Spreading Out-of-Distribution Detection on Graphs
Authors: Daeho Um
Jongin Lim
Sunoh Kim
Yuneil Yeo
Yoonho Jung
Venue: ICLR 2025 Poster (2025)
Keywords: out-of-distribution detection
graph neural networks
benchmark
healthcare
Primary area: learning on graphs and other geometries & topologies
TL;DR: For node-level OOD detection, we present a realistic benchmark setup with a new dataset and propose a novel aggregation scheme.

Abstract: Node-level out-of-distribution (OOD) detection on graphs has received significant attention from the machine learning community. However, previous approaches are evaluated using unrealistic benchmarks that consider only randomly selected OOD nodes, failing to reflect the interactions among nodes. In this paper, we introduce a new challenging task to model the interactions of OOD nodes in a graph, termed spreading OOD detection, where a newly emerged OOD node spreads its property to neighboring nodes. We curate realistic benchmarks by employing the epidemic spreading models that simulate the spreading of OOD nodes on the graph. We also showcase a ``Spreading COVID-19" dataset to demonstrate the applicability of spreading OOD detection in real-world scenarios. Furthermore, to effectively detect spreading OOD samples under the proposed benchmark setup, we present a new approach called energy distribution-based detector (EDBD), which includes a novel energy-aggregation scheme. EDBD is designed to mitigate undesired mixing of OOD scores between in-distribution (ID) and OOD nodes. Our extensive experimental results demonstrate the superiority of our approach over state-of-the-art methods in both spreading OOD detection and conventional node-level OOD detection tasks across seven benchmark datasets. The source code is available at https://github.com/daehoum1/edbd.

Published as a conference paper at ICLR 2025
SPREADING OUT-OF-DISTRIBUTION DETECTION ON GRAPHS
Daeho Um'
Jongin Lim
AT Center, Samsung Flectronics daeho. umu samsung. com
Al Center, Samsung Electronics jonny .1 im@samunn com
Sunoh Kim
Yuneil Yeo
Computer Engineering
Department of Civil and Environmental Engineering
Dankook University
UC Berkeley yunei y
suno838 dedakook.ac.
Yoonho Jung
Department of Electrical and Computer Engineering Scoul National University
jungyh1 98 snu. ac kr
ABSTRACT
Node-level nut-of-distribution (OOD) detection on graphs has received significant attention from the machine learning community. However, previous approaches are evaluated using unrealistic benchmarks that consider only randomly selected OOD nodes, failing to reflect the interactions among nodes. In this paper, we introduce a new challenging task model the interactions of OOD nodes in a graph, termed spreading OOD detection, where newly emerged OOD node spreads its property to neighboring nodes. We curate realistic benchmarks by employing the epidemic spreading models that simulate the spreading of OOD nodes on the graph. We also showcase a "Spreading COVID-19" dataset to demonstrate the applicability of spreading OOD detection in real- weal-world scenarios. Furthermore, to effectively detect spreading OOD samples under the proposed benchmark setup, we present a approach called energy distribution-basee detector (EDBD), which includes scheme. EDBD is designed to mitigate undesired
a novel energy-aggregatinn mixing of OOD scores between in-distribution (ID) and OOD nodes. Our extensive experimental results demonstrate the superiority of our approach over state-of-the art methods in both spreading OOD detection and conventional node-level OOD detection tasks across seven benchmark datasets. The source code is available at https:/ / / thub. com/ daeh /edbd.
INTRODUCTION
While neural networks have demonstrated surpassing performance in various machine learning tasks (He et al., 2016; Kipf & Welling, 2016), they often yield overconfident outputs for unseen data during training, leading crucial problems in safety-critical applications. To address this issue, con- siderable research has been conducted detecting new samples in -itoo--distribution (OOD) (Yang et al., 2021). However, the majority of current OOD approaches assume i.i... (independently and identically distributed) samples (Hendrycks & Gimpel, 2016; Lee et al., 2018; Liu et al., 2020), and research taking into account inter-dependency among samples has been under-explored.
Recently, several graph-based OOD detection methods (Zhao al., 2020b; Stadler et al., 2021: Wu et al., 2023; Song & Wang, 2022) have emerged to exploit inter-dependency among samples. These methods leverage the graph structure, where a node represents an individual sample and an edge models the connection between on the given graph. Despite their potential, existing graph-based OOD methods exhibit a critical limitation. Since there is no graph benchmark dataset designed for node-level OOD detection, these methods are commonly evaluated on synthetic graph datasets where OOD samples are assigned to
nodes. Then, they perform node-level OOD detection tasks
"Corresponding author

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Published as a conference paper at ICLR 2025
(Susceptible) OOD (Infected)
scod node
Go
g
g"
1=0
t=
t=2
1= 7
Figure I: Simulation of the spreading of OOD using an epidemic spreading model. Each infected OOD node attempts to infect susceptible neighbor with a specific probability. A blue represents successful infection case along an edge while red x signifies a failure case. Note that the spread can be initiated from multiple seed nodes in practice. The objective of spreading OOD detection is to discriminate OOD nodes well in every E TT..
randomly selected nodes.' The previous evaluation scenarios, characterized by the random node selection, fail to reflect interactions with the various nodes associated with OOD samples.
In real-world scenarios, newly introduced samples (i.e., OOD samples) typically engage in interactions with other connected nodes, spreading out through edges in a graph. For example, if a person is suddenly infected with virus, the virus spreads out to the person's neighboring individuals via interaction or contact. Computer viruses (Kephart & White, 1993), brand-new products (Amini & Li, 2011), and contaminants (Mei & Gong, 2018) are other examples of OOD sample spreading. Consequently, for enhanced applicability in real-world scenarios, it is imperative to incorporate a new benchmark setup for graph-based OOD detection that encompasses the spreading of OOD samples.
In this paper, we formulate new benchmark setup for graph-based OOD methods termed spreading OOD detection. Figure 1 illustrates the scenario of the proposed spreading OOD detection. First, we randomly select nodes to locate OOD samples on a graph with in-distribution (ID) nodes, as in previous graph-based OOD methods (Zhao et al., 2020b; Stadler et al., 2021; Song & Wang, 2022; Wu et al., 2023). These selected nodes serve as sources (e.g., the first infected person) for the spread of OOD. We model the spreading of OOD samples by graph diffusion process where new sample in OOD spreads from the source nodes to its neighboring nodes. Specifically, we utilize the epidemic spreading models (Allen, 1994; Peng et al., 2013) that treat spreading process Markov chain. Then, the goal of spreading OOD detection is to discriminate OOD nodes well across all time stamps on the Markov chain, without prior information about the spreading process. The proposed benchmark setup enables the evaluation of graph-based OOD methods on various graph datasets under realistic settings. To show the applicability of spreading OOD detection in real-world situations, we further present Spreading COVID-19 19, a new benchmark dataset simulating the spread of COVID-19.
Furthermore, we propose Energy Distribution-Bat Based Detector (EDBD), a novel graph-based OOD method that utilizes an energy (Liu et al., 2020) as an OOD score. Unlike the previous graph-based based OOD method (Wu et al., 2023) that updates initial energies via simple neighborhood aggregation solely based on the graph structure, we allow energies to directly control their aggregation process. Specifically, we exploit the initial energies as a temporary OOD indicator, and regulate aggregation on each node alleviate the mixing of energies between OOD nodes and ID nodes. This deliberate aggregation of EDBD enhances accurate OOD discrimination, ensuring that the influence of OOD nodes on neighboring ID nodes is appropriately tempered. Through extensive experiments, demonstrate the effectiveness of EDBD not only in spreading OOD detection but also the existing OOD detection settings.
Our key contributions summarized as follows:
We formulate spreading OOD detection, detection that incorporates interactions among OOD samples. We further establish new
benchmark setup for node-level OOD
Nodes belonging to randomly chosen classes (Zhao et al., 2020b; Stadler et al., 2021; Wu et 2023; Song Wang, 2022) or randomly chosen nodes (Stadler et al., 2021) designated as OOD.
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dataset called Spreading COVID-19 to benchmark the spreading OOD detection scenario in real-world data. We believe that our new benchmark setup and dataset offer a robust basis for comparing node-level OOD detection methods, considering their practical applicability to real-world scenarios.
We propose EDBD, a novel node-level OOD detector, which employs an attentive energy- aggregation scheme that prevents the mixing of energies between ID nodes and OOD nodes. Our EDBD framework demonstrates superior performance over existing state-of-the-ar methods in both spreading OOD detection and conventional OOD detection tasks on graphs.
2 RELATED WORK
OOD Detection. OOD detection and OOD generalization This paper focuses on OOD detection (Yang et al., 2021; Yoon et al., 2024; Um et al., 2024), where the goal is to identify unfamiliar samples not drawn from in-distribution (ID), which is the distribution of training data. A popular line of OOD detection research designs scoring functions, including maximum softmax probability (Hendrycks & Gimpel, 2016), ODIN score (Liang et al., 2018), Mahalanobis distance-basee score (Lee et al., 2018), and energy- based score (Liu et al., 2020). Besides, various approaches, such as outlier exposure introducing auxiliary OOD dataset (Hendrycks et al., 2019; Liu et al., 2020), have been proposed to tackle OOD detection. Meanwhile, OOD generalization (Shen et al., 2021) aims to develop machine learning model to perform well on OOD data. However, many OOD detection studies assume that inputs are i.i.d., thus predictions are based on a single input.
major problems related to OOD.
OOD Detection on Graphs. In contrast, graph-structured data contain dependencies between nodes, which are represented through the graph structure. Therefore, several methods have been developed to consider the graph structure in detecting OOD graphs (Li al., 2022; Liu et al., 2023). However, while these methods target graph-level OOD detection tasks, we focus on node-level OOD detection tasks. To address node-level OOD detection tasks on graphs, various techniques have been developed, such as Graph-based based Kernel Dirichlet distribution Estimation (GKDE) (Zhao et al., 2020b), Graph Posterior Network (GPN) (Stadler et al., 2021), and OOD Graph Attention Network (OODGAT) (Song& & Wang, 2022). Recently, GNNSAFE (Wu al., 2023) shows effectiveness in node-level OOD detection, which produces final OOD score for node by aggregating the initial OOD scores of neighboring nodes. However, in GNNSAFE, a node aggregates OOD scores from neighboring nodes, assigning equal importance to each, regardless of the OOD score distribution.
Machine Learning for COVID-19. The COVID-19 19 pandemic has driven the machine learning community to develop solutions and proactively prepare for future epidemics. In response, many datasets related to COVID-19 have been released, including those with medical images, textual data, and speech data (Zhao et al., 2020a; Shuja et al., 2021). Additionally, various studies have focused on modeling the spread of COVID-19 (Panagopoulos et al., 2021; Alguliyev et al., 2021) and applying machine learning techniques for its classification (Barstugan er al., 2020; Song et al., 2023). Nonetheless, our work stands apart due to our novel approach, which does not rely on specific information or features related to COVID-19 19 during training. This aspect makes our approach adaptable to new epidemics where initial information may be unavailable.
WHAT IS "SPREADING OOD DETECTION"
In this section, we formulate the problem of spreading OOD detection and describe the procedure of the spreading OOD benchmark setup in detail.
3.1 NOTATION
Let G = (V,E,X) be graph where V = {eI.... UN represents set of N nodes, & represents a set of edges, and X E NxF represents a feature matrix of F-dimensional node features. A E (0, N>N denotes adjacency matrix, where A., = 0 only if o; and vy are not connected each other. x; denotes the node feature of v, i.e., i-th row vector of x. (ox)) denotes the set of neighbors of vz.
3.2 PROBLEM STATEMENT
Figure 4 illustrates overview of the proposed spreading OOD detection. Formally, let D,n and D Dont represent in-distribution (ID) and out-of-distribute (OOD) data, respectively. For D Dins each sample in Din is associated with feature vector Xin. E R. and class label you. (1,..., C).
