a randomly
permuted version and a variation only one graph editing step apart. For all
graphs the Euclidean distance between original graph and edited graph was at
least greater than $0.3$. The randomly permuted version always had the same
embedding. Even for graphs out of training domain (geometric graphs with 128
nodes) all isomorphic and non-isomorphic graphs could be detected.
Additionally, we investigated how well the permuter model can assign a
permutation matrix to graph. As the high reconstruction performance in Table 1
suggest, most of the time, correct permutation matrices are assigned (See
Appendix E for a more detailed analysis). We find that permutation matrices
equivalently permute with the permutation of the input graph (See Appendix E).

Figure 3: Linear interpolation between two graphs in the embedding space.
Start graph’s edges are colored red. End graph’s edges are colored blue. Edges
present in both graphs are colored black. Thick lines are present in a decoded
graph, thin lines are absent.

#### Graph Classification

In order to quantitatively evaluate how meaningful the learned representations
are, we evaluate the classification performance of a Support Vector Machine
(SVM) on predicting the correct graph set label (29 classes as defined above)
from the graph embedding. As baseline we compare against SVMs based on two
classical graph kernel methods, namely Shortest Path Kernel (SP) [52] and
Weisfeiler-Lehman Sub-tree Kernel (WL) [53] as well as embeddings extracted by
the recently proposed contrastive learning model InfoGraph (IG) [36] and
averaged node embeddings extracted by a classical GAE. Our model, IG and GAE
where trained on the same synthetic dataset. In Table 2 we report the accuracy
for each model and find the SVM based on representations from our proposed
model to significantly outperform all baseline models. Notably,
representations extracted from a classical GAE model, by aggregating
(averaging) node-level embeddings into a graph-level embedding, perform
significantly worse compared to representations extracted by our method. This
finding is consistent with our hypothesis that aggregation of unsupervised
learned node-level (local) features might miss important global features,
motivating our work on graph-level unsupervised representation learning.

#### Graph Interpolation

The permutation invariance property of graph-level representations also
enables the interpolation between two graphs in a straight-forward way. With
classical GAEs such interpolations cannot be done in a meaningful way, as
interpolations between permutation dependent graph embeddings would affect
both graph structure as well as node order. In Figure 3 we show how successive
linear interpolation between the two graphs in the embedding space results in
smooth transition in the decoded graphs, successively deleting edges from the
start graph (red) and adding edges from the end graph (blue). To the best of
our knowledge, such graph interpolations have not been report in previous work
yet and might show similar impact in the graph generation community as
interpolation of latent spaces did in the field of Natural Language Processing
and Computer Vision.

###  4.2 Molecular Graphs

Next, we evaluate our proposed model on molecular graphs from the QM9 dataset
[54, 55]. This datasets contains about 134 thousand organic molecules with up
to 9 heavy atoms (up to 29 atoms/nodes including Hydrogen). Graphs have 5
different atom types (C, N, O, F and H), 3 different formal charge types (-1,
0 and 1) and 5 differ bond types (no-, single-, double-, triple- and aromatic
bond). Moreover, the dataset contains an energetically favorable conformation
for each molecule in form of Cartesian Coordinates for each atom. We transform
these coordinates to an (rotation-invariant) Euclidean distance matrix and
include the distance information as additional edge feature to the graph
representation (More details in Appendix F).

#### Graph Reconstruction and Generation

We define a holdout set of 10,000 molecules and train the model on the rest.
Up on convergence, we achieve on the hold out set a balanced accuracy of
$99.93\%$ for element type prediction, $99.99\%$ for formal charge type
prediction and $99.25\%$ for edge type prediction (includs prediction of non-
existence of edges). Distances between atoms are reconstructed with a root
mean squared error of $0.33$Å and a coefficient of determination of
$\text{R}^{2}=0.94$.
In the field of computational chemistry and drug discovery, the generation of
energetically reasonable conformations for molecules is a topic of great
interest [56]. Since our model is trained on approximating the data generating
process for molecules and their conformations, we can utilize the trained
model to sample molecular conformations. To retrieve a set of conformations,
we encode a query molecule into its latent representation and randomly sample
around this point by adding a small noise and decode the resulting
representations. We utilize _Multidimensional Scaling_ (MDS) [57] to transform
a decoded distance matrix back to Cartesian Coordinates. In Figure 5, we show
examples of molecular conformations sampled from the trained model. Under
visual inspection, we find that sampled conformations differ from encoded
conformations, while still being energetically reasonable (e.g., rotation
along rotatable bonds while avoiding clashes between atoms).

Dataset | PIGAE (ours) | ECFP
---|---|---
Classification (ROC-AUC $\uparrow$)
BACE | $0.824\pm 0.005$ | $0.82\pm 0.02$
BBBP | $\mathbf{0.81\pm 0.04}$ | $0.78\pm 0.03$
Regression (MSE $\downarrow$)
ESOL | $\mathbf{0.10\pm 0.01}$ | $0.25\pm 0.02$
LIPO | $\mathbf{0.34\pm 0.02}$ | $0.39\pm 0.02$

Figure 4: Downstream performance for
molecular property prediction tasks.

Figure 5: Example Molecular conformations sampled around original graph
representation.

#### Molecular Property Prediction

Finally, we evaluate the learned representations for molecular property
prediction. In order to accurately represent molecular graphs from different
parts of the chemical space, we train our proposed model on a larger dataset
retrieved from the public PubChem database [58]. We extracted organic
molecules with up to 32 heavy atoms, resulting into a set of approximately 67
million compounds (more details in Appendix F). We evaluate the
representations of the trained model based on the predictive performance of a
SVM on two classification and two regression tasks from the MoleculeNet
benchmark [59]. We compare representations derived from our pre-trained model
with the state-of-the-art Extended Connectivity Fingerprint molecular
descriptors (radius=3, 2048 dim) [60]. For each task and descriptor, the
hyperparameter $C$ of the SVM was tuned in a nested cross validation. The
results are presented in Table 5. Descriptors derived from our pre-trained
model seem to represent molecular graphs in a meaningful way as they
outperform the baseline on average in three out of four tasks.

##  5 Conclusion, Limitations and Future Work

In this work we proposed a permutation invariant autoencoder for graph-level
representation learning. By predicting the relation (permutation matrix)
between input and output graph order, our proposed model can directly be
trained on node and edge feature reconstruction, while being invariant to a
distinct node order. This poses, to the best of our knowledge, the first
method for non-contrastive and non-adversarial learning of permutation
invariant graph-level representations that can also be used for graph
generation and might be an important step towards more powerful representation
learning methods on graph structured data or sets in general. We demonstrate
the effectiveness of our method in encoding graphs into meaningful
representations and evaluate its competitive performance in various
experiments. Although we propose a way of reducing the computational
complexity by only attending on incoming messages in our directed message
self-attention framework, in its current state, our proposed model is limited
in the number of nodes a graph can consist of. However, recently, much work
has been done on more efficient and sparse self-attention frameworks [61, 62,
63]. In future work, we aim at building up on this work to scale our proposed
method to larger graphs. Moreover, we will investigate further into the
generative performance of the proposed model, as this work was mainly
concerned with its representation learning capability.

## Funding Disclosures

D.A.C. received financial support from European Commission grant numbers
963845 and 956832 under the Horizon2020 Framework Program for Research and
Innovation. F.N. acknowledges funding from the European Commission (ERC CoG
772230 “ScaleCell”) and MATH+ (AA1-6). F.N. is advisor for Relay Therapeutics
and scientific advisory board member of 1qbit and Redesign Science.

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