# GOODAT: Towards Test-time Graph Out-of-Distribution Detection

Luzhi Wang 1, Dongxiao He 1, He Zhang 2, Yixin Liu 2,
Wenjie Wang 3, Shirui Pan 4, Di Jin 1, Tat-Seng Chua 3 Corresponding author.

###### Abstract

Graph neural networks (GNNs) have found widespread application in modeling
graph data across diverse domains. While GNNs excel in scenarios where the
testing data shares the distribution of their training counterparts (in
distribution, ID), they often exhibit incorrect predictions when confronted
with samples from an unfamiliar distribution (out-of-distribution, OOD). To
identify and reject OOD samples with GNNs, recent studies have explored graph
OOD detection, often focusing on training a specific model or modifying the
data on top of a well-trained GNN. Despite their effectiveness, these methods
come with heavy training resources and costs, as they need to optimize the
GNN-based models on training data. Moreover, their reliance on modifying the
original GNNs and accessing training data further restricts their
universality. To this end, this paper introduces a method to detect Graph Out-
of-Distribution At Test-time (namely GOODAT), a data-centric, unsupervised,
and plug-and-play solution that operates independently of training data and
modifications of GNN architecture. With a lightweight graph masker, GOODAT can
learn informative subgraphs from test samples, enabling the capture of
distinct graph patterns between OOD and ID samples. To optimize the graph
masker, we meticulously design three unsupervised objective functions based on
the graph information bottleneck principle, motivating the masker to capture
compact yet informative subgraphs for OOD detection. Comprehensive evaluations
confirm that our GOODAT method outperforms state-of-the-art benchmarks across
a variety of real-world datasets.

## Introduction

Graph neural networks (GNNs) are potent representation learning methods that
focus on processing graph data (Wang et al. 2019a, b), and have been widely
used in financial networks (Zheng et al. 2021), binary code analysis (Wang et
al. 2023a), sarcasm detection (Wang et al. 2023b; Yu et al. 2023), etc.
Generally, GNNs provide strong support for accurate prediction of downstream
tasks by capturing the distribution of training data (Kipf and Welling 2017;
Zhang et al. 2021). However, when these well-trained GNN models are deployed
in open-world scenarios, they inevitably encounter graph samples from unknown
classes, the so-called graph “Out-of-Distribution (OOD)” data (Liu et al.
2023a). On the OOD data, well-trained GNNs may not be effective, as the
features and distribution patterns of these OOD graphs are not exposed during
GNN training (Bai et al. 2023). This situation can lead to prediction errors
when dealing with these unknown distribution samples, thereby reducing the
reliability of GNNs. In this scenario, an ideal GNN model should possess the
capability to identify and reject OOD samples, rather than misclassifying them
as belonging to the in-distribution (ID) classes.

Figure 1: Comparisons between GOODAT and other methods. To detect OOD samples,
(a) most GNN-based methods need to learn a detector from the training data
(Liu et al. 2023a); (b) other data-centric methods learn an MLP to modify the
training data while keeping the well-trained GNN fixed (Guo et al. 2023). (c)
In contrast, our test-time OOD detector directly works on the test data
without needing to consult the training data and change the parameters of the
well-trained GNN.

To effectively identify OOD graph samples, various graph OOD detection
approaches have emerged (Huang, Wang, and Fang 2022; Hoffmann, Galke, and
Scherp 2023). A subset of research (Liu et al. 2023a) centers around crafting
GNN-based OOD detection models that are meticulously tailored for graph OOD
detection tasks. While demonstrating effectiveness, they need to train
additional GNNs from scratch, resulting in a heavy resource expenditure.
Another type of methodology solves the graph OOD detection problem from a
data-centric perspective (Zheng et al. 2023a), implementing downstream tasks
by modifying data on top of well-trained GNNs. As a typical method, AAGOD (Guo
et al. 2023) imposes a multi-layer perception (MLP) based parametric matrix on
the adjacency matrix of each training graph. This is done without changing the
parameters of the well-trained GNN, effectively widening the differences
between OOD and ID graphs. By merely optimizing the MLP instead of retraining
the GNN, AAGOD mitigates the effort required for model design and parameter
training.

Despite the effectiveness of the aforementioned methods, both types of
existing graph OOD detection approaches rely on the training dataset, leading
to several limitations. Firstly, training a task-specific GNN-based model for
OOD detection usually requires significant computational resources and costs,
e.g., training an additional GNN from scratch (Jin et al. 2023c; Gui et al.
2022). Secondly, in certain scenarios or platforms where model architectures
are inaccessible due to privacy concerns (Zheng et al. 2023c), making
modifications or adjustments to the GNN architecture becomes impractical.
Meanwhile, in some cases (e.g., federated learning (Tan et al. 2023a)), the
original training dataset may also be inaccessible (Tan et al. 2023b), further
obstructing the training process of these OOD methods. In such instances,
these aforementioned OOD detection methods cannot be applied, exposing their
limited universality.

To address the above issues, we delve into the research problem of test-time
graph OOD detection. Concretely, a test-time graph OOD detection method solely
learns on testing data, without being dependent on training data. Moreover,
the method is not expected to redesign the backbone model or add additional
networks, ensuring its adaptability across diverse well-trained GNN models,
irrespective of the characteristics of the training data. Nevertheless,
developing such a test-time OOD detection method presents substantial
complexities due to the following challenges. Challenge 1: Inconsistent
learning objective. Most well-trained GNNs are trained for specific graph
learning tasks (e.g., graph classification) rather than OOD detection. In this
case, how to align these pre-existing models with the target of OOD detection
remains a difficulty. Challenge 2: Absence of labels. During test time, the
lack of graph labels poses a challenge in unsupervised detecting ID and OOD
graphs, compelling the need to design an unsupervised model. Challenge 3:
Unavailability of training data. Constrained by the test-time setting, access
to the training dataset is unfeasible. This scarcity of comprehensive
knowledge about the original training data of the GNN model imposes
significant limitations on our capacity to integrate an OOD detection model
into GNNs.

To solve the aforementioned challenges, we propose a novel data-centric method
to detect Graph Out-Of-Distribution At Test-time, namely GOODAT. To address
challenge 1, we first construct a plug-and-play graph masker consisting of
parameterized matrices. This enables us to compress informative subgraphs from
the original input graphs, thereby indicating their ID or OOD nature. This
lightweight masker can seamlessly integrate with any well-trained GNN,
endowing it with the capability to detect OOD samples. To handle challenge 2,
we design three unsupervised loss functions based on the graph information
bottleneck (GIB) principle, guiding the masker to capture compact yet
sufficient subgraphs for distinguishing ID and OOD graphs. To deal with
challenge 3, we fully exploit the test data to capture the ID graph patterns
of training data. Specifically, we operate under the assumption that all
graphs in the test dataset are inherently ID. Guided by these surrogate ID
labels, the OOD subgraph compressed by the ID label should significantly
differ from the ID subgraph compressed by the ID label. This distinction
serves as a reliable basis for effectively detecting OOD graphs. Fig. 1 shows
the differences between GOODAT and other methods. To sum up, the main
contributions of this paper are three-fold:

  * •

New paradigm. We pioneer the learning paradigm of test-time graph OOD
detection, unveiling a fresh perspective. This innovative paradigm sheds light
on lightweight, training data-independent, and plug-and-play solutions for
graph OOD detection, seamlessly applicable to any well-trained GNN models.

  * •

Novel method. We propose a simple yet effective method, GOODAT, to solve the
test-time graph OOD detection problem. Leveraging the information bottleneck
principle, GOODAT captures informative subgraphs from each input graph, thus
enabling the accurate identification of OOD samples within the test dataset.

  * •

Extensive experiments. We conduct experiments on multiple datasets and
scenarios to verify the effectiveness and superiority of GOODAT. Experimental
results show that GOODAT has achieved significant improvements in graph OOD
detection tasks compared to baselines.

Figure 2: Overview of GOODAT. In the GOODAT training process, a graph masker
$M$ is applied on the input test graph $G$, consisting of two parameterized
matrices. This graph masker $M$ is trained by utilizing three GIB-boosted
losses, taking the graph $G$ and its corresponding surrogate ID label $Y$ as
inputs. The informative subgraph $Z$ and the masked graph $Z^{\prime}$ are
obtained with the trainable parameters $M$ (e.g., $Z=G\odot M$). During the
inference phase of target GNNs, the OOD score of a test graph is obtained by
the graph masker and GIB-boosted losses to infer if the input graph is an OOD
graph.

## Related Works

### Graph Neural Networks

With graph-related problems rising in various real-world scenarios (Wu et al.
2022; Zheng et al. 2023b; Wu et al. 2023a), GNNs have emerged as a powerful
paradigm for tackling complex graph data (Liu et al. 2023c; Zheng et al.
2023d, 2022). GNNs have shown remarkable success across various domains (Zhang
et al. 2022), including social networks (Zheng et al. 2022), anomaly detection
(Liu et al. 2023b), binary code analysis (Jin et al. 2022), and recommender
systems (Jin et al. 2023b, a). Although many methods have been proposed to
improve GNN performance (Kipf and Welling 2017) (Xu et al. 2019), concerns
have emerged about other aspects (e.g., robustness (Zhang et al. 2023c),
privacy (Zhang et al. 2023a; Wu et al. 2024), fairness (Zhang et al. 2023b))
of GNN models. In the context of generalization, while GNNs perform well on ID
data, they may perform poorly on OOD data (Liu et al. 2023a).

### Graph OOD Detection

Graph OOD detection aims to detect whether the test graph is ID or OOD.
Recently, many studies have been proposed (Zhao et al. 2020; Stadler et al.
2021; Wu et al. 2023b). For example, GOOD-D (Liu et al. 2023a) uses graph
contrastive learning to provide an unsupervised view of graph OOD detection.
GraphDE (Li et al. 2022) models the graph generative process to learn latent
environment variables for detection. OODGAT (Song and Wang 2022) utilizes a
multi-head attention mechanism to compute node weights and transform them into
edge weights, aiding in the identification of OOD nodes. AAGOD (Guo et al.
2023) introduces a data-centric framework that enlarges the differences
between OOD and ID graphs. However, the above methods rely on the training
dataset to train OOD detection models, which is different from our GOODAT
method.

## Preliminaries and Problem Definition

Graphs and Graph Maskers. Given an undirected graph $G=(X,A)$,
$X\in\mathbb{R}^{n\times d}$ represents the node feature matrix with $n$ nodes
and each node has a feature dimension of $d$, and $A\in\mathbb{R}^{n\times n}$
indicates the adjacency matrix of $G$. A label $Y$ is associated with $G$,
where $Y=0$ indicates that the graph is ID, and $Y=1$ indicates that the graph
is OOD. A graph masker is defined as ${M=(M_{X},M_{A})}$, where
${M_{X}\in\mathbb{R}^{n\times d}}$ and ${M_{A}\in\mathbb{R}^{n\times n}}$ are
parameterized matrices for extracting the subgraph from the original graph
$G$. We can modify the graph masker by gradient descent on $M_{X}$ and
$M_{A}$. For example, we can obtain a subgraph ${Z=G\odot M=(X\odot
M_{X},A\odot M_{A})}$ from $G$, where $\odot$ denotes the Hadamard product.
Given $Z$, the remaining part of the test graph is donated as the masked graph
${Z^{\prime}=(X-X\odot M_{X},A-A\odot M_{A})}$. For a given test graph
$G=Z\cup Z^{\prime}$, the overlap between $Z$ and $Z^{\prime}$ is defined as
$Z\cap Z^{\prime}$, and $|Z\cap Z^{\prime}|$ represents its size.