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For Dout. is associated only with feature vector Nout E RF, and Xout and Xin sampled from distinct distributions. Following the conventional OOD benchmarks (Hendrycks & Gimpel, 2016), we split Pin into Dirain and Disst, and only Dirain is used for training. During training, train a neural classifier with features and labels in Difain, without a graph structure, to account for real- world spreading scenarios where classification often occurs at the sample-level (Perdisci et al., 2008; Kannan et al., 2011; Ward et al., 2020).
in
For evaluation, we assume a graph problem. Specifically, we define 9 to be a graph consisting of N ID samples in Diaet as nodes of the graph. The graph structure G represents deterdeeeyndncy among the nodes such as social relations among people. At = O, we randomly select nodes in the given graph G and set them as OOD nodes. These selected nodes serve as sources (e.g., the first infected person) for the spread of OOD. After the initialization, each ID node is probabilistically infected by neighboring OOD nodes for each time stamp 2 1, spreading OOD samples through the graph as increases." 2 When an ID node becomes an OOD node, its node feature is replaced by that of an OOD sample randomly chosen in Dout- Concretely, we denote the resulting graph at each time stamp & E {0, "0,,,,..777 TT} as g' = (V,E, where T is the end point of the spread and X' denotes a feature matrix containing OOD features for infected nodes at f.
EV, X) is given, and formulate a graph-based OOD
The goal of spreading OOD detection is to discriminate OOD samples (infected nodes) from ID samples (uninfected nodes) when G is given. Note that during evaluation, only G is provided without any additional information such as the position of the seed node or the current time stamp I. Thus, spreading OOD detection requires an OOD detector that performs well at any time stamp t. This benchmark setup does not necessarily require dedicated dataset; it can be easily implemented using existing graph datasets (see Appendix B.2.1). Spreading OOD detection provides realistic setup for node-level OOD detection, which lack standard evaluation setup except for random class split.
3.3 OOD SPREADING SCHEME
Let M E {0, 1) [0,11Nx(T++0 [x(++1) be a binary mask representing which nodes are OOD at where M.. represents that v is OOD at t. Note that some nodes are randomly designated as OOD nodes at = 0, and then the OOD samples spread along edges as I increases. To model the spread of OOD samples, (SI) and Sereppibbl--nffecte-Suseeeeeee Peng et al., 2013), which are widely used in epidemic-sprading studies. In this section, we primarily introduce the procedure using SIS model, since SI model can be considered as a special case of SIS model.
we employ Susceptible-Infect
(SIS) models (Allen, 1994;
SIS model can be formulated as a Markov process. Specifically, at each time stamp f, every node in g' can be in one of two states: susceptible or infected. During each time interval between consecutive time stamps, an infected node attempts to infect its susceptible neighbor with probability of y- Simultaneously, an infected node may be cured with probability of ô. The probability that o; will not be infected by its neighbors during the interval [l - 1, 4] is denoted as Gi.-- % will remain uninfected at if all of 2 's neighbors were either uninfected at - 1 or, if they were infected, they failed to infect v, during [t- - 1, t]. Thus, considering multiple neighbors of U; simultaneously, Si, can be expressed as follows:
G1,1
II (M, -7) + (1 M M)))) JEN(()
= II (1-7M) (1-7M,e-1) JEN())
is is uninfected at if it was uninfected at - and did not receive infections from its neighbors at i, or was infected at f - but was cured at t. That is, the probability that v; is uninfected can be expressed by (1 - M,,-- + + Hence, we determine the OOD configuration at t depending on that at t 1 as follows:
(2) for i E (I....,N) Nj. When a node infected, the original features of the node is replaced with features sampled from Dout- As t increases by 1, we iteratively spread OOD using Equation (2). At each time stamp f, as features change by infection, g' = (V, V is generated.
M Ber(1
M,, + &&,,))
Details of the spreading scheme (i.e. how infect node?) is descrihed in See. 3.3.
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Uniformly aggregating energies from neighbors
Based solely on the graph structure
GNNSAFF (Wu al. (2023))
node
Dissimilar
OOD
initial energy (B
Edge-level control
average
Similar
EDBA
(ours)
Node-level control
Energy consistency J
Total degree of aggregation from neighbors
Figure 2:
The concept of energy distribution-baan energy mixing between ID and OOD nodes. EDBA allows the energies themselves to control the entire aggregation process.
aggregation (EDBA) that prevents undesired
shown in the three graphs below, high energy similarity between two ID nodes (or OOD nodes) strengthens energy propagation at edge level, whereas low energy similarity between ID and OOD nodes weakens energy propagation. The energy histograms the bottom show energy variance among nodes, representing the energy distribution consistency among nodes. Moving from left to right, the histogram exhibits larger variance and subsequently lower consistency, resulting in decrease in the total degree of aggregation from neighboring nodes.
SI models are SIS models with 0, assuming that once node is infected, it cannot be cured. Thus, the spread in an SI model is also simulated through Eq by substituting zero for Figure illustrates the simulation process using model.
3.4 SPREADING COVID-19
We create Spreading COVID-19 dataset to demonstrate the applicability of spreading OOD detection real-world problems. Spreading COVID-19 simulates the propagation of COVID-19 on a human network. This dataset enables the detection of individuals infected with COVID-19 by identifying OOD nodes on the graph. In this dataset, ID data consists of four classes including normal (no illness), allergies, cold, and flu. An OOD class is COVID-19, which disease that has suddenly emerged and significantly impacted the world. For each class, we generate samples with 23-dimensional features, created based on information about symptoms of respiratory illnesses (AAFA; Clinic), such as chest tightness and rapid breathing. We utilize graph structure from the LastFM Asia graph (Rozemberezki Sarkar, 2020). We use SI and SIS models simulate the spread of COVID-19 and create episodes for both models, where episode represents {g' ]eeo, Figure in Appendix A illustrates the process of spreading OOD detection on mini version of the Spreading COVID-19 dataset under the single-seed setting. Further details on the Spreading COVID-19 dataset, including justification regarding dataset construction, are included in Appendix A, and we provide the entire dataset in the supplementary material.
4 PROPOSED METHOD
In this section, describe our method called Energy Distribution- BBasdd Detector (EDBD) for spreading OOD detection. We first briefly overview the proposed EDBD in Sec 4.1. We then present an aggregation scheme EDBD and two main components of the aggregation scheme Sec. 4.2, Sec. 4.3, and Sec. 4.4, respectively.
4.1 OVERVIEW
The key challenge in method is how design energy-aggregaie scheme enhance the discriminative ability between ID nodes and OOD nodes. We utilize an energy (LeCun et al., 2006) OOD score and aim refine energies through novel aggregation scheme. To obtain these energies, we first train a neural classifier on Diroin with cross-entropy loss. After training the neural classifier, we define an energy function of the neural classifier f(x) R² CC, which maps an F- dimensional input vector to C-dimensional logits. The energy function F(f(x)) TRC IR (LeCun
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et al., 2006) is defined by
B(f(x)) -T- log
_ ele(x)/T
(3)
where hyperparameter T is set to 1 in most cases and [x(x) denotes the y-th index of f(x), ie., the logit corresponding to the y-th class. The energy [(((x)) is shown to an efficient indicator for OOD detection (Liu et al., 2020), where a higher [xf(x)) implies that x is more likely to be OOD, Hence, we identify a node as an OOD node when its energy is high. Utilizing the graph structure, neighborhood aggregation can refine the initial energies to be more helpful in OOD detection (Wu et al., 2023). Here, we aim to design an aggregation scheme to enhance the discriminative ability of energies between ID and OOD nodes.
Unlike the previous method (Wu et al., 2023) that uniformly aggregates energies of neighboring nodes, our EDBD deliberately aggregates energies based on the energy distribution of neighboring nodes (see Figure 2), which avoids the undesired mixing of energies between ID and OOD nodes. In EDBA, the initial energies serve not only as targets for aggregation but also as controllers of the aggregation process (see Sec. 4.2). Specifically, EDBA consists of edge-level and node-level controllers. For the edge-level controller, EDBA employs the energy similarity matrix as the transition matrix used for aggregation (see Sec. 4.3). Namely, EDBA weakens the energy propagated from a node with high energy (potentially an OOD node) to connected node with low energy (potentially an ID node) during aggregation, and vice versa. For the node-level controller, EDBA employs the energy consistency matrix as the degree of total aggregation from neighbors (see Sec. 4.4). When energies are inconsistent across the neighboring nodes, ID and OOD nodes may coexist among them. In such cases, aggregating energy from neighboring nodes may lead undesirable outcomes. Hence, EDBA adjusts the degree of aggregation based on the energy consistency matrix.
4.2 ENERGY DISTRIBUTION-BASE AGGREGATION
EDBA controls energy aggregation both the edge-level and node-level to mitigate the mixing of energies between ID nodes and OOD nodes. When g
is provided in testing 101 by using Equation (3), where E (xf()))) Then, EDBA refines (()) through K-step aggregation. EDBA can be formulated by convex combination of (k-1) and transitioned E(k-1) (i.e., SE((-1)) as
phase, EDBD obtains the initial energies ( = [
(k) = (I-aC)E((-)) (CSE(k-)),
where S E RNN and C E IR NxN denote energy similarity matrix and energy consistency matrix, respectively, while a is a hyperparameter between 0 and and I E R an identity matrix. After completing K aggregation steps with EDBA, the final energies E(K) are obtained, which are used to detect OOD samples.
During aggregation, while S determines the degree of message passing that occurs on each edge (edge level), C determines the degree of reflecting transitioned energies from neighboring nodes for each node (node level). These S and C are constructed by utilizing (
4.3 ENERGY SIMILARITY MATRIX
In almost all cases, energy-based OOD detectors do not show perfect performance test data. This implies that high energies are not guaranteed for OOD samples. To refine imperfect ener- gies, GNNSAFE (Wu et al., 2023) demonstrates that aggregating energies from neighbors can be effective for OOD detection on graph-structured data. However, simple aggregation of energies from neighbors may impede OOD detection as an OOD node may have ID nodes in its neighbors, and vice versa. For example, from the central node's point of view in Figure 3(a), two nodes are ID among the three neighbors while the central node is OOD. Hence, aggregating energies from neighboring nodes with equal importance might lower the central node's energy, which can hamper its correct classification OOD.
O :ID
O :00D
(a)
(0)
Figure Examples of node and its neighbors on G

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Inspired by graph neural networks that use the attention mechanism (Veličković et al., 2018; Lim et al., 2021), we propose energy similarity- aggregation for edge-level control of aggregation. Energy aggregation at the i-th node v; for updating its energy can be seen weighted sum of the i-th node's energy and the energies of its neighbors. In our energy similarity-based aggregation, the weight of neighbor's energy is adjusted based its similarity the U; 's energy. Energy similarity- -assed aggregation assigns the larger weight of neighbor's ' energy as the energies between the neighbor and U, are similar. This ensures a more accurate energy assessment for OOD detection.
Formally, we construct a weighted adjacency matrix 5 E RNN N as follows:
5. = sim
(0)
if Ar, =1
(5)
if A 0,
where is the adjacency matrix of go and Sim(E)
E represents the similarity between E.")
(0)
and E E We define sim(E, _E as (e- (max(E(0) min E E) + (1-c) c) E E ()
where c E (0, 1) is hyperparameter. We then normalize S to S D- 5 where D is diagonal matrix with diagonal entries D = Σ, S.j. This row-stochastic S effectively aggregates neighboring energies based on the energy similarity between two connected nodes.
4.4 ENERGY CONSISTENCY MATRIX
Since OOD spreads by infecting nodes that are directly connected to them along edges, OOD nodes cluster together. Consequently, the remaining nodes, which are ID nodes, also form clusters. In the case where nodes are located within their cluster like Figure 3(c) and 3(d), energy aggregation can help refining each central node's 's energy to be closer to its own type since it is connected only with nodes of the same type (OOD/ID). In contrast, Figure 3(a) and 3(b) illustrate nodes located at the boundary between II) cluster and an OOD cluster. In these cases, aggregating the energy of neighbors may impede accurate OOD detection. Hence, energy consistency-bact aggregation is designed to vary the total amount of energies from neighbors for each node differently, depending on the node's 's location in relation to the clusters
To control aggregation at node-level based on the location of a node, we analyze the energy distribution of the node and its neighboring nodes. If a node is located within a cluster, the energy variance among these nodes will be low (i.e. the energies are consistent), as all the nodes belong to the same type (either OOD or ID). Conversely, for node at the boundary, this variance will be high (i.e. the energies are inconsistent). Thus, for each node, we construct a set denoted by S as follows,
S(n)) (E U (E) (E HEN())
(6)
for i E {L....,.N N]. We then calculate the standard deviation of the energies within S(y,) for all e [I....,N]] NJ. Let o, denote the standard deviation for the energies in S(v;). Using {o.):=,,...., we construct a normalized diagonal matrix E, where diagonal entries Σi,- = v1/max((o) ((a))ee(....))))