Graph Information Bottleneck (GIB). From the view of information theory, the
information bottleneck aims at compressing the original information to obtain
crucial information related to the label (Alemi et al. 2017). As for graph
information bottleneck, given a graph $G$ and its label $Y$, the graph
information bottleneck aims to compress the information of $G$ to obtain a
compressed graph $Z$, which maximizes the mutual information between $Y$ and
$Z$ and minimizes the mutual information between $Z$ and $G$ (Yu et al. 2021).
Specifically, assuming that $I(\cdot)$ indicates the Shannon mutual
information, the GIB can be defined as (Wu et al. 2020):

$\displaystyle\max_{Z}{I(Y,Z)-\lambda I(G,Z)},$ (1)

where $\lambda$ is a Lagrange multiplier.

Test-time Graph OOD Detection. For a test graph $G$, we assumed that it comes
from the ID graph distribution or the OOD graph distribution. We can define
the test-time graph OOD detection task as:

######  Definition 1 (Test-time graph OOD detection).

Given a well-trained GNN $f$ and a graph $G$ from the test dataset, the test-
time graph OOD detection aims to determine the source distribution of $G$
during the inference time of $f$ with an OOD detector D. Specifically, the
objective of the detection task is:

$\texttt{Detection label}=\begin{cases}1\ (OOD),&if\
\texttt{D}(f,G)\geqslant\eta\\\ 0\ (ID),&if\ \texttt{D}(f,G)<\eta\\\
\end{cases}$ (2)

where $\eta$ is a threshold, and the parameters of $f$ are fixed during the
OOD detection.

Figure 3: The core idea of GOODAT. ‘Inf.’ is the abbreviation of information.

## Methods

Fig. 2 shows the overview of our GOODAT method, whose effectiveness in
distinguishing ID and OOD graphs is demonstrated below. Given the surrogate
label, GIB is employed to find the subgraph with the highest correlation to
its label for each input graph. Hence, the subgraphs obtained from the ID and
OOD graphs can be distinguishable, since they come from different
distributions. During the test-time, a well-trained GNN $f$ tends to make
right predictions on ID graphs while making wrong predictions on OOD graphs,
since OOD graphs can be regarded as being in an unknown class. In this case,
the subgraph of OOD graphs extracted by the GIB principle can be significantly
different from the ones of ID graphs. As shown in Fig. 3, considering an ID
graph $G^{I}$ with the right label prediction (e.g., $Y=0$), the subgraph
obtained from GIB is denoted as $Z_{0}^{I}$. When the predicted label (e.g.,
$Y=0$) is wrong (e.g., for an OOD graph $G^{O}$), the extracted subgraph
$Z_{0}^{O}$ obviously differentiates from $Z_{0}^{I}$ as $Z_{0}^{I}$ is not
included in $G^{O}$. Otherwise, $G^{O}$ is not an OOD graph. Such
distinguishability between $Z_{0}^{I}$ and $Z_{0}^{O}$ can effectively
separate ID and OOD graphs, enabling our method to execute the test-time OOD
detection.

In this paper, three different GIB-boosted loss functions are presented to
enhance the capability of GIB on reasoning subgraphs that most correlate with
labels. The following are details on each of the three loss functions that
play different roles in our optimization.

### Subgraph GIB Loss

From the perspective of factual reasoning, the subgraph GIB loss facilitates
the extraction of informative subgraph $Z$ that related to the predicted
label. With the surrogate label $Y$ and obtaining the informative subgraph
$Z=G\odot M$ by applying the graph masker $M=(M_{X},M_{A})$, we propose to
utilize the following subgraph GIB loss to optimize the parameters $M_{X}$ and
$M_{A}$ in $M$. Specifically, we maximize the Shannon mutual information
between the subgraph $Z$ (embedding from $f$) and the label $Y$, while
minimizing the Shannon mutual information between the subgraph $Z$ and the
original test graph $G$ (embedding from $f$). We seek the most informative and
compressed subgraph representation by optimizing the following objective:

$\max_{Z}I(Z,Y)-\alpha I(Z,G),$ (3)

where $\alpha$ is the Lagrange multiplier to balance the two components.
Building on the work (Alemi et al. 2017), we transfer $I(Z,Y)$ and $I(Z,G)$
into our loss function.

(1) According to the definition of mutual information, we obtain $I(Z,Y)=\iint
p(Z,Y)\log\frac{p(Y|Z)}{p(Y)}dYdZ$. For the item
${p(Y|Z)=\int\frac{p(Y|G)p(Z|G)p(G)}{p(Z)}dG}$, which depends on the Markov
property (Sun et al. 2022) and is difficult to calculate, we propose using
$q(Y|Z)$ to approximate $p(Y|Z)$. Specifically, we use the Kullback-Leibler
divergence (Kullback and Leibler 1951) to measure their distance and make them
closer. Since ${D_{KL}(p(Y|Z)||q(Y|Z))\geqslant 0}$, we obtain

$\int p(Y|Z)\log p(Y|Z)dy\geqslant\int p(Y|Z)\log q(Y|Z)dY.\\\ $ (4)

According to Eq. (4), the lower bound of $I(Z,Y)$ is:

$\begin{aligned} I(Z,Y)&\geqslant\iint p(Y,Z)\log\frac{q(Y,Z)}{q(Z)}dZdY-\int
p(Y)\log p(Y)dY.\\\
&\geqslant\iiint\frac{p(Y,G)p(Z,G)}{p(G)}\log\frac{q(Y,Z)}{q(Z)}dZdYdG.\end{aligned}$

(5)

(2) For $I(Z,G)=\iint p(Z,G)\log\frac{p(Z,G)}{p(Z)p(G)}dZdG$, where it is not
easy to directly calculate $p(Z)$, we propose to use a learnable $\phi(Z)$ to
approximate $p(Z)$. Similarly, since $D_{KL}(p(Z)||\phi(Z))\geqslant 0$, we
can get an upper bound of $I(Z,G)$:

$\displaystyle I(Z,G)\leqslant\iint p(Z,G)\log\frac{p(Z,G)}{\phi(Z)p(G)}dGdZ.$
(6)

By combining the above inequalities about $I(Z,Y)$ and $I(Z,G)$, we derive a
lower bound of $I(Z,Y)-\alpha I(Z,G)$. Specifically,

$\displaystyle I(Z,Y)-\alpha I(Z,G)$ (7)
$\displaystyle\geqslant\iiint\frac{p(Y,G)p(Z,G)}{p(G)}\log\frac{q(Y,Z)}{q(Z)}dZdYdG$
$\displaystyle-\alpha\iint p(Z,G)\log\frac{p(Z,G)}{\phi(Z)p(G)}dGdZ.$

where $\alpha$ is a hyperparameter that balances informativeness and
compression. Thus, instead of directly solving $\max_{Z}I(Z,Y)-\alpha I(Z,G)$,
we employ the following subgraph GIB loss $\mathcal{L}_{s}$:

$\displaystyle\mathcal{L}_{s}=$
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}(\mathbb{E}_{\xi\sim p(\xi)}(-\log
q(Y_{i}|Z_{i}))+\alpha D_{KL}[p(Z_{i}|G_{i})||\phi(Z_{i})]),$ (8)
$\displaystyle\approx\mathcal{L}_{cls}(q(Y_{i}|Z_{i}),Y_{i})+\alpha
D_{KL}[p(Z_{i}|G_{i})||\phi(Z_{i})]),$

where $N$ represents the number of graphs, $\mathcal{L}_{cls}$ indicates the
classification loss. We use the Gaussian distribution to approximate $q$ and
$\phi$. Fig. 4 (a) illustrates the impact of subgraph GIB loss on the subgraph
compression process.

### Masked Graph GIB Loss

From the perspective of counterfactual reasoning, the masked graph GIB loss is
used to obtain the masked graph $Z^{\prime}$, which ensures the irrelevance
between the $Z^{\prime}$ and the label. Considering that $Z^{\prime}$ is
regarded as the noise part of the graph $G$, the optimization objective from
the GIB perspective can be formulated as:

$\max_{Z^{\prime}}\beta I(Z^{\prime},G)-I(Z^{\prime},Y),$ (9)

where $\beta$ is a Lagrange multiplier. Similar to the derivation of the
subgraph GIB loss, the masked graph GIB loss $\mathcal{L}_{m}$ can be defined
as:

$\displaystyle\mathcal{L}_{m}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}(\mathbb{E}_{\xi\sim p(\xi)}(\log
q(Y_{i}|Z_{i}^{\prime})-\beta
D_{KL}[p(Z_{i}^{\prime}|G_{i})||\phi(Z_{i}^{\prime})]),$ (10)
$\displaystyle\approx-\mathcal{L}_{cls}(q(Y_{i}|Z_{i}^{\prime}),Y_{i})-\beta
D_{KL}[p(Z_{i}^{\prime}|G_{i})||\phi(Z_{i}^{\prime})]).$

We use the Gaussian distribution to approximate $q$ and $\phi$. Fig. 4 (b)
illustrates the masked graph GIB loss.

According to an existing study (Tan et al. 2022), only considering the
subgraph GIB loss $\mathcal{L}_{s}$ leads to sufficient but not necessary $Z$,
while exclusively using the masked graph GIB loss $\mathcal{L}_{m}$ tends to
generate necessary but not sufficient $Z$. To overcome this limitation (i.e.,
a large $|Z\cap Z^{\prime}|$), employing $\mathcal{L}_{s}$ and
$\mathcal{L}_{m}$ simultaneously helps to dig out sufficient and necessary $Z$
(i.e., limited $|Z\cap Z^{\prime}|$).

### Graph Distribution Separating Loss

To further reduce the overlapping size (i.e., $|Z\cap Z^{\prime}|$), we
propose the graph distribution separating loss to separate the distributions
of $Z$ and $Z^{\prime}$ from the statistical view, where $Z$ and $Z^{\prime}$
are supposed to own a small joint probability density. Sklar (Sklar 1959)
declares that the joint distribution of $N$ random variables can be decomposed
into the respective marginal distributions of the $N$ variables and a Copula
function (Sklar 1973), so as to separate the randomness and coupling of the
variables. In this paper, we assume that $Z$ and $Z^{\prime}$ have the
following Gaussian form of marginal distributions, i.e.,

$\displaystyle
p_{Z}(Z)=\frac{1}{\sqrt{2\pi}|\sigma_{1}|}e^{\frac{-Z^{2}}{2\sigma_{1}^{2}}},p_{Z^{\prime}}(Z^{\prime})=\frac{1}{\sqrt{2\pi}|\sigma_{2}|}e^{\frac{-Z^{\prime
2}}{2\sigma_{2}^{2}}},$ (11)

where $\sigma_{1}$ and $\sigma_{2}$ represent the standard deviation of $Z$
and $Z^{\prime}$, respectively.

According to the Copula theory (Durante, Fernandez-Sanchez, and Sempi 2013),
there must be a function that couples the marginal probabilities. Therefore,
the joint probability function of $Z$ and $Z^{\prime}$ can be defined as:

$\displaystyle\mathcal{L}_{d}=\frac{1}{2\pi|\sigma_{1}\sigma_{2}|\sqrt{1-\rho^{2}}}e^{\left[-\frac{1}{2\left(1-\rho^{2}\right)}\times\left[\frac{Z^{2}}{\sigma_{1}^{2}}-2\rho\frac{Z\times
Z^{\prime}}{|\sigma_{1}\sigma_{2}|}+\frac{Z^{\prime
2}}{\sigma_{2}^{2}}\right]\right]},$ (12)

where $\rho$ indicates the correlation coefficient. Fig. 4 (c) illustrates how
the graph distribution separating loss influences the distinguishability of
graph distributions.

Figure 4: Illustrations of the three GIB-boosted losses in GOODAT.

### Training of GOODAT and OOD Graph Detection

By integrating the above three loss functions together, the overall loss
$\mathcal{L}_{g}$ for training $M$ can be expressed as:

$\mathcal{L}_{g}=\mathcal{L}_{s}+\mathcal{L}_{m}+\mathcal{L}_{d}.$ (13)

Note that in our method, all parameters in the target GNN $f$ are fixed, and
we employ the surrogate label $Y$ and embeddings of $G$, $Z$, and $Z^{\prime}$
to train the mask generator $M$.

When implementing OOD detection, we employ the subgraph GIB loss (Eq. 8) as
the graph OOD detection score. Specifically, for a test graph $G$, we first
use the well-trained graph masker $M$ to obtain its informative subgraph
$Z=G\odot M$. Next, we obtain the GIB loss on $Z$ w.r.t. $\mathcal{L}_{s}$,
which is used to determine whether $G$ is an OOD graph by Eq. (2).

## Experiments

Table 1: OOD detection results in terms of AUC score ($\%$). The best results
are highlighted with bold.

ID dataset | BZR | PTC-MR | AIDS | ENZYMES | IMDB-M | Tox21 | FreeSolv | BBBP | ClinTox | Esol | Avg.
---|---|---|---|---|---|---|---|---|---|---|---
OOD dataset | COX2 | MUTAG | DHFR | PROTEIN | IMDB-B | SIDER | ToxCast | BACE | LIPO | MUV | Rank
PK-LOF | $42.22{\scriptstyle\pm 8.39}$ | $51.04{\scriptstyle\pm 6.04}$ | $50.15{\scriptstyle\pm 3.29}$ | $50.47{\scriptstyle\pm 2.87}$ | $48.03{\scriptstyle\pm 2.53}$ | $51.33{\scriptstyle\pm 1.81}$ | $49.16{\scriptstyle\pm 3.70}$ | $53.10{\scriptstyle\pm 2.07}$ | $50.00{\scriptstyle\pm 2.17}$ | $50.82{\scriptstyle\pm 1.48}$ | $9.9$
PK-OCSVM | $42.55{\scriptstyle\pm 8.26}$ | $49.71{\scriptstyle\pm 6.58}$ | $50.17{\scriptstyle\pm 3.30}$ | $50.46{\scriptstyle\pm 2.78}$ | $48.07{\scriptstyle\pm 2.41}$ | $51.33{\scriptstyle\pm 1.81}$ | $48.82{\scriptstyle\pm 3.29}$ | $53.05{\scriptstyle\pm 2.10}$ | $50.06{\scriptstyle\pm 2.19}$ | $51.00{\scriptstyle\pm 1.33}$ | $10.0$
PK-iF | $51.46{\scriptstyle\pm 1.62}$ | $54.29{\scriptstyle\pm 4.33}$ | $51.10{\scriptstyle\pm 1.43}$ | $51.67{\scriptstyle\pm 2.69}$ | $50.67{\scriptstyle\pm 2.47}$ | $49.87{\scriptstyle\pm 0.82}$ | $52.28{\scriptstyle\pm 1.87}$ | $51.47{\scriptstyle\pm 1.33}$ | $50.81{\scriptstyle\pm 1.10}$ | $50.85{\scriptstyle\pm 3.51}$ | $8.1$
WL-LOF | $48.99{\scriptstyle\pm 6.20}$ | $53.31{\scriptstyle\pm 8.98}$ | $50.77{\scriptstyle\pm 2.87}$ | $52.66{\scriptstyle\pm 2.47}$ | $52.28{\scriptstyle\pm 4.50}$ | $51.92{\scriptstyle\pm 1.58}$ | $51.47{\scriptstyle\pm 4.23}$ | $52.80{\scriptstyle\pm 1.91}$ | $51.29{\scriptstyle\pm 3.40}$ | $51.26{\scriptstyle\pm 1.31}$ | $7.2$
WL-OCSVM | $49.16{\scriptstyle\pm 4.51}$ | $53.31{\scriptstyle\pm 7.57}$ | $50.98{\scriptstyle\pm 2.71}$ | $51.77{\scriptstyle\pm 2.21}$ | $51.38{\scriptstyle\pm 2.39}$ | $51.08{\scriptstyle\pm 1.46}$ | $50.38{\scriptstyle\pm 3.81}$ | $52.85{\scriptstyle\pm 2.00}$ | $50.77{\scriptstyle\pm 3.69}$ | $50.97{\scriptstyle\pm 1.65}$ | $8.1$
WL-iF | $50.24{\scriptstyle\pm 2.49}$ | $51.43{\scriptstyle\pm 2.02}$ | $50.10{\scriptstyle\pm 0.44}$ | $51.17{\scriptstyle\pm 2.01}$ | $51.07{\scriptstyle\pm 2.25}$ | $50.25{\scriptstyle\pm 0.96}$ | $52.60{\scriptstyle\pm 2.38}$ | $50.78{\scriptstyle\pm 0.75}$ | $50.41{\scriptstyle\pm 2.17}$ | $50.61{\scriptstyle\pm 1.96}$ | $9.3$
InfoGraph-iF | $63.17{\scriptstyle\pm 9.74}$ | $51.43{\scriptstyle\pm 5.19}$ | $93.10{\scriptstyle\pm 1.35}$ | $60.00{\scriptstyle\pm 1.83}$ | $58.73{\scriptstyle\pm 1.96}$ | $56.28{\scriptstyle\pm 0.81}$ | $56.92{\scriptstyle\pm 1.69}$ | $53.68{\scriptstyle\pm 2.90}$ | $48.51{\scriptstyle\pm 1.87}$ | $54.16{\scriptstyle\pm 5.14}$ | $5.4$
InfoGraph-MD | $\mathbf{86.14{\scriptstyle\pm 6.77}}$ | $50.79{\scriptstyle\pm 8.49}$ | $69.02{\scriptstyle\pm 11.67}$ | $55.25{\scriptstyle\pm 3.51}$ | $\mathbf{81.38{\scriptstyle\pm 1.14}}$ | $59.97{\scriptstyle\pm 2.06}$ | $58.05{\scriptstyle\pm 5.46}$ | $70.49{\scriptstyle\pm 4.63}$ | $48.12{\scriptstyle\pm 5.72}$ | $77.57{\scriptstyle\pm 1.69}$ | $4.5$
GraphCL-iF | $60.00{\scriptstyle\pm 3.81}$ | $50.86{\scriptstyle\pm 4.30}$ | $92.90{\scriptstyle\pm 1.21}$ | $61.33{\scriptstyle\pm 2.27}$ | $59.67{\scriptstyle\pm 1.65}$ | $56.81{\scriptstyle\pm 0.97}$ | $55.55{\scriptstyle\pm 2.71}$ | $59.41{\scriptstyle\pm 3.58}$ | $47.84{\scriptstyle\pm 0.92}$ | $62.12{\scriptstyle\pm 4.01}$ | $5.7$
GraphCL-MD | $83.64{\scriptstyle\pm 6.00}$ | $73.03{\scriptstyle\pm 2.38}$ | $93.75{\scriptstyle\pm 2.13}$ | $52.87{\scriptstyle\pm 6.11}$ | $79.09{\scriptstyle\pm 2.73}$ | $58.30{\scriptstyle\pm 1.52}$ | $60.31{\scriptstyle\pm 5.24}$ | $75.72{\scriptstyle\pm 1.54}$ | $51.58{\scriptstyle\pm 3.64}$ | $78.73{\scriptstyle\pm 1.40}$ | $2.6$
GTrans | $55.17{\scriptstyle\pm 5.04}$ | $62.38{\scriptstyle\pm 2.36}$ | $60.12{\scriptstyle\pm 1.98}$ | $49.94{\scriptstyle\pm 5.67}$ | $51.55{\scriptstyle\pm 2.90}$ | $61.67{\scriptstyle\pm 0.73}$ | $50.81{\scriptstyle\pm 3.03}$ | $64.02{\scriptstyle\pm 2.10}$ | $58.54{\scriptstyle\pm 2.38}$ | $76.31{\scriptstyle\pm 3.85}$ | $5.5$
AAGOD-GIN${}_{S}+$ | $76.75$ | $-$ | $-$ | $66.22$ | $59.00$ | $64.26$ | $-$ | $67.80$ | $-$ | $-$ | $-$
AAGOD-GIN${}_{L}+$ | $76.00$ | $-$ | $-$ | $65.89$ | $62.70$ | $57.59$ | $-$ | $57.13$ | $-$ | $-$ | $-$
Ours | $82.16{\scriptstyle\pm 0.15}$ | $\mathbf{81.84{\scriptstyle\pm 0.57}}$ | $\mathbf{96.43{\scriptstyle\pm 0.25}}$ | $\mathbf{66.29{\scriptstyle\pm 1.54}}$ | $79.03{\scriptstyle\pm 0.03}$ | $\mathbf{68.92{\scriptstyle\pm 0.01}}$ | $\mathbf{68.83{\scriptstyle\pm 0.02}}$ | $\mathbf{77.07{\scriptstyle\pm 0.03}}$ | $\mathbf{62.46{\scriptstyle\pm 0.54}}$ | $\mathbf{85.91{\scriptstyle\pm 0.27}}$ | $\mathbf{1.4}$

### Experimental Setup

#### Datasets

In this paper, we utilize the evaluation protocol proposed by (Liu et al.
2023a) which encompasses a graph OOD detection benchmark and a graph anomaly
detection benchmark. The graph OOD detection benchmark contains $8$ pairs of
molecule datasets, $1$ pair of bioinformatics datasets, and $1$ pair of social
network datasets. Each dataset pair within the same field exhibits a moderate
domain shift. Following the setting in previous studies (Liu et al. 2023a; Guo
et al. 2023), we allocate $90\%$ of ID samples for training GNNs, while the
remaining $10\%$ of ID samples and an equal number of OOD samples constitute
the test set. The graph anomaly detection benchmark comprises $15$ datasets
from TU benchmark (Morris et al. 2020). Anomalies encompass samples of the
minority or real anomalous class, with the rest categorized as normal data. In
our test-time OOD detection setting, we use the test set of each dataset as
the input of GOODAT for test-time training.

#### Baselines $\&$ Settings

We compared GOODAT with $13$ mainstream baseline methods for graph OOD
detection, which can be divided into four categories.

  * •

Graph Kernels + Detectors. This type of method uses graph kernels as the pre-
train function to learn graph embedding, and then inputs these graph
embeddings to OOD/anomaly detector. Graph kernels include the Weisfeiler-
Lehman kernel (WL) (Shervashidze et al. 2011) and propagation kernel (PK)
(Neumann et al. 2016). OOD/anomaly detectors include local outlier factor
(LOF) (Breunig et al. 2000), one-class SVM (OCSVM) (Manevitz and Yousef 2001),
and isolation forest (iF) (Liu, Ting, and Zhou 2008).