(0 < Σn < 1). Finally, we define C E N.NN by
C=I- EE
where 0 < 3 < 1 is a hyperparamter that controls the influence of Est-
We recall Equation (4) of EDBA expressed by ((k) (1- (L-aC)E(t-D + (CS((k-1), If v; has inconsistent energy distribution in its surroundings (i.e., high Σa R 1), C. approaches zero as B increases. This implies the i-th element of (CS becomes almost zero, that is, the transitioned
energy of 3ú becomes almost zero during aggregation. Then, EX(A) Ry E thha is, v preserves its energy value. In summary, if v; is located at the boundary, Eg tends to have a high value. Thus, by controlling S, we can make i; preserve its energy without energy mixing from its neighboring nodes.
EXPERIMENTS
The term label leave-out refers to the conventional node-level OOD detection task on graphs, which assumes nodes belonging to a subset of classes as ID and leaves out the other nodes for OOD. We conduct extensive experiments on 1) spreading OOD detection and 2) label leave-out.
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Table I: Performance label leave-out in benchmark datasets. The average FPR95 (%), average AUROC (%), and average AUPR (%) across 10 independent runs are reported with standard deviation
errors.
Dataset
CORA
AMAZON-PHOTE
Method
FPR95(L)
AUROC(())
AUPR(1)
FPR95(L)
AUROC(1)
AUPR(")
MSP
40.37 2.19
91.13 + 0.22
78.16 0.19
28.87 1.65
94.41 =: 0.61
92.44 + 0.75
ODIN
100.00
49.05 0.57
21.18 0.08
92.72
8.43
63.30 = 7.61
51.72 + 7.43
Mahulanobis
6.19
56.11 16.06
82.51 4.83
75.73
7.09
Energy
38.36 + 3.46
91.46 + 0.33
78.10 0.29
30.49
3.93
93.96 + 0.68
91.73
0.75
GKDE
60.88 2.25
87.15
72.12 1.10
91.60
8.81
60.00 11.43
56.61 + 12.77
GPN
44.04 5.88
87.48
6.38
81.21
7.40
35.54 1: 11.48
91.48
2.71
88.04
1: 3.41
OODGAT
85.21 + 1.66
64.81
0.87
02.65
1.01
13.33
0.46
97.27 + 0.33
95.01
+ 0.80
GNNSAFE
31.31 1.11
92.84
0.38
82.22
0.40
6.57 + 0.38
97.36
0.04
97.13
0.10
EDBD
30.48 1 1.11
92.95 1 0.38
82.31
0.38
5.82 1: 0.66
97.48
0.07
97.60 1 0.06
Dataset
AMAZON-COMPUTERS
COAUTHOR-CS
Method
FPR95(L)
AUROC(1)
AUPR(1)
FPR95(L)
AUROC(1)
AUPR())
MSP
70.77 2 3.54
76.81 = 2.31
71.01
= 2.23
29.07 = 3.53
91.15 = 0.73
97.73 # 0.28
ODIN
98.72 1: 2.56
53.36
1.91
45.93
2.83
100.00 4: 0.00
52.35 4.36
75.26 1: 1.96
Mahalanobis
71.09 + 1.90
73.14
1.27
62.63
2.43
64.40 14.83
81.73 3.53
84.30 + 15.80
Energy
58.40 1: 3.41
84.72
1.50
79.36
1.66
18.60 = 3.87
95.98 +: 0.71
98.39 1: 0.29
GKDE
90.64
58.50
10.46
8.10 49.25
88.02
1.77
95.50
GPN
80.55 + 16.98
74.08 + 15.09
09.27
26.68 + 11.56
93.54 2 3.35
97.40 + 1.47
OODGAT
86.16 7.35
73.55
5.48
84.17 2.90
13.16 1.13
96.83
0.21
96.58
0.10
GNNSAFE
39.94 6.84
89.75
1.79
85.63 3.36
11.31
1.69
97.44
0.35
99.06
0.13
EDRD
35.59 + 6.94
92.45 + 0.98
90.34 + 1.45
10.19 + 1.63
97.68 + 0.35
99.14 0.13
5.1 EXPERIMENTAL SETUP
Evaluation Metric. For label leave-out, we measure three metries: 1) the false positive rate OOD samples when the true positive rate of ID samples is at 95% (FPR95); 2) the area under the receiver operating characteristic curve (AUROC); 3) the area under the precision-recall curve (AUPR). In spreading OOD detection, the goal is to perform well under any spreading situations regardless of i. Hence we define new metrics calculated per episode. FPR95-T represents the averaged FPR95 across each graph in {G* (1...... where T is the length of the episode. Similarly, we define AUROC-T and AUPR-T, which are the averaged AUROC and AUPR values, respectively, calculated across each graph at different time stamps within the episode.
Datasets. For label leave-out, following the setup in Wu et al. (2023), perform experiments four benchmark datasets: Cora (Sen et al., 2008), Amazon-Photo (Shchur et al., 2018), Amazon- Computers (Shchur et al., 2018), and Coauthor-CS (Sinha et al., 2015). For spreading OOD detection, we utilize three benchmark datasets, including Cora, LastFM Asia (Rozemberczki & Sarkar, 2020), and the proposed Spreading COVID-19 dataset. We use SI and SIS models for epidemic spreading simulations. For each (dataset, epidemic model) pair, we create 15 episodes, each of which conclude when over 75% of nodes are infected. We then allocate five episodes for the validation set and ten episodes for the test set. For the benchmark datasets used in spreading OOD detection, we employ Bernoulli distribution (i.e., x ~ Ber(O..)) to generate the features of OOD nodes.
Baselines. We compare EDBD with eight state-of-the-an methods, which can be divided into two categories: (1) OOD detection methods for i.i.d. sampled data and (2) OOD detection methods for graph-struse graph-structtred data. Category (1) includes MSP (Hendrycks & Gimpel, 2016), ODIN (Liang et al., 2018), Mahalanobis (Lee et al., 2018), and Energy (Liu et al., 2020), while Category (2) comprises GKDE (Zhao et al., 2020b), GPN (Stadler et al., 2021), OODGAT (Song & Wang, 2022) and GNNSAFE (Wu al., 2023). GKDE and OODGAT are evaluated solely on label leave-out since they requires identical graph structures during both the training and testing phases.
Backbone Encoder. For fair comparisons, we use the same backbone for all baselines and the proposed method. For spreading OOD detection, we utilize a multi-layer perceptron (MLP) since the connectivity among samples not provided during training. For label leave-out, where both training and testing are conducted on given graph, we exploit GCN (Kipf & Welling, 2016) as a backbone. For all experiments, hyperparameters are tuned on validation sets. Further experimental details are provided in Appendix B.
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Published as a conference paper at ICLR 2025
Table 2: Performance on Spreading OOD detection in the Spreading COVID-19 dataset. The average
FPR95-T (%), average AUROC-T (%), and average AUPR-T (%) across 10 episodes are reported with standard deviation errors.
Single-seed setting
Epidemic model
SI
SIS
Method
FPR9S-T (4)
AUROC-T (1)
AUPR-T (1)
FPR95-T (4)
AUROC-T
AUPR-T (1)
MSP
94,25 + 4.56
38.51 = 5.41
70.17 = 4.56
97
1.38
34.45
5.87
70.01 2:7.3
ODIN
93.19 = 5,00
61.82 #: 11.50
78.08 =: 6.05
89.87 4: 5.5.7
63.56
10.63
80.26 2: 5.54
Muhulanobis
63.64
44.93
47.29 + 42.24
79.46
15.28
72.71 + 36.33
46.61
34.07
78.95 + 12,00
Energy
80.46
87.94
GPN
90.29
2.36
30.35 + 13.00
79.55 63.45
87.07 + 14.50
67.
12,68
83.03 4.40
GNNSSAE
69.76
15.25
80.65
6.07
87.10
1.12
77.
76
+ 14.01
76.02
12.65
65.53
=: 5.30
54.67
17.58
8.35
19.75
76.22
14.58
5.94
Multi-seed setting
Epidemic model
SIS
Method
FPR95-T (L)
AUROC-T (1)
AUPR-T (1)
EPR95-T
AUROC-T (1)
AUPR-T (*)
+:
1.87
37.49
2.58
ODIN
92.61 + 3.42
63.15 + 12.07
72.47 6.67++66.66 6.61
91 90 4.09
64.30 + 10:37
09.06 - 6.80
Mahalanobis
90.11 1: 0.53
27.88
= 0.67
53.78 0:3.2.0.3 0.95
90.55 0.50
31.09 1: 0.63
54.166 0.62
Energy
49.78
85.94 4.92
10.33
GPN
87.72 + 6.17
62.81 9.26
75.13
5.38
89. + 5.18
61.85
8.89
71:30 5.25
GNNSAFE
63. 2 11.09
82.21 5.91
86.11 2: 4.33
63.3 1.14
77.80
4: 5.16
80. 3,96
49.62 15.22
83.29
6.68
5.01
59.93 10.47
6.16
Table 3: Performance on Spreading OOD detection in benchmark datasets. The average FPR95-T (%), average AUROC-T (%), and average AUPR-T (%) across 10 episodes are reported with standard deviation errors.
Rpidemic
Dataset
CORA
LASTEM ASIA AUROC-T
model
Method
FPR95-1
AUROC-T
AUPR-T
FPROS-T (L)
AUPR-T
90.71
ODIN
100.00+ 0.00
46.65
10.85
74.03 + 4.70
100.00 0.00
17.35
6.96
68.61
4.08
Mahalanohis
65.32 44.90
65.50
32.93
$9.21
11.85
50.20 + 48.43
87.34
15.97
96.66
3.92
Energy
96.
54.44
001.6 1.71
GPN
100.00 0.00
53.71 1.38
73.93
2.67
100.00
0.00
19.70
75.05
18.16
GNNSAFE
31.15 10.74
93. 45 2.18
97.59
0.71
64.83 a 9.52
$0.73 01.73.1:4 4.22
93.94
1.05
EDBD
14.99
14.53
97.54
1.81
98.68
0.85
46.68
14.83
1.71 94.10
98.11 0.41
MSP
80.47
92.91
03:22
1.50
57.19
18:25
ODIN
8.997
52. 12.1444
11.24
09:25 1: 6.79
100.00
0.00
20.71
9.38
61.54 =: 5.78
Mabalanohis
74.28 2 40.81
68 0T
+ 25 25.99
S8.072 = 10.64
86 = 28,
83:20 6.94
94.87
1.96
SIS
Energy
38.95
GPN
100.00 0.00
47.38 4.09
69:93 4: 6.55
100.00 + 0.00
22.41 13.41
$7.50
9.84
GNNNAAE
47.30
11:24
90.91 2.78
93. 19 1.54
73.94 8.07
W4.73+-4.55 4.55
90.17 1.64
EDBD
3261
1222
34.19
52.37
19333
5.2 LABEL LEAVE-OUT RESULTS
The results of label leave-out, the conventional OOD detection setting graphs, are demonstrated in Table In this setting, EDBD outperforms state-of-the-ar methods across all metries. Class homophily, which is the phenomenon where nodes within the same class tend to connect with each other, is inherent in many graph-structured datasets. Thus, since label leave-out treats a subset of classes OOD, OOD nodes in this setting may also exhibit dense connections among themselves, similar to the pattern observed in spreading OOD detection. We confirm that EDBD achieves state-of thetthe-rrtt performance across all the datasets.