  * •

Graph Neural Networks + Detectors. This type of method employs graph-level
self-supervised GNN models as pre-trained models alongside OOD detectors to
accomplish the detection. The self-supervised GNN models include InfoGraph
(Sun et al. 2020) and GraphCL (You et al. 2020). The detectors include iF and
Mahalanobis distance (MD) (Sehwag, Chiang, and Mittal 2021).

  * •

Test-time Training Methods. This type of method achieves graph OOD
generalization at test-time, without any modification to the parameters of
well-trained GNNs. The most recent method that falls into this category is
GTrans (Jin et al. 2023c). Since GTrans is not explicitly designed for graph
OOD detection, its loss value is adopted as the OOD score in our experiments.

  * •

Data-centric Methods. One quintessential method of this category is AAGOD (Guo
et al. 2023). AAGOD employs a graph adaptive amplifier module, which is
integrated into a well-trained GNN to facilitate graph OOD detection. Unlike
the test-time training approach, AAGOD employs the training set for the OOD
detection training process. As of the submission of this paper, AAGOD has not
released the code, so we conduct comparisons based on the experimental results
outlined in the AAGOD evaluations. AAGOD exists in two versions: AAGOD-
GIN${}_{S}+$ and AAGOD-GIN${}_{L}+$, which correspond to distinct OOD
evaluation methods. Unreported experimental results are denoted by ‘$-$’.

We employ a GIN (Xu et al. 2019) as the well-trained GNN encoder. We use the
Adam optimizer (Kingma and Ba 2014) for optimization. All experimental
procedures are conducted on a GeForce GTX TITAN X GPU device with $24$ GB
memory. We repeated all experiments five times and report the average score
and standard deviation.

Table 2: Anomaly detection results in terms of AUC score ($\%$). The best
results are highlighted with bold.

Method | PK-OCSVM | PK-iF | WL-OCSVM | WL-iF | InfoGraph-iF | GraphCL-iF | GTrans | Ours | Improve
---|---|---|---|---|---|---|---|---|---
PROTEINS-full | $50.49{\scriptstyle\pm 4.92}$ | $60.70{\scriptstyle\pm 2.55}$ | $51.35{\scriptstyle\pm 4.35}$ | $61.36{\scriptstyle\pm 2.54}$ | $57.47{\scriptstyle\pm 3.03}$ | $60.18{\scriptstyle\pm 2.53}$ | $60.16{\scriptstyle\pm 5.06}$ | $\mathbf{77.92{\scriptstyle\pm 2.37}}$ | $28.37\%$
ENZYMES | $53.67{\scriptstyle\pm 2.66}$ | $51.30{\scriptstyle\pm 2.01}$ | $\mathbf{55.24{\scriptstyle\pm 2.66}}$ | $51.60{\scriptstyle\pm 3.81}$ | $53.80{\scriptstyle\pm 4.50}$ | $53.60{\scriptstyle\pm 4.88}$ | $38.02{\scriptstyle\pm 6.24}$ | $52.33{\scriptstyle\pm 4.74}$ | $-$
AIDS | $50.79{\scriptstyle\pm 4.30}$ | $51.84{\scriptstyle\pm 2.87}$ | $50.12{\scriptstyle\pm 3.43}$ | $61.13{\scriptstyle\pm 0.71}$ | $70.19{\scriptstyle\pm 5.03}$ | $79.72{\scriptstyle\pm 3.98}$ | $84.57{\scriptstyle\pm 1.91}$ | $\mathbf{95.50{\scriptstyle\pm 0.99}}$ | $12.92\%$
DHFR | $47.91{\scriptstyle\pm 3.76}$ | $52.11{\scriptstyle\pm 3.96}$ | $50.24{\scriptstyle\pm 3.13}$ | $50.29{\scriptstyle\pm 2.77}$ | $52.68{\scriptstyle\pm 3.21}$ | $51.10{\scriptstyle\pm 2.35}$ | $61.15{\scriptstyle\pm 2.87}$ | $\mathbf{61.52{\scriptstyle\pm 2.86}}$ | $0.60\%$
BZR | $46.85{\scriptstyle\pm 5.31}$ | $55.32{\scriptstyle\pm 6.18}$ | $50.56{\scriptstyle\pm 5.87}$ | $52.46{\scriptstyle\pm 3.30}$ | $63.31{\scriptstyle\pm 8.52}$ | $60.24{\scriptstyle\pm 5.37}$ | $51.97{\scriptstyle\pm 8.15}$ | $\mathbf{64.77{\scriptstyle\pm 3.87}}$ | $2.31\%$
COX2 | $50.27{\scriptstyle\pm 7.91}$ | $50.05{\scriptstyle\pm 2.06}$ | $49.86{\scriptstyle\pm 7.43}$ | $50.27{\scriptstyle\pm 0.34}$ | $53.36{\scriptstyle\pm 8.86}$ | $52.01{\scriptstyle\pm 3.17}$ | $53.56{\scriptstyle\pm 3.47}$ | $\mathbf{59.99{\scriptstyle\pm 9.76}}$ | $12.01\%$
DD | $48.30{\scriptstyle\pm 3.98}$ | $71.32{\scriptstyle\pm 2.41}$ | $47.99{\scriptstyle\pm 4.09}$ | $70.31{\scriptstyle\pm 1.09}$ | $55.80{\scriptstyle\pm 1.77}$ | $59.32{\scriptstyle\pm 3.92}$ | $76.73{\scriptstyle\pm 2.83}$ | $\mathbf{77.62{\scriptstyle\pm 2.88}}$ | $1.16\%$
NCI1 | $49.90{\scriptstyle\pm 1.18}$ | $50.58{\scriptstyle\pm 1.38}$ | $50.63{\scriptstyle\pm 1.22}$ | $\mathbf{50.74{\scriptstyle\pm 1.70}}$ | $50.10{\scriptstyle\pm 0.87}$ | $49.88{\scriptstyle\pm 0.53}$ | $41.42{\scriptstyle\pm 2.16}$ | $45.96{\scriptstyle\pm 2.42}$ | $-$
IMDB-B | $50.75{\scriptstyle\pm 3.10}$ | $50.80{\scriptstyle\pm 3.17}$ | $54.08{\scriptstyle\pm 5.19}$ | $50.20{\scriptstyle\pm 0.40}$ | $56.50{\scriptstyle\pm 3.58}$ | $56.50{\scriptstyle\pm 4.90}$ | $45.34{\scriptstyle\pm 3.75}$ | $\mathbf{65.46{\scriptstyle\pm 4.34}}$ | $15.86\%$
REDDIT-B | $45.68{\scriptstyle\pm 2.24}$ | $46.72{\scriptstyle\pm 3.42}$ | $49.31{\scriptstyle\pm 2.33}$ | $48.26{\scriptstyle\pm 0.32}$ | $68.50{\scriptstyle\pm 5.56}$ | $71.80{\scriptstyle\pm 4.38}$ | $69.71{\scriptstyle\pm 2.21}$ | $\mathbf{80.31{\scriptstyle\pm 0.85}}$ | $11.85\%$
COLLAB | $49.59{\scriptstyle\pm 2.24}$ | $50.49{\scriptstyle\pm 1.72}$ | $\mathbf{52.60{\scriptstyle\pm 2.56}}$ | $50.69{\scriptstyle\pm 0.32}$ | $46.27{\scriptstyle\pm 0.73}$ | $47.61{\scriptstyle\pm 1.29}$ | $49.76{\scriptstyle\pm 1.37}$ | $44.91{\scriptstyle\pm 0.87}$ | $-$
HSE | $57.02{\scriptstyle\pm 8.42}$ | $56.87{\scriptstyle\pm 10.51}$ | $62.72{\scriptstyle\pm 10.13}$ | $53.02{\scriptstyle\pm 5.12}$ | $53.56{\scriptstyle\pm 3.98}$ | $51.18{\scriptstyle\pm 2.71}$ | $58.49{\scriptstyle\pm 2.68}$ | $\mathbf{63.05{\scriptstyle\pm 0.90}}$ | $0.53\%$
MMP | $46.65{\scriptstyle\pm 6.31}$ | $50.06{\scriptstyle\pm 3.73}$ | $55.24{\scriptstyle\pm 3.26}$ | $52.68{\scriptstyle\pm 3.34}$ | $54.59{\scriptstyle\pm 2.01}$ | $54.54{\scriptstyle\pm 1.86}$ | $48.19{\scriptstyle\pm 3.74}$ | $\mathbf{69.41{\scriptstyle\pm 0.04}}$ | $25.65\%$
p53 | $46.74{\scriptstyle\pm 4.88}$ | $50.69{\scriptstyle\pm 2.02}$ | $54.59{\scriptstyle\pm 4.46}$ | $50.85{\scriptstyle\pm 2.16}$ | $52.66{\scriptstyle\pm 1.95}$ | $53.29{\scriptstyle\pm 2.32}$ | $53.74{\scriptstyle\pm 2.98}$ | $\mathbf{63.27{\scriptstyle\pm 0.04}}$ | $15.90\%$
PPAR-gamma | $53.94{\scriptstyle\pm 6.94}$ | $45.51{\scriptstyle\pm 2.58}$ | $57.91{\scriptstyle\pm 6.13}$ | $49.60{\scriptstyle\pm 0.22}$ | $51.40{\scriptstyle\pm 2.53}$ | $50.30{\scriptstyle\pm 1.56}$ | $56.20{\scriptstyle\pm 1.57}$ | $\mathbf{68.23{\scriptstyle\pm 1.54}}$ | $17.82\%$
Avg. Rank | $6.3$ | $5.9$ | $4.4$ | $5$ | $4.2$ | $4.3$ | 4.4 | $\mathbf{2.1}$ |

Table 3: Ablation study results in terms of AUC score ($\%$). The best results
are highlighted with bold.