5.3 SPREADING OOD DETECTION RESULTS
Table 2 shows the results of spreading OOD detection on the Spreading COVID- dataset. The results of spreading OOD detection on existing benchmark datasets are shown in Table 3. Our EDBD achieves state-of-the-art performance regardless of the epidemic model on the Spreading COVID-19, Cora, and LastFM Asia datasets. The proposed EDBD achieves the best performance across the evaluation metrics, regardless of the epidemic model, both under the single-seed setting and the multi- seed setting. GNNSAFE, which refines energies via neighborhood aggregation, suffers performance degradation compared to Energy, a method agnostic to graph structure, in terms of FPR95-T. This indicates that aggregating without consideration of energy distribution results in a performance drop. In contrast, EDBD consistently achieves superior performance across all experiments, demonstrating the effectiveness of the proposed ditryyydisrrbuuionnaaaaa aggregation scheme. Through the results

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Published as a conference paper at ICLR 2025
Table 4: Ablation study of EDBD spreading OOD) detection and label leave-out, conducted Cora. and C denote energy similarity matrix and energy consistency matrix, respectively. Label leave-out
Dataset
Amazon-Photo
Coauthor-CS
S
C
(_)
AUROC(1)
AUPR(1)
FPR95(J)
AUROC(1)
AUPR(1)
X
6.57
0.38
97.36 + 0.04
0.11
11.31 + 1.69
97.44
99.06 + 0.13
X
6,43 0.70
97.47 + 0.07
97.58
1.08:00.0000
10.98 = 1.85
97.55
0.36
99.11 0.12
/
6.05 1: 0.62
97.41
+: 0.06
97.22
1 0.02
10.66 + 1.65
97.59
0.31
99.10 + 0.09
/
5.82 1 0.66
97.48 0.07
97.60 1 0.06
10.19
1 1.63
97.68 1 0.35
99.14 1 L0133
Spreading OOD on COVID-19
Model
SIS
S
C
FPR95-T(4)
AUROC-T(")
AUPR-T(T)
FPR95-T(L)
AUROC-T(1)
AUPR-T(())
x
X
69.76 +: 15.23
80.63 1: 6.07
87.40 0.10
77.76 : 14.64
76.02 12.65
85.53 4.33
X
56.25 + 18.02
81.54 8.30
5.57
70.59 = 19.60
76.11 14.31
54.86 + 17.60
81.58
+ 8.32
88.16 5.59
68.03
19.51
76.16 + 13.97
85.88 5.85
54.67 1 17.58
81.60
8:35
88.22 5.59
67
19.75
76.22 14.58
85.94
5.94
Spreading OOD Cora
Model
SI
SIS
S
C
PPR95-T(1)
AUROC-T(1)
AUPR-T(1)
FPR95-T(1)
AUROC-T(1)
AUPR-T(1)
x
31.15 10.74
93.45 + 2.16
97.59 + 0.71
47.55
11.24
2.78
93.19 1.54
X
/
20.42
14.48
96.77 1 1.80
98.59 1 0.80
38.92
10.04
93.32 1 1.69
94.07 1 1.41
V
x
15.63 + 13.91
97.51 + 1.78
98.61 1: 0.83
32.84 + 7.31
94.06 + 1.23
94.10 4: 1.39
/ V
14.99 + 14.53
97.54 + 1.81
98.68 + 0.85
32.61 + 7.18
94.17 + 1.22
94.19 1.39
the benchmark datasets, confirm that EDBD is consistently effective in spreading OOD detection under various settings.
5.4 ABLATION STUDY
We conduct an extensive ablation study to analyze the effectiveness of the components in EDBD. We perform the ablation study under six experimental settings, including different tasks, datasets, and epidemic models. Table 4 demonstrates the results of the ablation study. We exclude $ by replacing sim(Es E
E ( in Equation (5) with This implies that EDBD utilizes only graph structure during the energy aggregation process without considering energy similarity between two connected nodes. To exclude we simply in Equation (4) zero. The first rows of the tables in Table where both and C, correspond to the performance of GNNSAFE, as it relies solely on the graph structure during energy aggregation. We confirm that the two components significantly contribute to performance improvement. The combination of two components results in the best performance.
Appendix provides complexity analysis terms of time and memory (in Appendix C.I), statistical analysis (in Appendix C.6), performance according to time stamps (in Appendix C.2), and extensive experiments that demonstrate the robustness of EDBD under realistic scenarios, including dynamic graph structures (in Appendix C.3) and missing features (in Appendix C.4).
CONCLUSION
In this paper, we introduce spreading OOD detection, a problem that facilitates the evaluation of node-level OOD detection under realistic settings. To highlight the significance of this problem, we then present the Spreading COVID-19 dataset, which allows node-level OOD detection methods to identify COVID-19 19 without prior knowledge of the virus. Moreover, we propose EDBD that leverages flexible aggregation depending on each node's 's circumstance. Our aggregation scheme not only applicable to refining energy but also can be extended to the refinement of any scalar value with noise on a graph. However, our aggregation scheme is effective when connected nodes tend to have similar scalar values. In contrast to graph-level OOD detection, there is no dataset specifically designed for node-level OOD detection. We hope that our work will inspire the creation of datasets for practical node-level OOD detection research within the machine learning community. Since employ SI and SIS models in this work, spreading OOD detection using complex epidemic models is left for future research.
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Published as a conference paper at ICLR 2025
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Shaofeng Zou, Venugopal V Veeravalli, Jian Li, and Don Towsley. Quickest detection of dynamic events in networks. IEFE Transactions on Information Theory, 66(4):2280-2295, 2019.
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A
SPREADING COVID-19 DATASET
Training
COVID-19
Evaluation
2018
2019
OOD Class
Nocmal
ID
Cold
Classifier
Flu
Spread
Classifier
Allery
OOD
ID Classes
Figure 4: Illustration of spreading OOD detection on the Spreading COVID-19 dataset. A classifier can only be trained using ID data. During testing, OOD nodes emerge on a network and serve as seed nodes for spreading. OOD begins spread from these seed nodes by infecting neighboring nodes. The objective of spreading OOD detection is to discriminate OOD nodes from ID nodes in graph where the spread is occurring. Spreading OOD detection generalization of the existing OOD detection setting that tranforms randomly selected nodes into OOD nodes. (All emojis designed by OpenMoji.)
Spreading COVID-19 is dataset for spreading OOD detection. In this dataset, ID classes are normal (no illness), allergies, cold, and flu. An OOD class is COVID-19. For each class, we generate samples with 23-dimensional features, created based on information about symptoms of respiratory illnesses (AARA; Clinic). Features include the following symptoms: (sore throat, cough, musele ache, rapid breathing, chest tightness, chill, runny nodes, stuffy nose, fever, nausea, vomiting, diarrhea, shortness of breath, difficulty breathing, loss of taste, loss of smell, itchy nose, itchy eyes, itchy mouth, itchy inner ear, sneezing, pink eye, tiredness). These features are sampled based on the occurrence of each symptom in the respective diseases. For each illness class, we set the sampling probability for its corresponding symptoms (i.e., features) to 0.5. For normal class, we set the sampling probability equally to 0.01 across all the symptoms. Through these processes, Derain Deai Diest and Dout are prepared. Dtrain and Deal contain 450 and 50 samples for each ID class, respectively.
We utilize the graph structure from the LastFM Asia graph (Rozemberczki & Sarkar, 2020), which has 7, 7,62 nodes. The LastFM Asia dataset is licensed under a Creative Commons Attribution 4.0 international (CC BY 4.0) license. This license allows for the sharing and adaptation of the dataset for any purpose, provided that the appropriate credit is given. We then assign samples randomly sampled from Diest to each node, representing individual. We randomly assign 906 samples nodes for each ID class. To generate an episode of COVID-19 spreading. we randomly select a seed node and replace its features with those randomly sampled from Dout to simulate the node being infected with COVID-19, Using the SI and SIS models, the initial seed node attempts to infect neighboring nodes. We set 7 0.5 for the SI model and (7, (0.5, 0.1) for the SIS model. For each model, we create 15 episodes. We conclude an episode when over 75% of nodes are infected. As a result, we obtain dataset for a single-seed setting. For a multi-seed setting, we repeat the aforementioned procedures by setting the number of initial seed nodes to 100.
We provide the Spreading COVID-19 dataset in the supplementary material. This zip file con- tain four types of elements: (1) "D'"grapheedge_nne 'x_cold.npy'' x_coviddnpy':: 'x_flu.npy', "x "x_normal..py generated symptom features for each class (3) "graph_x.npy' ID-class features of the graph before COVID-19 spreading is applied; (4) npy format files in the 'episodes' folder: These files are matrices of shape N by T, where the t-th column represents the infection status of the nodes at time stamp t. In this column, a value of 1 indicates infected, and a value of 0 indicates not infected. The license for the Spreading COVID-19 dataset is Creative Commons Attribution 4.0 international (CC BY 4.0).
the graph structure; (2) 'x_allergy.npy'
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Justification Using LastFM Asia for the Graph Structure. Publicly available COVID-19 19 data with offline contact networks among individuals is lacking while there are many datasets containing infection populations over time and individual symptoms. To the best of our knowledge, except for a few small-sized networks, there are no publicly available datasets for offline social networks meet COVID-19 dataset. Dunhar al. (2015) provides rationale that online networks share very similar structural characteristics with offline networks. Based on this rationale, for the most realistic alternative to the offline social network, we utilize the online social network (LastFM Asia) to simulate the pandemic in our Spreading COVID-19 dataset. As demonstrated in Singh et al. (2021), our setting more promising than using randomly generated networks as used in Alrasheed al. (2020); Rafiq et al. (2023).
Node Degree Distribution (Bin Size
150+ Merged)
5000
4000
Frequency
LastFM Asia
2000
1000
140
10
20
30
40
50
60
70
100
110
120
130
140
150
Degree
Node Degree Distribution (Bin Size 10)
40-
Frequency
High School Contact
20
Degree
Figure 5: Node degree distributions of the LastFM Asia network and the high school contact network.
Comparison between LastFM Asia and human contact network. The edges of the LastFM Asia dataset are constructed based on mutual follower relationships among users. Due to the lack of large human contact networks, we utilize the structure of this dataset to simulate the spread of COVID-19 the proposed Spreading COVID-19 19 dataset, based on the rationale that online networks share structural characteristics similar to offline networks. To further validate the of the graph structure the LastFM Asia dataset, we thoroughly compare it with the High School Contact network (Mastrandrea et al., 2015).
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Figure 5 compares the node degree distributions of the two worldw networks. Human contact networks often feature super-spreaders with high node degrees. As expected, the High School Contact network contains super-spreaders with node degrees of 50 more, as shown in the figure. Similarly, the LastEM Asia network also includes high-degree nodes, with some exceeding degrees of 140, which can act super-spreaders. Although the network sizes of the two datasets differ (327 nodes for the High School Contact network and 7624 nodes for the LastFM Asia network), both networks exhibit similar characteristics in terms of the presence of super-spreaders.
Furthermore, we compare the average episode duration, defined as the number of steps required for 75% of nodes to become infected. For the SI, SIS, and SEIR Recovered) models, the High School Contact network requires 8.3, 10.1, and 18.4 steps, respectively, while the LastEM Asia network requires 10.0, 11.1, and 24.6 steps, respectively. Considering the difference in network sizes, we can confirm that the human contact network did not lead to faster spreading compared the LastEM Asia network.
-eeeppiible-xoosed--ee Infectious-
Although we demonstrate that the LastFM Asia dataset and the human contact network share similar properties in terms of graph structure, we further simulate COVID-19 spreading on the High School Contact network and perform spreading OOD detection to validate the effectiveness of EDBD in those scenarios as well (in Appendix C.8).
Justification on Parameter Setting for Epidemic Models. We selected the parameter values for the SI and SIS models based on recent work investigating COVID- 19 dynamics (Eikenberry et al., 2020). In continuous- time SIS models, the ranges of the transmission rate 7 and the recovery rate 6 are estimated to be [0.5, (da 1) and [1/30, 1/3] (da respectively. When converting the ranges of 7 and 5 from continuous to discrete-time models, with the time interval set to one day, they become [0.393, 0.777] and [0.033, 0. 0.283]. respectively. Therefore, for the SIS model, we selected 0.5 for y and 0.1 for within these ranges. The SI model is the SIS model that approximates d of 0.033 to zero. Moreover, it is easily feasible construct new datasets by simply adjusting ? and 5 within the ranges.
Difference between the Spreading COVID-19 dataset and general infectious disease datasets.
While general infectious disease datasets (Singh et al., 2021; Alrasheed et al., 2020; Rafiq et al., 2023) focus solely on the spreading patterns of infectious diseases, ours includes person-level symptom- based features. Furthermore, we provide ID classes for each sample: normal (no illness), allergies, cold, and flu. This setup enables the formulation of COVID-19 detection as OOD detection, where classifiers trained only on ID data attempt to detect the new disease, COVID-19. To the best of our knowledge, the Spreading COVID-19 19 dataset is the first COVID-11-relate dataset to include both sample-level features and class labels.
B EXPERIMENTAL DETAILS
B. DATASET DETAILS
We conduct experiments on seven benchmark datasets, including Cora, Amazon Photo, Amazon- Computers, Coauthor-CS, Spreading COVID-19, Cora with spreading OOD, and LastFM Asia with spreading OOD. Table 5 shows the statistics of datasets used in this paper. All the datasets are provided in Pytorch Geometric (Fey & Lenssen, 2019). The Cora dataset a citation network, where nodes and edges represent publications and citation links, respectively. The LastFM dataset is social network, with nodes as individuals and edges reflecting social connections between them. Amazon-photo is recommendation network, where nodes represent goods, and an edge connects two nodes only when the goods are frequently bought together. Coauthor- is a co-authorship graph, where nodes are authors and an edge connects two authors if they co-authored a paper. The Cora, Amazon-Photo Amazon-Computerr, Coauthor-CS datasets are MIT-licensed. The LastFM Asia dataset is licensed under Creative Commons Attribution 4.0 international (CC BY 4.0) license.
B.2 IMPLEMENTATION DETAILS
We conduct all the experiments single NVIDIA GeForce RTX 2080 Ti GPU with 11GB memory and an Intel Core 15-6600 CPU @ 3.30 GHz. For training, we leverage Adam optimizer (Kingma & Ba, 2014) and set the maximum number of epochs 200. We report test performance an epoch
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Table 5: Dataset statistics.
Dataset
#Nodes
#Edges
#Features
#Classes
CORA
2,708
5,429
1.433
7
LASTEM ASIA
7,624
27,806
128
18
AMAZON-PHOTO
7,650
238,162
745
AMAZON-COME COMPUUNCOMPUTERR
13,752
491,722
767
10
COAUTHOR-CS
18,333
163,788
6805
15
which yields the lowest validation loss. Learning rates are selected within {0.01, 0.000, 0.0001} by using a grid search. For fair comparison, for MSP, ODIN, Mahalanobis, Energy, GNNSAFE, and EDBD, we use the same encoder backbone, a GCN with two layers and hidden dimension of 64. For GKDE and GPN, employ their public implementations and adhere to the hyperparameters reported in their respective paper.
B.2.1 SPREADING OOD DETECTION
There is no dataset specifically designed for node-level OOD detection where OOD samples are separated. Thus, to perform spreading OOD detection on the Cora and LastFM Asia datasets, we first create additional OOD samples. We employ a Bernoulli distribution with p = 0.1 to generate Ber(0.1)). Next, we partition samples within the original datasets into Dtraan D and Diest, We exclude 20 nodes per each class. These excluded nodes are then divided into Derain and D at ratio of 9:1. On a graph consisting of the remaining nodes in Diest, spread OOD samples randomly sampled from Dout. Utilizing the SI or SIS model, we generate 15 episodes. We set ạ = 0.5 for the model and (7, = (0.5, 0.1) 1) for the SIS model. Among these episodes, episodes are designated the validation set and 10 episodes for OOD detection. Similar to Spreading COVID-19 19, for each dataset and epidemic model pair, we generate 15 episodes, each concluding when more than 75% of the nodes infected. Among the 15 episodes, we allocate five episodes for the validation set and ten episodes for the test set.
features of these OOD samples (i.e.,
the test set
B.2.2 LABEL LEAVE-OUT
We follow the setting of label leave-out in Wu et al. (2023). The number of ID classes for Cora, Amazon-Photo, and Coauthor-CS is set to 3, 3, and 10, respectively. For a training/valddationnesst split on ID nodes the Cora datasets, we adhere to the split used in Kipf & Welling (2016). For splits ID nodes in the Amazon- Phooto and Coauthor-CS datasets, validation, and test nodes with proportions of 0.1, 0.1, and 0.8, respectively.
random splits for training,
B.2.3 EDBD IMPLEMENTATION
We employ a grid search for hyperparameter tuning and -specfccc hyperparameters (a, e) are selected from ((a, B,e){a (0.1, 0.2, 0.3, 0.5), 3 E {1, 2,2 1 1.e (0.01, 0.05, 0.1, }5]] based on validation sets. The number of aggregation, K, is chosen from {1,2].
B.2.4 IMPLEMENTATION OF BASELINES
For the baselines except for OODGAT, we utilize the implementations provided the GitHub repository released by Wu et al. (2023). All implementations strictly follow the descriptions provided the respective paper. For OODGAT, we utilize the implementation provided the GitHub repository Both publicly available repositories do not contain any statements regarding licenses. Across all the baselines, we adhere to the hyperparameter tuning strategies and settings described in their respective papers.
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Table 6: Training and inference times (seconds).
CORA
AMAZON-PHOTO
Method
Tr. time
In. time
Tr. time
In. time
MSP
0.006
0.010
0.007
0.021
ODIN
0.013
0.020
0.014
0.047
Mahalanobis
0.006
0.125
0.008
0.251
Energy
0.010
0.012
0.014
0.023
GKDE
0.007
0.018
0.008
0.026
GPN
0.273
0.959
1.035
3.004
GNNSAFE
0.010
0.013
0.014
0.024
FastEDBD
0.010
0.018
0.014
0.035
EDBD
0.010
0.062
0.014
0.438
Table 7: Memory usage for different datasets.
Dataset
Memory usage (GB)
CORA
1.199
SPREADING COVID-19
1.203
LASTEM ASIA
1.195
SIAZON-PHOTEE
1,451
AMAZON- CEYOO-COMPUTERS
1.609
COAUTHOR-CS
2.285
OGBN-ARXIV
3,551
C ADDITIONAL EXPERIMENTS
C.I COMPLEXITY ANALYSIS
Here discuss the time complexity EDBD. Energy similarity-based aggregation calculates the energy similarity for each edge, leading to O(EE) complexity. Then, energy consistency-base aggregation measures energy consistency for each node by considering neighboring nodes. Hence, the complexity of energy consistency-baset aggregation is O(N2). EDBD operates with time complexity of O(EE + N2). To quantitatively demonstrate that EDBD, which has outstanding performance, is also efficient, compare training and inference times of EDBD with those of baselines.
We empirically find that energy consistency-basee aggregation is the computational bottleneck of EDBD. In Appendix 5.4, we confirm that EDBD without energy consistency- -based aggregation also demonstrates good performance, although there is performance loss in FPR95/FPR95-T compared to the original EDBD. Therefore, we suggest a light version of EDBD, called FastEDBD, for situations where fast inference necessary. Table shows training and inference times of methods including FastEDBD, Since methods using energy (Energy, GNNSafe, and EDBD) simply train classifiers with cross-entropy loss, they commonly have short training times. While EDBD exhibits long inference times, FastEDBD demonstrates substantial reductions in inference times. We recommend choosing between EDBD and FastEDBD based on what is more important a given situation: performance or inference time.
The memory complexity of EDBD O((0) + (B((+ C)) + O(N) + O(EE)) where 0 denotes trainable parameters. We conduct additional experiments the OGBN-Arxiv dataset, which contains 169,343 nodes, representing a large-scale graph. We confirm that EDBD operates well on the OGBN- Arxiv dataset. The table presents the memory usage of EDBD across various datasets. As shown in the table, EDBD maintains reasonable memory usage across different datasets, including the large-scale OGBN-Arxiv. This demonstrates EDBD's 's scalability in handling large-scale graphs.
We further compare the complexity of all methods used in this paper. Table compares the input and memory complexity of scCR with other state-of-the-are methods. Although EDBD utilize additional
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Table 8: Comparison of Big-O memory complexity. and B represent the batch trainable parameters and the batch size, respectively.
Method
Big-O
MSP
ODIN
0[(8)=([[(F+C)
Mahalanobis
-(0)+++0(((+++)))0000
Energy
0((1)=0(B[[+C))
GKDE
GNNSAFE
O(88) 1000=+
EDBD
FPR-95 ( ())
AUROC (t)
AUPR (t)
40
100
30
SI
10
2
10
12
14
16
10
18
100
100 00000000
96
95
SIS
92
90
85
80
10 12 14
18
20
24
28
10
20 24 26
2
10
18 20
28
MSP
Energy
GNNSAFE
EDBD
Figure 6: Performance Spreading OOD detection according to time stamp t, measured by FPR-95 (%), AUROC(%), and AUPR(%).
matrices S R and C RNNN S is weighted adjacency matrix with non-zero values corresponding to the number of edges, and C is a diagonal matrix. Thus, the memory complexity of EDBD is (()) O(B(F + C)) + O(N) OEE), which is the same as that GNNSAFE.
C.2 PERFORMANCE SPREADING OOD DETECTION ACCORDING TO
The goal of spreading OOD detection is to effectively discriminate OOD nodes in every time stamp. Therefore, we use FPR95-T, T. AUROC-T, and AUPR-T, which are metrics averaged over T. and we confirm that EDBD shows the outperforming performance in terms of the three metries. However, to thoroughly analyze the performance gains achieved by EDBD, it is necessary to evaluate the performance each time stamp t. Figure 6 demonstrates the performance of node-level OOD methods on the Cora dataset according to time stamp f, in terms of FPR95, AUROC, and AUPR. The three most competitive baselines, MSP, Energy, and GNNSAFE, are compared with our EDBD. We find that EDBD consistently surpasses the comparing methods across time stamp I, regardless the epidemic model and used metric. We observe that performance show significant fluctuations in the latter part of time stamp t. This is due to the varing end points of 10 episodes. Since the performance calculated as the average of the 10 episodes, fluctuations arise due to episodes that conclude earlier.
C.3 HIGHLY DYNAMIC GRAPH STRUCTURES
We conduct additional experiments evaluating the robustness of EDBD highly dynamic graph structures. In these experiments, we vary the dynamicity of the graph by removing a certain percentage of existing edges and generating the same number of edges each timestamp of the
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Table 9: Performance of EDBD Spreading OOD detection under dynamic graph structure scenar-
ios.
Spreading OOD detection on Cora (SI)
Disappeared ratio
New ratio
FPR95-T(()
AUROC-T(1)
AUPR-T(()
0
0.00%
0.00%
14.99 + 14.53
97.54 = 1.81
98.68 + 0.85
0.01
10.6%
10.6%
18.04 + 14.52
98.31 + 0.99
97.23 + 1.73
0.02
18.7%
18.7%
22.04 13.33
96.71
98.40 = 0.91 0.91
0.05
39.5%
39.5%
23.43 + 15.14
96.28
98.13 0.77
0.1
63.5%
63.5%
23.37 14.72
96.26 +
97.94 + 0.78
0.2
86.6%
86.6%
27.23 + 15.19
95.50
2.13
97.82 = 0.90
0.5
99.6%
99.6%
31.21 + 14.79
94.81 + 2.47
97.64 + 0.78
Spreading OOD detection on Cora (SIS)
P
Disappeared ratio
New ratio
FPR95-T(J)
AUROC-T(1)
AUPR-T(1)
0.00%
0.00%
32.61 7.18
94.17 1.22
94.19 1.39
0.01
12.3%
12.3%
26.98 + 9.44
95.45
0.91
95.89 - 0.66
0.02
21.7%
21.7%
29.93 7.49
95.02 0.89
95.63 0.68
0.05
44.1%
44.1%
28.69 + 5.02
04.96
95.55 0.83
0.1
69.5%
69.5%
33.23 = 11.43
9..4
95.33 = 0.86
0.2
90.2%
90.2%
31.90 6.68
04.11
95.56 0.85
0.5
99.7%
99.7%
42.28 + 9.86
92.64 + 1.63
94.97 + 0.87
Table 10: Performance of Feature Propagation (FP)+EDBD Cora under feature missing scenarios for varying missing rates (rm).