$\mathcal{L}_{s}$ |  $\mathcal{L}_{m}$ |  $\mathcal{L}_{d}$ | BZR | PTC-MR | AIDS | ENZYMES | IMDB-M | Tox21 | FreeSolv | BBBP | ClinTox | Esol
---|---|---|---|---|---|---|---|---|---|---|---|---
COX2 | MUTAG | DHFR | PROTEIN | IMDB-B | SIDER | ToxCast | BACE | LIPO | MUV
✓ | - | - | $39.10{\scriptstyle\pm 1.59}$ | $67.49{\scriptstyle\pm 3.48}$ | $72.28{\scriptstyle\pm 1.03}$ | $61.55{\scriptstyle\pm 3.25}$ | $79.01{\scriptstyle\pm 0.02}$ | $68.89{\scriptstyle\pm 0.01}$ | $68.58{\scriptstyle\pm 0.10}$ | $65.20{\scriptstyle\pm 1.06}$ | $56.50{\scriptstyle\pm 0.16}$ | $55.70{\scriptstyle\pm 1.38}$
- | ✓ | - | $73.53{\scriptstyle\pm 0.04}$ | $72.70{\scriptstyle\pm 0.04}$ | $82.34{\scriptstyle\pm 0.18}$ | $52.74{\scriptstyle\pm 0.01}$ | $69.98{\scriptstyle\pm 0.02}$ | $68.87{\scriptstyle\pm 0.01}$ | $67.65{\scriptstyle\pm 0.08}$ | $63.77{\scriptstyle\pm 0.24}$ | $66.71{\scriptstyle\pm 0.05}$ | $85.82{\scriptstyle\pm 0.08}$
- | - | ✓ | $72.11{\scriptstyle\pm 0.28}$ | $72.72{\scriptstyle\pm 0.06}$ | $92.52{\scriptstyle\pm 0.02}$ | $51.95{\scriptstyle\pm 0.11}$ | $79.02{\scriptstyle\pm 0.02}$ | $68.90{\scriptstyle\pm 0.01}$ | $68.62{\scriptstyle\pm 0.06}$ | $76.98{\scriptstyle\pm 0.04}$ | $54.03{\scriptstyle\pm 0.04}$ | $80.23{\scriptstyle\pm 0.05}$
✓ | ✓ | - | $\mathbf{83.06{\scriptstyle\pm 1.45}}$ | $81.83{\scriptstyle\pm 0.12}$ | $96.33{\scriptstyle\pm 0.29}$ | $65.67{\scriptstyle\pm 2.31}$ | $79.02{\scriptstyle\pm 0.03}$ | $68.92{\scriptstyle\pm 0.01}$ | $68.77{\scriptstyle\pm 0.10}$ | $76.65{\scriptstyle\pm 0.25}$ | $63.25{\scriptstyle\pm 0.56}$ | $85.90{\scriptstyle\pm 0.44}$
✓ | - | ✓ | $48.22{\scriptstyle\pm 3.09}$ | $67.09{\scriptstyle\pm 2.58}$ | $70.64{\scriptstyle\pm 3.28}$ | $60.46{\scriptstyle\pm 3.40}$ | $79.02{\scriptstyle\pm 0.02}$ | $68.89{\scriptstyle\pm 0.01}$ | $68.67{\scriptstyle\pm 0.14}$ | $66.21{\scriptstyle\pm 0.89}$ | $56.34{\scriptstyle\pm 0.17}$ | $54.60{\scriptstyle\pm 1.29}$
- | ✓ | ✓ | $73.58{\scriptstyle\pm 0.04}$ | $72.73{\scriptstyle\pm 0.05}$ | $82.83{\scriptstyle\pm 0.20}$ | $52.73{\scriptstyle\pm 0.02}$ | $79.01{\scriptstyle\pm 0.02}$ | $68.88{\scriptstyle\pm 0.01}$ | $67.73{\scriptstyle\pm 0.07}$ | $63.70{\scriptstyle\pm 0.14}$ | $\mathbf{66.76{\scriptstyle\pm 0.08}}$ | $85.76{\scriptstyle\pm 0.09}$
✓ | ✓ | ✓ | $82.16{\scriptstyle\pm 0.15}$ | $\mathbf{81.84{\scriptstyle\pm 0.57}}$ | $\mathbf{96.43{\scriptstyle\pm 0.25}}$ | $\mathbf{66.29{\scriptstyle\pm 1.54}}$ | $\mathbf{79.03{\scriptstyle\pm 0.03}}$ | $\mathbf{68.92{\scriptstyle\pm 0.01}}$ | $\mathbf{68.83{\scriptstyle\pm 0.02}}$ | $\mathbf{77.07{\scriptstyle\pm 0.03}}$ | $62.46{\scriptstyle\pm 0.54}$ | $\mathbf{85.91{\scriptstyle\pm 0.27}}$

### Performance of Graph OOD Detection

We conduct a comparative analysis of our proposed method against $13$
competing approaches across $10$ graph OOD detection datasets. The results, in
terms of AUC scores, are summarized and presented in Table 1. From this
comprehensive comparison, we garner the following insights: 1) GOODAT
showcases a remarkable performance by outperforming all baseline methods on
$8$ datasets. Furthermore, when considering the average rank across all
methods, our proposed approach stands as the leader. This underscores the
effectiveness of GOODAT in accurately detecting OOD samples within diverse
graph-structured datasets. 2) While we may not have achieved the absolute best
results on two datasets, our performance is in close proximity to the optimum
results. One plausible explanation for this outcome is the relatively high
edge density observed in these datasets, which potentially reduces the effect
of GIB-boosted loss for subgraph compression. 3) Although the test-time graph
OOD generalization method GTrans is also effective for OOD detection, our
method has achieved overall advantages on all datasets. This demonstrates that
directly substituting of graph OOD generalization for graph OOD detection may
lead to sub-optimal performance. 4) Compared to the data-centric method,
AAGOD, our proposed approach shows superiority across all datasets. This
implies that our method achieves better results only using a test dataset,
showcasing enhanced efficacy.

### Performance of Graph Anomaly Detection

To assess the potential applicability of GOODAT on graph anomaly detection
tasks, we conduct experiments on $15$ datasets, following the evaluation
protocol in (Liu et al. 2023a; Ma et al. 2022). We select $6$ graph anomaly
detection methods and a test-time training method as baselines, and the
experiment results are summarized in Table 2. The AAGOD baseline is not
included here as its code is not released. Experimental results indicate that
GOODAT’s applicability extends seamlessly to the anomaly detection scenario,
showcasing remarkable performance. This superior performance can be attributed
to the inherent strengths of our method, which effectively enhances the
distinctions in anomalous samples by utilizing GIB-boosted losses with ID-
label guidance. Moreover, we observe that GOODAT exhibits significant
advantages in anomaly detection over the use of GTrans. This observation
emphasizes the universality of GOODAT in contrast to other test-time-oriented
techniques.

### Ablation Study

GOODAT incorporates three core loss functions: subgraph GIB loss
$\mathcal{L}_{s}$, masked graph GIB loss $\mathcal{L}_{m}$, and graph
distribution separation loss $\mathcal{L}_{d}$. To evaluate the effectiveness
of each of them, an ablation study is conducted and the results are summarized
in Table 3, where a ‘$\checkmark$’ indicates the presence and a ‘$-$’ denotes
the absence of a component. As a result of analyzing the table, several key
observations emerge. Firstly, utilizing all components concurrently yields
optimal results on $8$ out of $10$ datasets, with decent results on the
remaining $2$ datasets. This demonstrates the effectiveness of integrating
multiple loss functions to enhance graph OOD detection. Secondly, the distinct
contributions of each loss function underscore their effectiveness as
individual components. Finally, combining the two loss functions often leads
to performance improvements, outperforming isolated evaluations of individual
components.

### Parameter Sensitivity Analysis

In GOODAT, two hyperparameters $\alpha$ (in Eq. 8) and $\beta$ (in Eq. 10) are
employed to control the involvement degree of subgraph GIB loss and masked
graph GIB loss, respectively. We conduct a parameter sensitivity experiment on
the PTC-MR/MUTAG dataset, where $\alpha$ is selected from
$\\{0.1,0.3,0.5,0.7,0.9\\}$ and $\beta$ is selected from
$\\{0.01,0.03,0.05,0.07,0.09\\}$. The results are depicted in Fig. 5 (a). The
results reveal that, when $\beta$ is fixed, optimal outcomes are achieved with
$\alpha$ values within the range of $0.1$-$0.3$. This implies that a slight
level of compression on subgraphs proves the most effective results. Likewise,
when $\alpha$ is held constant, $\beta$ values in the range of $0.3$-$0.5$
yield optimal results. This observation suggests that moderate compression of
masked graph components enhances model effectiveness.

### Visualization

(a) Influence on the parameter $\alpha$ and $\beta$.

(b) OOD and ID G-Emb.

(c) Sub and Masked G-Emb.

Figure 5: Parameter sensitivity analysis and visualization.

To visually demonstrate the impact of our methods, we visualize distributions
of subgraph embeddings to show the distinction between ID and OOD graphs. Fig.
5 (b) shows the embedding distributions of OOD subgraphs and ID subgraphs. We
observe that the ID subgraph and OOD subgraph can be clearly distinguished,
which indicates that GOODAT can detect OOD graphs intuitively. Fig. 5 (c)
shows the distribution of subgraphs (i.e., $Z$) and masked graphs (i.e.,
$Z^{\prime}$), where the distinction between subgraphs and masked graphs is
also obvious. This demonstrates the effectiveness of our proposed graph
distribution separation loss.

## Conclusions

In this paper, we make the first attempt toward detecting graph out-of-
distribution (OOD) samples at test time. To achieve this, we introduce a
pioneering method, named GOODAT, which is a data-centric, unsupervised, and
plug-and-play solution. With a graph masker applied to the input test graph,
GOODAT identifies the clear differentiation between OOD graphs and ID graphs.
We design three GIB-boosted losses to optimize the graph masker. Comprehensive
experimentation demonstrates the superiority of GOODAT compared to baseline
methods across diverse real-world benchmark datasets.

## Acknowledgments

This work is supported by National Key Research and Development Program of
China (2023YFC3304503), National Natural Science Foundation of China
(92370111, 62276187, 62272340).

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# VLBI observations of supernova PTF11qcj:
Direct constraints on the size of the radio ejecta

N. T. Palliyaguru 11affiliation: Department of Physics and Astronomy, Texas
Tech University, Lubbock, TX 79409-1051 (USA) **affiliationmark:  A. Corsi
11affiliation: Department of Physics and Astronomy, Texas Tech University,
Lubbock, TX 79409-1051 (USA) M. Pérez-Torres 22affiliation: Instituto de
Astrofísica de Andalucía -Consejo Superior de Investigaciones Científicas
(CSIC), PO Box 3004, 18008, Granada, Spain. E. Varenius 33affiliation:
Department of Earth and Space Sciences, Chalmers University of Technology,
Onsala Space Observatory, 439 92 Onsala, Sweden. 44affiliation: Jodrell Bank
Centre for Astrophysics, The University of Manchester, Oxford Rd, Manchester
M13 9PL, UK. H. Van Eerten55affiliation: Department of Physics, University of
Bath, Claverton Down, Bath BA2 7AY, UK<EMAIL_ADDRESS>
###### Abstract

We present High Sensitivity Array (HSA) and enhanced Multi Element Remotely
Linked Interferometer Network (eMERLIN) observations of the radio-loud broad-
lined type Ic supernova PTF11qcj obtained $\sim 7.5$ years after the
explosion. Previous observations of this supernova at 5.5 yrs since explosion
showed a double-peaked radio light curve accompanied by a detection in the
X–rays, but no evidence for broad H$\alpha$ spectral features. The Very Long
Baseline Interferometry (VLBI) observations presented here show that the
PTF11qcj GHz radio ejecta remains marginally resolved at the sub-
milliarcsecond level at $\approx 7.5$ yrs after the explosion, pointing toward
a non-relativistic expansion. Our VLBI observations thus favor a scenario in
which the second peak of the PTF11qcj radio light curve is related to strong
interaction of the supernova ejecta with a circumstellar medium of variable
density, rather than to the emergence of an off–axis jet. Continued VLBI
monitoring of PTF11qcj in the radio may strengthen further this conclusion.