Task
Label leave-out Cora
Spreading OOD Detection (SI)
F'
(L)95(L)
AUROC(T)
AUPR(*)
FPR95-T(J)
AUROC-T(t)
AUPR-T(t)
30.48 +: 1.11
92.95
0.38
82.31 0.38
14.99
14.53
97.54 1.81
98.68 1: 0.85
0.2
21.15 + 3.23
94.44 0.08
+.97
26.00 1.000::.0 5.67
95.52
1.09
95.16 + 1.27
0.4
21.63 + 2.08
0.31 94.38
+.199
26.51 6.14
95.50 + 1.13
95.16 + 1.26
0.6
32.30
5.67
9..49
0.33
26.52 6.13
95.44 1.17
95.16 + 1.26
0.8
43.28
7.04
91.98
0.72
0.47
27.60 7.36
95.36 1.24
95.16 1 1.26
47.18 1 9.96
90.57 1.57:1111 1.75
3.40
27.01 5.93
95.48 1.05
95.16 1 1.26
0.99
19.54 4: 11.02
1.16:1..1 +
80.10 -
30.29 + 8.37
94.88 1.41
95.09 + 1.27
spreading OOD process. Specifically, each time stamp of spreading OOD, p% of the existing edges are removed and the same number of new edges are generated. Disappeared ratio and New ratio represent the percentage of edges that have disappeared and the percentage of new edges formed, respectively, when the OOD spread has finished. Table 9 shows the results of the experiments. Despite the minor degradation, EDBD maintains a reasonable performance level even under highly dynamic conditions, showcasing its robustness across various missing scenarios.
C.4 MISSING FEATURES
Since missing data can occur not only in edges but also in features. Hence, we further consider incomplete features in graph-structured data, which are also prevalent in real-world scenarios. In this case, diffusion-baar imputation methods (Rossi et al., 2022; Um et al., 2023; 2025b;a) (distinct from generative diffusion models (Ho et al., 2020; Dhariwal & Nichol, 2021; Yeo & Um, 2025)) that have demonstrated their effectiveness in graph struturrr data can assist EDBD by imputing missing features. Table 10 presents the performance of EDBD using features imputed by FP, diffusion- based imputation method, under missing feature scenarios. We can confirm that EDBE using FP yields reasonable performance in OOD detection tasks even with 99% missing features.
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Table 11: Performance comparison of neighhorhood mean, GNNSAFE, and our EDBD. Spreading OOD detection on Cora (SI)
Method
FPR95-T(4)
AUROC-T(1)
AUPR-T(1)
Neighborhood mean
41.87 + 9.22
86.95 + 3.52
95.45 + 0.83
GNNSAFE
31.15 + 10.74
93.45 + 2.16
97.59 + 0.71
EDBD
14.99 + 14.53
97.54 + 1.81
98.68 + 0.85
Spreading OOD detection on Cora (SIS)
Method
FPR95-T(L)
AUROC-T(t)
AUPR-T(1)
Neighborhood mean
56.36 = 10.77
86.12 4.48
91.49 = 1.79
GNNSAFE
47.15 + 11.24
90.91 + 2.78
93.19 + 1.54
EDBD
32.61 7.18
94.17 + 1.22
94.19 + 1.39
Label leave-out on Cora AUROC(()
Method
FL)95(()
AUPR(1)
Neighborhood mean GNNSAFE
32.86 + 1.58
92.67 + 0.37
82.20 + 0.35
31.31 + 1.11
92.87 0.38
82.22 + 0.40
EDBD
30.48 + 1.11
92.95 0.38
82.31 0.38
C.5 COMPARISON WITH NEIGHBORHOOD MEAN
As an entropy-like function for class scores from a trained neural network, the energy function is verified to be an effective indicator to distinguish OOD samples (Liu et al., 2020). In graph-structured data, connected two nodes often share similar properties, such as features and classes. Therefore, whether node is OOD can be aided by its neighboring nodes. Motivated by this, GNNSAFE (Wu et al., 2023) refines the energy of each node by simply aggregating the energies from its neighboring nodes through multiple aggregation steps. Here, GNNSAFE performs energy aggregation based solely on the graph structure. However, an ID node can have OOD nodes in its neighbors and vice versa. In this case, this simple aggregation scheme of GNNSAFE, may lead undesired energy mixing between ID and OOD nodes. The conceptually simplest way, neighborhood aggregation can be seen as a special case of GNNSAFE with a single aggregation step.
In contrast, our EDBD employs energy as an OOD indicator for node during the energy aggregation process. This implies that before the aggregation process, EDBD briefly classifies nodes into OOD nodes and ID nodes to prevent. the energy mixing between OOD nodes and ID) nodes. EDBD allows the energies to control the entire aggregation process themselves, leading to significant performance improvement in both spreading OOD detection and conventional OOD detection tasks graphs.
To verify that EDBD outperforms the conceptually simplest aggregation strategy, which is referred to neighborhood mean, conduct additional experiments in spreading OOD detection and conventional OOD detection tasks. Table 11 shows performance comparison of neighborhood mean, GNNSAFE, and EDBD in label leave-out (conventional OOD detection) and spreading OOD detection using SI and SIS models. As shown in the table, EDBD demonstrates its superiority. irrespective of tasks and metrics.
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Table 12: p-values comparing EDBD the runner-up in each setting. denotes state-of-the-art, not a runner-up.
For Table
Dataset
Metric
Runner-up
Runner-up's
Ours
p-value
FPR95(L)
GNNSAFE
31.31 + 1.11
30.48 + 1.11
0.119
Cora
AUROC(1)
GNNSAFE
92.84 + 0.38
92.95 0.38
0.583
AUPR(1)
GNNSAFE
82.22 0.40
82.31 + 0.38
0.656
FPR95(4)
GNNSAFE
6.57 0.38
5.82 + 0.66
6.61 x 10
Photo
AUROC(1)
GNNSAFE
0.04 97.36
97.48 0.07
6.90 x 10-4
AUPR(1)
GNNSAFE
97.13 0.10
97.60 0.06
1.73 x 10-6
FPR95(J)
GNNSAFE
39.94 6.84
35.59 6.94
0.137
Computers
AUROC(1)
GNNSAFE
89.75 1.79
92.45 + 0.98
3.60 X 10-2
AUPR(1)
GNNSAFE
85.63 + 3.36
90.34
3.36 x 10-3 -3
FPR95(J)
GNNSAFE
11.31 + 1.69
10.19 1.63
0.152
Coauthor-CS
AUROC(1)
GNNSAFE
97.44 = 0.35
97.68 + 0.35
0.123
AUPR(1)
GNNSAFE
99.06 = 0.13
99.14 0.13
0,257
For Table (single-seed)
Model
Metric
Runner-up
Runner-up's
Ours
p-value
FPR95-T(L)
Energy
54.82 + 17.43
54.67 + 17.58
0.981
SI
AUROC-T(1))
Energy
80.46 8.54
81.60 1 8.35
0.784
AUPR-T(t)
Energy
88.00 5.52
88.22 + 5.59
0.948
FPR95-T(L)
Energy
67.94 + 18.98
67.42 19.75
0.959
SIS
AUROC-T(1)
Energy
73.94 + 14.49
76.22 + 14.58
0.769
AUPR-T(t)
Energy
85.51 = 5.70
85.94 + 5.94
0.896
For Table 2 (multi-seed)
Model
Metric
Runner-up
Runner-up's
Ours
p-value
FPR95-T(J)
Energy
49.78 14.73
49.62 + 15.22 0.982
SI
AUROC-T(t)
Energy
82.66 + 6.79
83.29 + 6.68 0.850
AUPR-T(1)
GNNSAFE
86.11 + 4.33
86.11 + 5.01
1.000
FPR95-T(L)
Energy
60.40 10.33
59.93 + 10.47
0.921
SIS
AUROC-T(t)
Energy
80.18 + 6.24
80.68 1 6.16
0.885
AUPR-T(t)
Energy
81.63 + 4.84
81.70 + 4.92
0.973
For Table 3 (SI)
Dataset
Metric
Runner-up
Runner-up's
Ours
p-value
(PR95-T(L)
Energy
22.79 + 18.21
14.99 + 14.53
0.290
Cora
AUROC-T(1)
Energy
96.36 + 2.12
9..54
0.269
AUPR-T(t)
Energy
98.03 + 1.07
98.68 + 0.85
0.191
FPR95-T(L)
MSP
52.98 + 0.41
46.68 + 14.83
0,236
LastEM Asia
AUROC-T(1)
MSP
93.71 1.80
0.589
AUPR-T(1)
MSP
97.96 + 0.44
98.11 0.41
0.557
For Table (SIS)
Dataset
Metric
Runner-up
Runner-up's
Ours
p-value
FPR95-T(L)
Energy
38.95 9.16
32.61 + 7.18
2.72 x 10
Cora
AUROC-T(T)
Energy
93.27 + 1.34
94.17 = 1.22
0.189
AUPR-T(1)
Energy
93.59 1.55
94.19 1.39
0.447
FPR95-T(U)
MSP
57.19 + 18.25
52.37 = 19.93
0.501
LastEM Asia
AUROC-T(1)
MSP
91.48 1 2.88
91.78 3.04
0.814
AUPR-T(1)
Mahalanobis
94.87 1 1.96'
94.40 1.52
0.618
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28
28
28
28
SI
20
20
20
20
16
12
12
B
12
12
0.1
1/4
1/3
1/2
0.01
0.05
0.1
0.5
40
40
40
40
38
38
38
SIS
36
36
36
34
34
34
34
B
0.1
0.2
0.3
0.5
1/4
1/3
1/2
0.01 0.05 0.1 0.5 0.75
2
Figure 7: Performance of EDBD in spreading OOD detection, measured by FPR95-T (where lower values indicate better performance), on the Cora dataset evaluated under two settings using SI and SIS models, respectively, with varying values of a, B, e, and K, under two settings using SI and SIS models, respectively. The blue dashed lines represent existing state-of-the-art performance.
Table 13: Performance on Spreading OOD detection the High School Contact dataset. The average FPR95-T (%), average AUROC-T (%), and average AUPR-T (%) across 10 episodes reported
with standard deviation errors.
Epidemic model
SI
SIS
Method
TPR95-T (L)
AUROC-T (t)
AUPR-T (1)
IPR95-T (L)
AUROC-T (1)
AUPR-T (1)
MSP
96.93 2.94
8.36
62.83 + 17.40
96.95 + 2.67
37.77 = 7.19
61.70 + 15.19
ODIN
90.92 : 10.03
63.40 1: 8.39
73.51 + 15.63
91.21 + 9.72
62.23 + 7.19
72.10
15.19
Mahalanobis
71.13 33.90
64.08
18.19
72.07
22.60
72.30
62.80 28.62
70.95 - 21.96
Energy
55.82 13.72
82.57 7.89
89.36
7.22
67.21 + 8.69
79.21 + 6.87
86.05 9.00
GPN
81.94 +: 3.36
71.28 =: 1.69
75.74
1.41
86.40 + 2.30
66.92 = 1.20
71.85 = 0.65
GNNSAFF
59.05 15.41
82.90 5.70
89.49
7.40
68.39
14.80
80.96 6.28
86.92 9.91
EDBD
53.88 19.95
8.21 83.
89.63 + 7.09
66.90
12.97
81.10
87.36 8.42
C.6 STATISTICAL ANALYSIS
We conduct additional experiments to evaluate the statistical significance of our EDBD's 's superior performance. Table 12 shows p-values comparing EDBD to the runner-up each setting for all results in Tables 1, 2. and 3. As shown in the table, the p-values for Table 2 and Table 3 are higher compared to those Table However, spreading OOD detection tasks involve performing spreading simulations for each run, resulting in varying spreading patterns for OOD. These different spreading patterns significantly impact the difficulty level each run, causing substantial variations in the overall performance of methods and consequently leading to large standard deviations. When the standard deviation high, the statistical test does not show a significant difference between methods, resulting in higher p-values, even if the mean performance differs between EDBD and baselines. Nevertheless, while the runner-ups change depending on the setting, EDBD consistently outperforms in all settings and metrics, except for AUPR-T in spreading OOD detection on LastFM Asia using SIS model. This consistency demonstrates its generalizability across various graph OOD-related tasks.
C.7 HYPERPARAMETER SENSITIVITY
We conduct additional experiments to provide comprehensive analysis of the impact of different hyperparameters, including a, 3, e, and K on the performance EDBD. Under two settings using SI and SIS models, respectively, we report FPR95-T in spreading OOD detection the Cora dataset. Figure shows the results. When varing each hyperparameter, the other hyperparameters are set to their optimal values. Compared to existing state-of-the performance (22.79% and 38.95% for the SI and SIS settings, respectively), EDBD models consistently exceed by a considerable margin regardless of the values of a, and K. While EDBD with {0.75) commonly shows worse
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Table 14: Performance Spreading OOD detection in the Spreading COVID-19 dataset using SIS models with various combinations of (7, 8). The average FPR95-T (%), average AUROC-T (%), and average AUPR-T (%) across 10 episodes are reported with standard deviation errors.
(1.6)
(0.7.0.1)
(0.3.0.1)
Method
FPR95-T (1)
AUROC-T (t)
AUPR-T (1)
FPR95-T (U)
AUROC-T (?)
AUPR-T (t)
MSP
98.22 = 1.34
33.12 6.09
72.66 3.52
08.43.1
1.41
34.66 8.26
66.97 = 16.67
ODIN
91,085..
66.86 6.0.9
82/41 #.91
93.01
1.68
65.34 8.26
76.63 = 12:29
Mahalanobis
79.52 + 18.92
00.93 42.25
75.08 + 14.05
79.64
19.21
57. 36-43
72.54
20.73
Energy
64.44 +: 20.22
74.62 +: 15.00
87.54 1 5.46
27.08 71.33
72.02 1 14.51
80.24
11.66
GPN
58
7.95
GNNSAFE
T3.13 = 16.52
75.96 + 13.13
88
5.22
10.52 86.88
73.90 12.81
80.12 = 11.45
15.09
87.00
70.02
73.65
(7.6)
(0.5,
(0.5, 0.05)
Method
FPR95-T
AUROC-T
AUPR-T (1)
FPR95-T
AUROC-T (1)
AUPR-T (1)
MSP
98.01 # 1.52
37.40 6.56
66.52 11.38
98.19 19.19:2: 1.52
35.22 7.51
T3.05 = 9.67
ODIN
4.09
62.60 6.56
75.37 9.07
92.64
64.78 7.51
82.37 +
Mahalanohis
78.94 + 16.92
62.60 6.56
9.07 75.37
78.19 19.03
61.02 + 42.42
77.81 + 7.84
Energy
75.25 16.88
70.70 12.32
80.22
67.77
73.99 15.31
87.37
GPN
96.28 + 1.88
60.23
15.36
75.82
8.60
95.08 2.00
61.56 + 18.04
82.90
10.26
GNNSAFE
82.27
12.96
73.58
12.32
80.26
8.55
77.43 1 17.52
78.18 1 14.14
87.65 1 7.65
EDBD
73.72 = 17.
73.68 = 12.03
80.35 + 3.00
66.8 06682222 + 23.73
78.45
14.80
87
7.94
Table 15: Performance Spreading OOD detection for RSV. The average FPR95-T (%), average AUROC-T (%), and average AUPR-T (%) across 10 episodes are reported with standard deviation
Epidemic model
SIS
Method
FPR95-T (L)
AUROC-T (t)
AUPR-T (1)
FPR95-T (1)
AUROC-T (t)
AUPR-T (1)
MSP
85.48 +
63.93 6.13
3.15
86.33 + 6.74
63.09
5.59
79.87
ODIN
99.97 0.05
36.07 1007::0 6.13
70,01 = 4.34
99.97 + 0.05
36.91 = 5.59
69.34 1..4133 3.13
Mahalanobis
75.35
50.79 38.40
79.24 + 14.42
75.55 + 32.59
50.72
78.30 = 15.19
Energy
72.18 15.73
9.33
$4.00 5.29
75.31 13.36
71.34 8.96
52.99 3.96
GPN
92.67 8.44
53.38 13.79
81,18 2.13
93.17 1 7.47
54.15 12.89
81.1
GNNSAFB
80.99 = 16.19
74.10:19.07
84.69 5.07
74.54 4: 14.23
71.71 8
83.60
3.50
EDBD
70.94 21.05
74.18 + 9.51
84.74 + 5.45
73.96
18.11
72.75 +: 9.00 83.69 + 4.06
performance compared to the existing state-of-the-art performance, the small value of e indicates that energy aggregation focuses less on energy similarity and more on the graph structure. This result highlights the importance of considering energy distribution in the energy aggregation process. As mentioned in Appendix B.2.3, we determine the hyperparameters of EDBD for each setting using grid search within fixed ranges, based on performance on the validation set.
C.8 SPREADING COVID-19 ON A CONTACT NETWORK
To validate EDBD's s effectiveness in highly realistic scenarios, we conduct additional experiments using contact network dataset collected in high school in France (Mastrandrea et al., 2015). Since the original dataset comprises numerous networks collected over short time intervals, we aggregate them into a single network. We then measure the performance of methods spreading OOD detection by replacing only the graph structure in the Spreading COVID-19 19 dataset with this network.
Table 13 presents the results when SI and SIS models are used, respectively. As shown in the table, our EDBD consistently outperforms state-of- of-te-of-the-rr methods across all metrics and epidemic models. These results demonstrate its effectiveness even in highly realistic scenarios.
C.9 SIS MODELS WITH VARIOUS PARAMETER COMBINATIONS
We adjust the parameters of the SIS model and compare the performance of the methods across various SIS models by performing spreading OOD detection the Spreading COVID-19 dataset. Specifically, we conduct experiments by modifying the original (y, 6) values of (0.5, 0.11) to (0.7, 0.1) and (0.3, 0.1). Additionally, performed experiments by changing the original (v, values to
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Table 16: Performance of FDDB+siimilarit-bassee link imputation in spreading OOD detection
Cora under edge missing scenarios. The average FPR95-T (%), average AUROC-T (%), and average AUPR-T (%) across 10 episodes are reported with standard deviation errors.
Method
EDBD
EDBD with Similarity-Ban Based Link Imputation
Missing rate
FPR95-T (4)
AUROC-T (1)
AUPR-T (1)
FPR95-T (4)
AUROC-T (1)
AUPR-T (1)
0
14.99 + 14.53
97.54 + 1.81
98.68
0.85
14.99 + 14.53
97.54 1.81
98.68 0.85
0.05
18.49 = 14.78
97.07 2.10
98.55 0.99
21.37 = 17.38
96.47 + 2.55
98.48 = 0.98
0.1
18.17 = 15.76
97.00
98.53 0.75
17.76 = 16.75
97.01 + 2.16
98.62 0.88
0.2
19.74 17.07
96.88 + 1.94
9
16.74 17.60
97.22 2:44
98.55 = 0.75
0,3
23.43 + 16.36
96.06 1:00::20 2.79
97.47 1.09
15.88 + 16.26
97.40 2.10
98.78 + 0.69
0.4
40.80 1: 25.99
91.84 1: 10.05
2.60
16.20 + 17.09
97.12 1 2.35
98.66 1 0.90
0.5
41.23 = 32.91
90.62 + 12.02
95.98 + 2.47
23.00 = 30.39
95.40 + 5.77
98.63 = 0.82
Y
S
E
R
Exposed
Infected
Cured
Figure 8: A state transition diagram of SEIR model. S, E, I, and Rt represent the four possible state of node, susceptible, exposed, infectious, and recovered, respectively.
(0.5, 0.2) and (0.5, 0.05). Table 14 presents the results. As shown in the table, except for AUPR-T under (7, 6) (0.7, EDBD consistently outperforms state-of-the-an methods and demonstrates robustness against various spreading patterns.
C.10 SPREADING OOD DETECTION FOR RSV
We conduct additional experiments to detect urgent contagion, Respiratory Syncytial Virus (RSV), using spreading OOD detection. RSV is currently garnering significant attention due to surge in infections and its particularly severe impact infants and the elderly (Langedijk & Bont, 2023; Moline, 2024). We generate samples with features based on information about symptoms of respiratory illness (Healthline; Sample; Wallace), following the same procedure specified in Appendix A. We replace COVID-19 samples in the Spreading COVID-19 dataset with RSV samples, and compare the performance of methods in spreading OOD detection. Table 15 presents the results for RSV spreading scenarios using SI and SIS models. As shown in the table, EDBD outperforms state-of-the-ar methods in both settings across all metrics. These results on the recent contagion show the generalizability of EDBD.
C.11 INCOMPLETE GRAPH STRUCTURES
Since EDBD utilizes the graph structure for energy aggregation, conduct additional experiments to evaluate its performance under edge-missing settings. As performance degradation is expected when edges are incomplete, degradation. This link imputation technique creates single edge for each node by connecting the node with the highest feature similarity. Table 16 the results in spreading OOD detection the Cora dataset. As shown in the table, while the performance of EDBD worsens as the missing rate increases. EDBD with the similarity-based link imputation effectively mitigates performance degradation. We confirm that EDBD with siiillrrttybasee link imputation maintains reasonable performance even with an edge missing rate of 50%.
apply simple missing link imputation technique mitigate the
C.12 SPREADING OOD DETECTION USING SEIR MODELS
To validate the effectiveness of EDBD in OOD spreading using complex epidemic models, we conduct additional experiments. We adopt ceceepttbbe-xxpsseeee SrpoosedIIffectousss et al., 2018) as a complex epidemic model. As illustrated in Figure 8, which shows the state transition diagram of SEIR, each node exist in of four states: susceptible, exposed, infectious, or
(SEIR) model (Rachah
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Table 17: Performance Spreading OOD detection in the Spreading COVID-19 dataset using SEIR model. The average FPR95-T (%), average AUROC-T (%), and average AUPR-T (%) across 10
episodes are reported with standard deviation errors.
Epidemic model
SEIR
Method
FPR95-T (4)
AUROC-T (1)
AUPR-T (t)
MSP
98.02 1 1.16
35.46 6.21
67.50 4.03
ODIN
92.15 3.80
64.54 6.00
78.32 2.15
Mahalanobis
79.56 18.93
=0.25
82.78 15.47
Energy
69.11 + 17.63
73.56
83.52 $
GPN
92.27 2.98
61.76
GNNSAFE
76.09 16.75
75.93 12.38
84.21
EDBD
68.27 + 18.53
76.31 + 13.32
84.23 + 6.58
Infected by neighbor
Resisted infection
Infected by neighbor
Resisted infection
Si.
Not cured
S
E.2
D 1-1
Size
B
ST model
Cured (b) SIS model
Figure 9: Illustration of (a) model and (b) an SIS model, viewed from the perspective of single node. S and I represent the two possible state of node, susceptible and infected, respectively, In both models, Ç.., depends on (epidemic transmission rate via an edge connected an infected node) and the graph structure. This is because a susceptible node may be simultaneously exposed to infection attempts by multiple infected neighbors.
recovered. SEIR model, a susceptible node can transition only to the exposed state, an exposed node can transition only to the infectious state, and infectious node transition only to the recovered state. The probabilitics of these transitions are denoted by 7, a,, and 6. Based on recent research investigating COVID-19 dynamics (Eikenberry et al., 2020), we 7 a, and 5 to 0.5, 0.2, and 0.1, respectively. We follow the experimental setup for spreading OOD detection as specified in Appendix 1.2.1. Table 17 demonstrates the results on spreading OOD detection in the Spreading COVID-19 dataset using the SEIR model. As shown in the figure, EDBD outperforms state-of-the-art methods across all metries, demonstrating its superiority even in complex spreading scenarios.
D STATE TRANSITION DIAGRAM OF SI AND SIS MODELS
Spreading processes using SI and SIS models can be modeled as a Markov chain because the configuration of N nodes at the + 1-th time stamp depends solely on their configuration at the t-th time stamp. Since each node can be in the two states, there exist 2 possible configurations of this system each time stamp. Figure 9 shows the state transition diagrams of SI model and SIS model, respectively.
E THEORETICAL ANALYSIS
The key concept of EDBD is to enable the energies (OOD scores) to control their own update process through energy aggregation. In contrast, GNNSAFE, which performs energy aggregation based solely on the graph structure, cannot prevent the undesired mixing of energies between ID and OOD nodes. We provide theoretical analysis to explain why GNNSAFE leads this undesired mixing of energies. Before proceeding, it is important to note that the goal energy aggregation is to enhance the separation between the energies of ID nodes and those of OOD nodes.