$\star$$\star$affiliationtext: Email:

##  1\. Introduction

Supernovae (SNe) of type Ib/c are believed to mark the deaths of massive stars
that are stripped of their hydrogen (type Ib), and possibly helium (type Ic),
envelope before explosion (Filippenko, 1997). A sub-class of Ib/c SNe dubbed
broad-line (BL) Ic, estimated to constitute only $\approx 5\%$ of the Ib/c
population (Woosley & Bloom, 2006; Gal-Yam, 2017), is of particular interest
due to its relation to long-duration gamma-ray bursts (GRBs), the most
relativistic stellar explosions we know of in the universe (Piran, 2004;
Mészáros, 2006). While all GRB-associated SNe are of type BL-Ic (Woosley &
Bloom, 2006; Hjorth & Bloom, 2012, but see Cano et al. (2014) for the peculiar
case of SN 2013ez), not all BL-Ic events make a GRB (e.g., Berger et al.,
2003; Soderberg et al., 2006; Corsi et al., 2016). Thus, the question of what
physical ingredients enable some stripped-envelope massive stars to launch a
relativistic jet remains open (e.g., Modjaz et al., 2016).

As first demonstrated by the well-known case of SN 1998bw/GRB 980425 (Galama
et al., 1998; Kulkarni et al., 1998), radio observations are particularly well
suited to identify those BL-Ic SNe that may harbor GRBs (hereafter referred to
as engine-drive SNe), since radio synchrotron emission traces the fastest
moving ejecta (e.g., Berger et al., 2003). At the same time, because non-
thermal radio photons are produced in the interaction of the SN shock with the
circumstellar material (CSM), bright radio emission can also be the smoking
gun for (non-relativistic) ejecta interacting with a high-density CSM (e.g.,
Chevalier, 1998; Chevalier et al., 2004; Chevalier & Fransson, 2006). Although
strong CSM interaction is not commonly observed in BL-Ic SNe, a few cases
exists such as SN 2007bg (Salas et al., 2013), PTF11qcj (Corsi et al., 2014;
Palliyaguru et al., 2019), SN2018gep (Ho et al., 2019a) and, possibly,
AT2018cow (Rivera Sandoval et al., 2018; Smartt et al., 2018; Ho et al.,
2019b; Margutti et al., 2019).

Here, we focus on PTF11qcj, a radio-loud BL-Ic SN extensively monitored via
our approved programs on the Karl G. Jansky Very Large Array (VLA; Corsi et
al., 2014; Palliyaguru et al., 2019). The extraordinary radio luminosity of
PTF11qcj ($\sim 10^{29}\rm{erg\,s^{-1}\,Hz^{-1}}$) is reminiscent of the
GRB–associated SN 1998bw (Kulkarni et al., 1998). As discussed in Corsi et al.
(2014) and Palliyaguru et al. (2019), our radio monitoring over the first
$\approx 5.5$ years since explosion has revealed an unusual double–peaked
radio light curve. The radio emission observed during the first light curve
peak ($t\lesssim 200$ d) can be modeled within the standard synchrotron self-
absorption (SSA) model for a spherical SN shock expanding in the CSM. This
model yields an estimated speed of $\approx 0.3-0.5\,c$ for the fastest SN
ejecta (Corsi et al., 2014), placing PTF11qcj in an intermediate class between
“ordinary” BL-Ic SNe and engine-driven ones like SN 1998bw or SN 2009bb
(Soderberg et al., 2010). The simple, spherically symmetric model of SN shock
interaction with a smooth CSM (simple power-law density profile), however,
cannot explain the second radio peak. As discussed in Palliyaguru et al.
(2019), two more complex scenarios can be invoked to interpret this peculiar
behavior of PTF11qcj: (i) A spherical SN shock going through a medium with
extreme CSM density variations, perhaps related to eruptive progenitor mass
loss; (ii) A radio-emitting SN shock (first peak) followed by radio emission
from an emerging off-axis GRB jet, initially pointed away from our line of
sight (second peak).

In Palliyaguru et al. (2019) we have shown that, while modeling of our VLA
dataset within scenario (i) can indeed explain the second radio peak, the
presence of an off–axis jet (scenario (ii)) cannot be ruled out just based on
light curve measurements. However, scenarios (i) and (ii) make rather
different predictions for the angular size of the PTF11qcj ejecta at very late
times. Motivated by these considerations, here we present Very Long Baseline
Interferometry (VLBI) observations of PTF11qcj aimed at setting direct
constraints on the size (angular diameter) of its radio ejecta. These
observations ultimately provide a direct test for the presence of relativistic
expansion, as expected in the case of an off-axis GRB jet.

Our paper is organized as follows. In Section 2, we present the HSA and
eMERLIN observations of PTF11qcj. In Section 3, we discuss these observations
within the light curve and radio ejecta size predictions of the two scenarios
mentioned above. Finally, in Section 4 we summarize our results and conclude.

##  2\. Observations and data reduction

Table 1HSA results for PTF11qcj (project codes BP229A and BP229B), see Section 2.1 for discussion. | 1.66 GHz | 15.37 GHz
---|---|---
Observing date (UT) | 2018-12-08 | 2019-04-28
Project code | BP229A | BP229B
Observing time including calibrators (h) | 8 | 8
Image off-source RMS noise ($\mu$Jy/beam) | 125 | 24
CLEAN restoring beam (mas)2 | $4.06\times 2.66$ | $0.54\times 0.24$
CLEAN beam position angle (deg) | -29 | -21
Peak flux density (mJy) | $4.32\pm 0.92$ | $0.494\pm 0.054$
Integrated flux density (mJy) | $5.8\pm 1.4$ | $0.681\pm 0.085$
Right Ascension [J2000] | 13h13m41.47512s | 13h13m41.47490s
Declination [J2000] | +47∘17′56.8017′′ | +47∘17′56.7988′′
Deconvolved fitted major axis | - |  $300\pm 71\,\mu$arcsec
Deconvolved fitted minor axis | - |  $76\pm 76\,\mu$arcsec
Deconvolved fitted Pos. ang. | - | $125\pm 19^{\circ}$
Table 2eMERLIN results for PTF11qcj (project code DD8011), see Section 2.2 for discussion.  | 1.5 GHz | 5.07 GHz
---|---|---
Observing date (UT) | 2019-08-29 | 2019-08-01
Project code | DD8011 | DD8011
Observing time including calibrators (h) | 6 | 5
Image off-source RMS noise ($\mu$Jy/beam) | 42 | 37
CLEAN beam (mas2) | $59\times 19$ | $622\times 78$
CLEAN beam position angle (deg) | 40 | 42
Peak flux density (mJy) | $5.82\pm 0.87$ | $4.15\pm 0.62$
Integrated flux density (mJy) | $5.78\pm 0.87$ | $4.45\pm 0.67$
Right Ascension [J2000] | 13h13m41.4745s | 13h13m41.4746s
Declination [J2000] | +47∘17′56.795′′ | +47∘17′56.800′′

###  2.1. HSA observations

We observed the field of PTF11qcj at 1.66 GHz (project code BP229A, PI:
Palliyaguru) and 15.37 GHz (BP229B, PI: Palliyaguru) with the High Sensitivity
Array (HSA) on 2018 December 08.37 UTC and 2019 April 28.97 UTC. The HSA
included, in both bands, the Very Long Baseline Array (VLBA, USA) and the
Effelsberg 100 m antenna (Germany). At 1.66 GHz, we also used the VLA for
improved sensitivity. Both observations covered 128 MHz continuum bandwidth
and were correlated at the National Radio Astronomy Observatory (NRAO) Array
Operations Center (AOC) in Socorro (New Mexico, USA) with averaging times of 2
s and 1 s at 1.66 GHz and 15.37 GHz, respectively. In both observations, we
correlated the target data at R.A. $13^{\rm h}13^{\rm m}41.5100^{\rm s}$, Dec.
$+47^{\circ}17^{\prime}57.600^{\prime\prime}$ (J2000; Corsi et al., 2014). In
setting up our observations, we used standard phase referencing, where scans
on target are interleaved with scans on a nearby compact complex gain
calibrator with known position. At 1.66 GHz and 15.37 GHz, we used J1310+4653
and J1358+4737, respectively, as our complex gain calibrators. The VLBA
observations were correlated using the NRAO’s implementation of the DiFX
software correlator (Deller et al., 2011).

We performed all calibration and imaging procedures using the 31DEC19 release
of the Astronomical Image Processing System (AIPS; Greisen, 2003) and
ParselTongue (Kettenis et al., 2006). PTF11qcj was clearly detected at both
frequencies. At 1.66 GHz, in particular, given the relatively large separation
between PTF11qcj and the complex gain calibrator J1358+4737 ($\approx
7.6^{\circ}$) we used self-calibration to correct for the residual phase
errors towards PTF11qcj and obtain a reliable flux density measurement. The
final VLBI images are presented in Figure 1.

Figure 1.— VLBI images of PTF11qcj at 1.66 GHz (top) and 15.37 GHz (bottom).
Note the different scale on the axes.

We fitted Gaussian intensity profiles to the target images to obtain the flux
density and position of PTF11qcj at each frequency. Our results are summarized
in Table 1. The reported flux density uncertainties are the quadrature sum of
the instrumental calibration uncertainty plus the uncertainty of the Gaussian
fit. At 15.37 GHz, we adopt an instrumental uncertainty of 15%. At 1.66 GHz,
we use an uncertainty of 20% to also account for residual errors due to the
large separation between the target and the complex gain calibrator.

To estimate the source size we fit Gaussian intensity distributions to the
images. We find that the source appears marginally resolved along the major
axis of our 15.37 GHz observations, while the minor axis is consistent with an
unresolved source. The deconvolved major (minor) FWHM size of the fitted
Gaussian model is $300\pm 71\,\rm\mu arcsec$ ($76\pm 76\,\rm\mu arcsec$), with
position angle $125\pm 19$ deg (see also Table 1). This corresponds to a
diameter of $(5.2\pm 1.2)\times 10^{17}$ cm $((1.3\pm 1.3)\times 10^{17}$ cm).
Given the relatively weak radio emission from PTF11qcj, although this results
gives us an estimate of the source size, we refrain from speculation about the
non-symmetrical nature of this result. PTF11qcj radio ejecta is not resolved
at 1.66 GHz.

###  2.2. eMERLIN observations

We observed the field of PTF11qcj at 5.07 GHz and 1.51 GHz with all six
eMERLIN antennas on 2019 August 01.55 UTC and 2019 August 29.47 UTC via our
DDT project DD8011 (PI: Perez-Torres). We used J1310+4653 as our complex gain
calibrator, 1 s integration time, and 512 MHz continuum bandwidth in both
bands. We used the standard eMERLIN calibrators 3C286 and OQ208 for flux
density calibration and bandpass calibration, respectively.

We calibrated and edited the correlated data using the eMERLIN CASA pipeline
version 1.1.11 (Moldon, 2018). We applied self-calibration in both bands to
correct for significant residual phase errors and minor amplitude errors. We
used WSClean (Offringa et al., 2014) to deconvolve the calibrated data and
produce the final images. PTF11qcj is clearly detected in both bands.

We report in Table 2 the peak and total flux densities, along with the
position of PTF11qcj, at each frequency, calculated by fitting Gaussian
intensity profiles to the images. The quoted flux density uncertainties
correspond to the sum in quadrature of the systematic, i.e., instrumental
uncertainty (15%) and the image off-source RMS noise.

##  3\. Modeling

The complete radio light curves of PTF11qcj are shown in Figure 2. These
include the latest flux measurements (integrated fluxes from Tables 1 and 2)
along with data published in Corsi et al. (2016) and Palliyaguru et al.
(2019). Hereafter, we discuss the possible interpretation of these light
curves within the the standard synchrotron self-absorption (SSA) scenario for
radio SNe (see Soderberg et al., 2005, and references therein). We also
consider an alternative interpretation within an SSA (first radio light curve
peak) plus off-axis GRB (second radio light curve peak) scenario. Finally, we
discuss the direct size constraints obtained via our VLBI observations in the
context of both these scenarios.

Figure 2.— Radio light curves of PTF11qcj obtained with the VLA, HSA and eMERLIN at six different frequencies. The latest HSA and eMERLIN data are shown by triangles. The first radio peak is modeled within a standard SSA model as described in Corsi et al. (2014) (Model 0 in Table 3; solid curves). The re-brightening phase is modeled in two different scenarios: (i) within the standard SSA model (dotted curves; Model 3 in Table 3) and (ii) within an off-axis afterglow model (long–dashed, blue curves; see Section 3.2). The sum of the best fit CSM model in the first peak and off-axis jet emission from the second peak is also shown (dot–dashed curve). The dash-dot-dot-dotted line in the bottom right panel shows the effects of synchrotron cooling within the SSA scenario as discussed in Section 3.1. The measurement of the shock size from HSA data rules out the off–axis jet scenario.  Table 3Best fit parameters for the standard SSA model described in Section 3.1.  Parameter | Model 0 | Model 1 | Model 2 | Model 3
---|---|---|---|---
$r_{0}$ (cm) |  1.1$\times 10^{16}$ |  1.1$\times 10^{16}$ (fixed) |  1.1$\times 10^{16}$ (fixed) |  1.1$\times 10^{16}$ (fixed)
$\xi$ | 0.23 | 0.19 | 0.19 | 0.34
$\alpha_{r}$ | 0.79 | 0.79 (fixed) | 0.79 (fixed) | 0.61
$t_{e}$ | 55842 (fixed) | 55842 (fixed) | 55842 (fixed) | 55842 (fixed)
$s$ | 2.0 (fixed) | 1.4 | 1.1 | 0.0 (fixed)
$B_{0}$ (G) | 6.7 | 5.8 | 3.2 | 1.35
$p$ | 3.0 (fixed) | 3.0 (fixed) | 3.0 (fixed) | 3.0 (fixed)
$\alpha_{B}$ | -1.0 | -0.76 | -0.64 | -0.39
$\gamma_{m,0}$ | 7.3 | 7.8 | 10.5 | 28.1
$\alpha_{\gamma}$ | -0.4 | -0.4 | -0.4 | -0.8
$n_{e,0}\,(\rm cm^{-3})$ | $1.5\times 10^{5}$ | $1.0\times 10^{5}$ | $2.4\times 10^{4}$ | $1.6\times 10^{3}$
$\alpha_{n_{e}}$ | -1.6 | -1.1 | 1-0.88 | 0.0
$\dot{M}_{0}\,(\rm M_{\odot}\,yr^{-1})$ | $1.2\times 10^{-4}$ | $8.4\times 10^{-5}$ | $1.9\times 10^{-5}$ | $6.4\times 10^{-6}$
$\alpha_{\dot{M}}$ | 0.0 | 0.48 | 0.71 | 1.2
$\chi^{2}$/dof | 1793/90 | 1288/39 | 602/38 | 211/38

###  3.1. Light curves within the SSA scenario

Similarly to what done in Corsi et al. (2016) and Palliyaguru et al. (2019),
we can model the radio emission of PTF11qcj in the SSA scenario (Soderberg et
al., 2005). As described in Soderberg et al. (2005), the temporal evolution of
the shock radius, $r$, minimum Lorentz factor, $\gamma_{m}$, and magnetic
field, $B$, are parameterized as:

$\displaystyle r=r_{0}\left(\frac{t-t_{e}}{t_{0}}\right)^{\alpha_{r}},$ (1)
$\displaystyle B=B_{0}\left(\frac{t-t_{e}}{t_{0}}\right)^{\alpha_{B}},$ (2)
$\displaystyle\gamma_{m}=\gamma_{m,0}\left(\frac{t-t_{e}}{t_{0}}\right)^{\alpha_{\gamma}},$
(3)

with $\alpha_{r}$, $\alpha_{B}$ and $\alpha_{\gamma}$ the temporal indices of
the three quantities respectively, $t_{e}$ the explosion time, and $t_{0}$ an
arbitrary reference epoch since explosion, here set to $10$ d. Within the
standard assumptions, one has (see Equations (9)-(10) in Soderberg et al.,
2005):

$\displaystyle\alpha_{\gamma}=2(\alpha_{r}-1),$ (4)
$\displaystyle\alpha_{B}=\frac{(2-s)}{2}\alpha_{r}-1.$ (5)

Here, $s$ describes the density profile of the shocked CSM where the density
of the radiating electrons within the shocked CSM is given by (Chevalier,
1982)

$n_{e}=n_{e,0}\left(\frac{t-t_{e}}{t_{0}}\right)^{\alpha_{n_{e,0}}}\propto
r^{-s}.$ (6)

The above relations follow from assuming that the energy density of shocked
particles (protons and electrons) and amplified magnetic fields are a constant
fraction ($\epsilon_{e}\approx\epsilon_{B}\approx 0.33$ under the hypothesis
of equipartition) of the post-shock energy density $U\propto n_{e}v^{2}$,
where:

$v=v_{0}\left(\frac{t-t_{e}}{t_{0}}\right)^{\alpha_{r}-1},$ (7)

is the shock speed. The density of electrons in the shocked CSM is related to
the progenitor mass-loss rate via the relation (see Equation (13) in Soderberg
et al., 2005):

$\dot{M}=\frac{8\pi}{\eta}n_{e,0}m_{p}r^{2}_{0}v_{w}\left(\frac{t-t_{e}}{t_{0}}\right)^{\alpha_{r}(2-s)}=\dot{M}_{0}\left(\frac{t-t_{e}}{t_{0}}\right)^{\alpha_{\dot{M}}},$
(8)

where we have assumed a nucleon-to-proton ratio of 2, $v_{w}\sim 1000$ km/s is
the velocity of the stellar wind, and where $\eta$ (typically in the range
$\eta\sim 2-10$) characterizes the thickness of the radiating electron shell
as $r/\eta$.

In the GHz radio band, the observing frequencies $\nu$ are typically such that
$\nu_{m}<<\nu$, where (see Equation (A6) in Soderberg et al., 2005):

$\nu_{m}=\gamma^{2}_{m}\left(\frac{eB}{2\pi m_{e}c}\right)$ (9)

is the characteristic synchrotron frequency of electrons with Lorentz factor
$\gamma_{m}$. In the above Equation, $m_{e}$ is the electron mass, and $c$ is
the speed of light. In this frequency range, assuming the synchrotron cooling
frequency is higher than the observing frequency, and neglecting synchrotron
cooling effects, the SSA emission from the shocked electrons reads:

$f_{\nu}(t)={\cal
F}\left(\frac{t-t_{e}}{t_{0}}\right)^{(4\alpha_{r}-\alpha_{B})/2}(1-e^{-\tau^{\xi}_{\nu}(t)})^{1/\xi}\nu^{5/2},$
(10)

where ${\cal F}$ is a normalization constant that depends on the parameters
$(r_{0},B_{0},p)$ with $p$ the power-law index of the electron energy
distribution (see Equations (A11)111The functions defined in (A11) are not
time–dependent and therefore included in the normalization constant and (A13)
in Soderberg et al., 2005), and where the optical depth $\tau$ is given by
(see Equations (20) and (A14) in Chevalier & Fransson, 2003; Soderberg et al.,
2005, respectively):

$\tau(t)={\cal
T}\left(\frac{t-t_{e}}{t_{0}}\right)^{(p-2)\alpha_{\gamma}+(3+p/2)\alpha_{B}+\alpha_{r}}\,\nu^{-(p+4)/2},$
(11)

with ${\cal T}$ a normalization constant that depends on the parameters
$(r_{0},B_{0},p,\gamma_{m,0},\eta)$. We note that for $\nu_{m}<<\nu$, the
value of $\gamma_{m,0}$ is left largely unconstrained by the observations, and
thus typically fixed so that $\nu_{m,0}\sim 1$ GHz. In addition, the thickness
of the shell is typically set to a value $\eta>1$ (Li & Chevalier, 1999).
Thus, for a given choice of $\eta$ and $\nu_{m}$, the SSA model is a function
of the parameters ($r_{0},\xi,\alpha_{r},t_{e},s,B_{0},p)$.

Within the SSA scenario, and with the data collected here, we can further test
the hypothesis first presented in Palliyaguru et al. (2019) that the double-
peaked radio light curve of PTF11qcj is due to strong interaction with CSM of
variable density. Our results are shown in Figure 2, and are reported in Table
3. Model 0 in Table 3 is the best fit SSA model for the first peak ($t\lesssim
215$ days since explosion) as reported in Corsi et al. (2016). The last is
obtained by setting $t_{e}=55842$, $p=3$ (as typically expected for Type Ib/c
SNe; see Chevalier & Fransson, 2006), $\eta=10$, $\nu_{m,0}=1$ GHz, and
varying $r_{0}$, $\xi$, $\alpha_{r}$, $s$, and $B_{0}$. Model 0 is also
plotted in Figure 2 (solid line). Model 1 is a fit to the second radio peak
($t\gtrsim 587$ days since explosion) where we keep $r_{0}$ and $\alpha_{r}$
fixed to their best fit values for the first peak so as to ensure a smooth
radial evolution between the first and second radio light curve peaks, and
allow $\xi$, $s$, and $B_{0}$ to vary. This fit is similar to the one reported
in Palliyaguru et al. (2019) but updated to include the eMERLIN and HSA data
presented here.

Compared to Palliyaguru et al. (2019), the reduced $\chi^{2}$ for Model 1 is
substantially higher, indicating a worsening of the goodness of fit. Model 2
is a fit with a model identical to Model 1 but where the 15 GHz HSA data has
been excluded. The improved $\chi^{2}$ value for Model 2 compared to Model 1
indicates a significant discrepancy between data and model at the highest
radio frequencies, suggesting a steepening in the highest frequency light
curve which may be caused by the passage of the cooling frequency in band.
Indeed, the effects of synchrotron cooling may become important at the late
timescales considered here. Within the SSA scenario, the synchrotron cooling
frequency can be written as (see Equation (A16) in Soderberg et al., 2005):

$\nu_{c}=\frac{18\pi\,m_{e}\,c\,e}{(t-t_{e})^{2}\sigma^{2}_{T}\,B^{3}},$ (12)

where $\sigma_{T}$ is the Thomson cross-section and $e$ is the electron charge
(Rybicki & Lightman, 1986; Soderberg et al., 2005). For $\nu>\nu_{c}$, the
flux density becomes (Soderberg et al., 2005):

$f_{\nu}(t)\propto\nu^{-p/2}\,t^{(6\alpha_{r}+(8-5p)\alpha_{B}+2(p-2)\alpha_{\gamma}-4p+2)/2}.$
(13)

Model 1 in Table 3 predicts $\nu_{c}\approx 50\,\rm GHz$ at $t-t_{e}\approx
7.5$ yrs, which is above the highest frequency of our observations, resulting
in large residuals with the HSA data point. Thus, in Table 3 we also report
the results of a fit where we set $t_{e}=55842$, $p=3$, $s=0$, $\eta=2$,
$\nu_{m,0}=3$ GHz, $r_{0}=1.1\times 10^{16}$ cm, but allow $\alpha_{r}$,
$B_{0}$, and $\xi$ to vary (Model 3). As shown in Figure 3, with this choice
the cooling frequency falls below 16 GHz at $t-t_{e}\approx 1345$ d ($\approx
3.6$ yrs since explosion), thus causing a steeping of the light curve at this
frequency. In Figure 2 we plot Model 3 with dotted lines, and the expected
steeping ($f_{\nu}(t)\propto t^{-2.6}$ according to Equation (13)) of the 16
GHz light curve due to synchrotron cooling with a dash-dot-dot-dotted line.
Model 3 substantially improves the goodness of fit for the second radio peak
compared to Models 1 and 2.

In summary, while the SSA fits described in this Section have large
limitations due to the simplifications that characterize the SSA model,
overall they are indicative of the fact that (i) a non-constant CSM profile is
needed to explain the PTF11qcj radio light curves, and (ii) synchrotron
cooling may be playing a role at late times. We also note that in Model 3 a
flat ($s=0$) radial profile of the environment (ISM-like) is favored (see
Chevalier, 1982, for a discussion of the $s=0$ case). Finally, as shown in
Figure 4, the combined radial evolution of the shock as implied by Model 0 for
the first peak of the PTF11qcj light curves and by Model 3 for the second
peak, implies that the shock has decelerated while encountering the CSM
discontinuity. A similar case is of SN2014C, where VLBI observations revealed
that the SN has substantially decelerated at late times after encountering a
higher density shell (Bietenholz et al., 2020).

Figure 3.— Cooling frequency vs time during phase 2. The horizontal dotted
lines mark the observation frequencies of 2.5 GHz, 13.5 GHz and 16 GHz. For
$\alpha_{r}=0.61$, $\nu_{m,0}=3\,\rm GHz$ and $\eta=2$ (Model 3), $\nu_{c}$
(solid line) crosses $16\,\rm GHz$ at $\sim 1345$ days since explosion.
Figure 4.—  Radius vs time during phases 1 and 2. During phase 2, the radial
evolution is modeled using Model 3 in table 3. The date of the 16 GHz VLBI
data is marked by the vertical dashed line. Our modeling does not include the
time while encountering the CSM discontinuity and shows that the shock has
decelerated during phase 2.

###  3.2. SSA plus off-axis GRB scenario

Similarly to what done in Palliyaguru et al. (2019), we also use numerical
simulations for off-axis GRB jets by van Eerten et al. (2012) to model the
second peak of PTF11qcj light curve within a constant density environment,
considering the new HSA and eMERLIN data points. Within this scenario, the
first radio peak is the radio emission from the SN ejecta while the second
radio peak is due to the delayed emission from an initially off-axis jets that
enters our line of sight after spreading and decelerating. These two-
dimensional hydrodynamic simulations of the GRB jets take into account the
Blandford-Mckee solution (Blandford & McKee, 1976) in the relativistic regime,
the Sedov-von Neumann–Taylor (SNT) solution (Taylor, 1950) in the late non-
relativistic regime, and a transition in between regimes (Zhang & MacFadyen,
2009).

In Table 4 we report the best fit results for the isotropic equivalent kinetic
energy of the explosion $E_{\rm iso}$, the jet half-opening angle,
$\theta_{0}$, ISM density $n_{\rm ISM}$, and the observer angle, $\theta_{\rm
obs}$, when the latest four data points from HSA and eMERLIN, as well as the
Chandra X-ray data point reported in Palliyaguru et al. (2019), are added to
the fit. These fits also assume equipartition of energy between particles and
magnetic fields such that $\epsilon_{e}=\epsilon_{B}=0.33$. We set the
electron energy index, $p=2.5$, which is typical for GRB afterglows, as
expected from theoretical considerations (Kirk et al., 2000; Achterberg et
al., 2001). The best fit off-axis jet light curves are shown in Figure 2
(light blue, long-dashed lines).

Table 4Best fit results in the off-axis GRB scenario. The radius ($R_{\perp}$) and the corresponding angular diameter from modeling, and the expected radius from Equation (15) ($R_{\perp,exp}$) and the corresponding angular diameter are also listed.  Model | $E_{\rm iso}$ | $n$ | $\theta_{0}$ | $\theta_{\rm obs}$ | $p$ | $\chi^{2}$ | $R_{\perp}$ | angular size | $R_{\perp,exp}$ | expected angular size
---|---|---|---|---|---|---|---|---|---|---
| (erg) |  ($\rm cm^{-3}$) | (rad) | (rad) |  |  | (cm) | (mas) | (cm) | (mas)
Model 1 | $1\times 10^{53}$ | $1\times 10^{-5}$ | 0.2 | 0.4 | 2.5 | $1285/41\approx 31$ | $3.0\times 10^{19}$ | 35 | $4.8\times 10^{19}$ | 55

###  3.3. Size constraints: CSM-interacting vs off-axis GRB scenario

Within the SSA scenario, Models 0, 1, and 2 described in Section 3.1 all imply
a shock radius around the time of the 16 GHz HSA observation (2759 days post
explosion) of $r\sim 10^{18}$ cm. This in turn corresponds to an angular
diameter of $\sim 1$ mas at the redshift $z=0.028$ of PTF11qcj, larger than
the size constraints set by our HSA observations (see Table 1). On the other
hand, Model 3 gives $r\approx 3\times 10^{17}$ cm at 2759 days post explosion
(see Figure 4), which corresponds to an angular diameter of $\approx 0.4$ mas,
compatible with the HSA observations reported in Table 1. We note however that
within the SSA model, which assumes spherical symmetry, we are not able to
model any potential asymmetry in the shock geometry.

Within the off-axis GRB scenario for the second radio peak, the ejecta will no
longer produce a symmetric image on the sky. Instead, the resolved image will
highlight the front edge of the jet heading to the observer, producing an
elongated and curved shape. For model 1 in Table 4, this curved front will
have traveled $R_{\perp,travel}=3.0\times 10^{19}$ cm in projected distance
from the origin of the explosion. The projected width of the image is
$R_{\perp,w}=1.4\times 10^{19}$ cm, and its projected height
$R_{\perp,h}=2.5\times 10^{18}$ cm. These correspond to a projected angular
distance from the origin of the explosion of 35 mas, a projected angular width
of 8 mas, and a projected angular height of 14 mas. Thus, if the second radio
light curve peak of PTF11qcj was due to an off-axis GRB, we should have
resolved a much larger image in our HSA observations.

We note that while the off–axis GRB model considered here assumes a top-hat
jet, a structured jet such as the one considered in Nakar & Piran (2017) would
imply sizes that can be bracketed by the the two extreme cases of a
spherically symmetric blast wave, and a non-spreading relativistic cone.
Indeed, for a spherically symmetric Sedov-Taylor blast wave, the analytical
expression for the shock radius at a late–time $t$ is given by (van Eerten,
2018):

$\displaystyle R=C_{R}(k)\left(\frac{E_{j}}{10^{51}\rm
erg}\right)^{\frac{1}{5-k}}\left(\frac{\rho_{ref}}{m_{p}}\right)^{\frac{-1}{5-k}}\times$
$\displaystyle\left(\frac{R_{ref}}{c\times
10^{6}}\right)^{\frac{-k}{5-k}}\left(\frac{t}{10^{8}}\right)^{\frac{2}{5-k}}$
(14)

where $C_{R}(k)=1.5$ parsec for the power law index of the CSM density profile
$k=1.4$ (Model 1, Table 3), $E_{j}$ is the total energy in the ejecta,
$\rho_{\rm ref}$ and $R_{\rm ref}$ are the circumburst medium density and
radius at a reference time (which we set to 10 days), respectively (van
Eerten, 2018). For the best fit parameters of Model 1 in Table 3, we find the
expected radius would be $8.9\times 10^{17}\rm cm\,(1\,\rm mas)$, larger than
our HSA constraint.

For a relativistic jet expanding into the ISM, the apparent radius at time $t$
may be calculated using the analytical expression (Oren et al., 2004):

$R_{\perp,exp}=5\times
10^{16}\left(\frac{E_{51}}{n}\right)^{1/6}T_{j}^{-1/8}t^{5/8},$ (15)

where $T_{j}=(E_{51}/n)^{1/3}(\theta_{0}/0.1)^{2}(1+z)$ is the jet break time,
$\theta_{0}$ is the jet half opening angle, $n$ is the ISM number density
(Oren et al., 2004). For the best fit parameters of $E_{\rm iso}$,
$\theta_{0}$, $n_{\rm ISM}$ listed in Table 4, the expected radius at $2759$
days post explosion is $4.8\times 10^{19}$ cm. This corresponds to an angular
size of $\sim 55$ mas at the redshift $z=0.028$ of PTF11qcj, much larger than
our HSA constraint.

Based on these results, the off-axis hypothesis for the origin of the second
radio peak of the PTF11qcj light curves is disfavored.

##  4\. Summary and Conclusion

We have presented HSA and eMERLIN observations of PTF11qcj obtained $\sim$7.5
yrs post explosion. The source is marginally resolved in the HSA data at 15
GHz, indicating a diameter of $(300\pm 71)\,\rm\mu as$. This corresponds to an
average expansion velocity of $\approx 0.036\pm 0.008\,c$.

Modeling of the light curves within an SSA scenario requires the interaction
of the shock with a non-smooth CSM whose radial profile changes from a stellar
wind profile ($n_{e}\propto r^{-2}$) to a constant density medium. This
variable CSM profile also affects the temporal evolution of the shock radius.
We also find that synchrotron cooling may be playing a role at the highest
radio frequencies. Within this SSA model with variable CSM, the derived size
of the shock at the time of our HSA observations can be reconciled with our
measurements. Our results are thus consistent with the CSM interaction model
(considering that such model is highly simplified). The size constraints
derived from our HSA observations also seem to disfavor an off-axis
relativistic jet scenario for the radio re-brightening of PTF11qcj.

In conclusion, the VLBI observations reported in this paper favor the original
interpretation of PTF11qcj as a strongly CSM-interacting radio SN (see Corsi
et al., 2014; Palliyaguru et al., 2019). However, our conclusions are limited
by the simplifications inherent in the spherically symmetric SSA model adopted
here. We encourage more detailed theoretical modeling aimed at interpreting
PTF11qcj complex radio light curve and available VLBI data.

A.C. acknowledges support from the National Science Foundation CAREER Award
#1455090, and from the Chandra GO Award #GO7-18065X. N.T.P acknowledges
support from NSF NANOGrav Physics Frontier Center (NSF Grant No. PFC-1430284)
and start-up funds to J.D. Romano from TTU. M.P.-T. acknowledges financial
support from the State Agency for Research of the Spanish Ministry of Science,
University, and Education (MCIU) through the ”Center of Excellence Severo
Ochoa” award for the Instituto de Astrofísica de Andalucía (SEV-2017-0709) and
through grants AYA2015-63939-C2-1-P and PGC2018-098915-B-C21. The National
Radio Astronomy Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc. e-MERLIN
is a National Facility operated by the University of Manchester at Jodrell
Bank Observatory on behalf of STFC. This work made use of the Swinburne
University of Technology software correlator, developed as part of the
Australian Major National Research Facilities Programme and operated under
licence.

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