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Since energies positive values, first assume that the energies of ID nodes and OOD nodes follow gamma distributions. Given that the energies ID nodes tend to be smaller than those of OOD nodes, define the energy distributions
follows: (1) ID energies have variance o²; (2) OOD energies have mean Mout and variance o², where Pin < Pout- Additionally, assume that the energies of nodes in each type III (independent and identically distributed). The energy update rule of GNNSAFE is defined as:
Pan and
XE (0) + -11-1) (V(x))
Σ
0 E
(8)
JEN((v)
where À E [0, 1] is hyperparamter and ((x)) denotes the of neighbors of node U..
Consider an ID node v2 with four neighbors, and analyze the following two cases: First, all four neighbors are ID nodes. Second, neighbors are II) nodes, and the other OOD nodes.
In the first case,
E E[E 1 XB 1
A
Σ E
Aptin + (1 A)Pin Min-
(EA(
That is, the mean of the updated energy remains the same as the original mean, Pin- In terms of variance, given the i..dd property, the neighbors' energies (0) are independent of each other and of E Therefore, the variance can be expressed as:
Var - 22 Var(E (1) + (1 2)
(0) jEN((ss
Σ
(10)
We simplify the neighbor term and factor out a2:
Var Var(E =0² 12 (22+ (1- 2)2
(11)
4
Since E [0, 1], the term X² + (1-X)² 1 2
+ 1/4 1. This implies:
Var
< ~²
(12)
This confirms that, in first case, GNNSAFE ensures that the mean remains the same while standard deviation decreases, probabilistically enhancing the discriminative energy of the ID node u; compared to the energies of OOD nodes.
For the second case, node v; has two ID neighbors and two OOD neighbors. Let analyze the mean and variance of the updated energy E(!) Using the energy update rule:
E (1) E
+1 -) 1
Σ
E|E [1
(13)
JEN((e.
Substituting the expectations for ID (fein) and OOD (Houa) nodes, and noting that two neighbors are ID nodes and two are OOD nodes:
E Apin (1 (1-A)) -1)= (2 22pot)
(14)
Simplification gives:
EE Apin (1 1) 2 (Jin Mout) Pin (1- 2
(Mout Msn))
(15)
By GNNSAFE, the mean of the ID node's energy shifts towards the higher energy values of the OOD nodes due to their influence the aggregation term. This undesired mixing reduces the separation between ID and OOD energies, potentially decreasing the effectiveness of OOD detection.
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The variance of [ can be expressed as:
Var(E X² Var(EE))) + +1 - V/
1
Σ
E
(16)
JEN(o.)
For the neighbor aggregation term:
Var 4 Σ |EE,
E
Σ
)) (20² + +20²)
4
o²
(17)
16
16
16
4
JEN(s))
jEN(w)
Finally,
-12- x)2 =0² 12 (1 A)2
+ + -
(18)
4
As in the first case, the variance decreases compared to the initial variance o², but the influence of the OOD neighbors shifts the energy distribution. However, the shift in the mean undermines the goal of energy aggregation, which is to enhance the separation between ID and OOD energies.
This analysis can be easily extended to the general case where the ID node u; has neighbors, among which Rin are ID nodes and nout are OOD nodes (nim + nout = n).. Let us derive the mean and variance for the updated energy E for this general case.
The energy update rule for E, (1) is given by:
E T AI E 101 + (1- 1)= Σ E
jEN(v.)
Apain + (1 - A)= (Nin Min + TooutHouu))
Aptin + (1 (1-A) A) () ("in Min Hout
Pout
Pin (1 Nout
(Hout Min).
(19)
ID node's energy due to the influence of OOD neighbors. The higher the proportion of OOD neighbors (Nout/ /n) or the larger the difference (Hout Msn), the greater the shift in the mean towards the OOD energy values. This undesired shift undermines the separation between ID and OOD energies.
Here, the term (1 X) Ross (Mout (Un) quantifies the shift the mean
The variance of E(!) is:
Var (a(() = Var(((())) + - 2ar
JEN((N)
= + - V1
Σ
JEN(e.
= No00 + (1 (1= 1
(Er(E(00) (0)
n
jEN(2:;
1
No00 + (1 2)
(nmo² + +
n
0²
= 2302 + (1~1)2
o² 12 (1 2)
(20)
n
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Since n 2 1, the term X² compared to the initial variance a²
[1-A)²
S 1, confirming that the variance of E is always reduced
For the general case, the mean E (1) is shifted towards OOD energies when OOD neighbors are present, reducing the separation between ID and OOD energies. The variance Var(E, decreases, but the shift in the mean dominates, leading to the undesired mixing.
This theoretical analysis highlights why GNNSAFE's reliance on graph structure can lead to the undesired mixing of ID) and OOD energies, reducing its effectiveness in maintaining energy separation. Furthermore, this analysis shows that ID nodes aggregating energy from surrounding ID nodes and OOD nodes aggregating energy from surrounding OOD nodes are beneficial for updating energies. Unlike GNNSAFE, EDBD prevents undesired energy mixing by allowing ID nodes to focus on their neighboring ID nodes and OOD nodes to focus their neighboring OOD nodes during energy aggregation.
F DISCUSSIONS
What is the main challenge of spreading OOD detection? The main challenge of spreading OOD detection is that only g with the graph structure and X are given without any additional information such as what the current t is, how much spreading has occurred, and the location of the seed node. This is inspired by the fact that in most real-world OOD-related problems, information about the spreading is not provided. Accordingly, depending on É and the spreading pattern of an episode, the best method can be different. Due to this challenge of the spreading OOD detection task, the goal is to perform OOD detection well under any spreading situations regardless of t. Hence, new three metrics for spreading OOD detection are FPR95- T, AUROC-T, and AUPR-T, TT, which are the averaged FPR95, AUROC, and AUPR across each graph in {G' (EEE..........
Despite this main challenge of the spreading OOD detection task, EDBD can address various spreading situations through its edge-level and node-level aggregation controllers, which are the energy similarity matrix and the energy consistency matrix, respectively. EDBD does not rely solely on the graph structure for energy aggregation; instead, it performs energy aggregation adaptively for each node according to its specific situation. Specifically, these two controllers tailor energy aggregation for each node based on the energy distribution of the node and its neighboring nodes. This energy-based energy aggregation process prevents undesired energy mixing between in-distribution (ID) nodes and OOD) nodes.
Comparison of spreading OOD detection and dynamic event detection. While dynamic event detection (Zou et al., 2019; Zhang et al., 2022; Sridhar & Poor, 2022) seems similar to our spreading OOD detection in that they address the problem of dynamic events spreading through graph, there are clear differences between dynamic event detection and our work from three perspectives. Firstly, dynamic event detection relies on prior knowledge of the distribution of anomalies (i.e., OOD), such as what the distribution is or the shape of the distribution. In contrast, spreading OOD detection assumes that any information about OOD samples is inaccessible, which is the nature of anomalies or OOD in many real-world scenarios. Secondly, dynamic event detection assume that they completely know the distribution from which normal samples (i.e., ID samples) are drawn. However, following OOD detection tasks (Liu et al., 2020; Rawat et al., 2021), we utilize only observed ID features and partial labels for ID samples without fully knowing the distribution. Lastly, studies on dynamic event detection (Zou et al., 2019; Zhang et al., 2022; Sridhar & Poor, 2022) use statistical models related to this distribution, while we use a neural network trained on ID data to determine OOD based on class scores.
Does EDBA take into account information about temporal dynamics? EDBA does not take into account information about temporal dynamics. In many real-world OOD-related problems, the full details of the temporal dynamics, such as when OOD spreading begins, are often unknown. Therefore, in the task of spreading OOD detection, which reflects realistic scenarios, information about temporal dynamics is not provided, although different time slices are used for evaluation. Spreading OOD detection assumes that each time slice is independent, meaning that information from other time slices is inaccessible. Consequently, EDBA, tailored for spreading OOD detection, does not consider temporal dynamics.
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Published as a conference paper at ICLR 2025
Why are human contact studies not considered? Human contact (Mossong et al., 2008; Balean et al., 2009; Pastor-Satorral et al., 2015; Mastrandrea et al., 2015) a highly important topic in the context of the spread of infectious diseases and can play key role in transmission pathway identification and disease dynamics modeling. However, we emphasize that this paper focuses on introducing realistic benchmarks by simulating spreading, which can model the spread of brand-new products, contaminants, as well as infectious diseases. Although we used simple models, such as SI and SIS models, for OOD spreading cover these broad scenarios, incorporating human contact modeling would be beneficial for conducting highly realistic simulations, particularly in the context of infectious diseases.
Why OOD detection is not approached as binary classification? OOD detection aims to identify samples that were not seen during the training phase (i.e, OOD samples) when performing classification into II) (in-distribution) classes. Since it is not always feasible to use samples from all possible classes during training, OOD detection addresses a practical scenario. In contrast, binary classification requires samples from both classes to be provided during the training phase. By leveraging the OOD detection approach, we can handle completely new samples (e.g., COVID-19, brand-new products, or computer viruses) without any prior information.
OOD Detection vs. Outlier Detection OOD detection and outlier detection may appear similar but fundamentally different problems, with the main difference lying in their target for detection. OOD detection identifies unseen test samples that differ from the training data distribution. In contrast, outlier detection directly processes all observations and aims to identify outliers within contaminated dataset (Yang et al., 2021). In other words, the term "in-distribution" (ID) in outlier detection refers to the majority of the observations, whereas for OOD detection, ID refers specifically to the training data.
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Reviews:
[{"review_id": "HTH5MkIbfQ", "reviewer": "Reviewer_cFt1", "summary": "This research addresses a significant gap in node-level out-of-distribution (OOD) detection by introducing a novel task that considers the interactions among nodes in a graph. Previous methods have relied on unrealistic benchmarks that randomly select OOD nodes, which do not reflect real-world scenarios where OOD properties can spread among neighboring nodes. The key contributions of this paper include:\n1. Introduction of Spreading OOD Detection: This new task models how newly emerged OOD nodes can influence their neighbors, simulating more realistic scenarios of OOD detection.\n2. Realistic Benchmark Creation: The authors develop benchmarks using epidemic spreading models to simulate the dynamics of OOD node interactions within graphs. \n3. “Spreading COVID-19” Dataset: A specific dataset is introduced to illustrate the practical applications of spreading OOD detection, showcasing its relevance to real-world situations.\n4. Energy Distribution-Based Detector (EDBD): The paper presents a new detection approach that employs an energy-aggregation scheme aimed at reducing the overlap in OOD scores between in-distribution (ID) and OOD nodes, enhancing detection accuracy.\n5.Extensive Experimental Validation: Results from comprehensive experiments demonstrate that the proposed EDBD approach outperforms existing state-of-the-art methods in both spreading OOD detection and traditional node-level OOD detection tasks across seven benchmark datasets.\nOverall, this work advances the field of OOD detection by incorporating the complexities of node interactions in graph structures, providing both theoretical and practical contributions.", "questions": "Theoretical Foundations of EDBD:\nQuestion: What theoretical framework supports the effectiveness of the energy-aggregation scheme in EDBD?\nSuggestion: A more detailed theoretical justification for why this approach mitigates the mixing of OOD scores would strengthen your argument and improve the method's credibility.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "This research addresses a significant gap in node-level out-of-distribution (OOD) detection by introducing a novel task that considers the interactions among nodes in a graph. Previous methods have relied on unrealistic benchmarks that randomly select OOD nodes, which do not reflect real-world scenarios where OOD properties can spread among neighboring nodes. The key contributions of this paper include:\n1. Introduction of Spreading OOD Detection: This new task models how newly emerged OOD nodes can influence their neighbors, simulating more realistic scenarios of OOD detection.\n2. Realistic Benchmark Creation: The authors develop benchmarks using epidemic spreading models to simulate the dynamics of OOD node interactions within graphs. \n3. “Spreading COVID-19” Dataset: A specific dataset is introduced to illustrate the practical applications of spreading OOD detection, showcasing its relevance to real-world situations.\n4. Energy Distribution-Based Detector (EDBD): The paper presents a new detection approach that employs an energy-aggregation scheme aimed at reducing the overlap in OOD scores between in-distribution (ID) and OOD nodes, enhancing detection accuracy.\n5.Extensive Experimental Validation: Results from comprehensive experiments demonstrate that the proposed EDBD approach outperforms existing state-of-the-art methods in both spreading OOD detection and traditional node-level OOD detection tasks across seven benchmark datasets.\nOverall, this work advances the field of OOD detection by incorporating the complexities of node interactions in graph structures, providing both theoretical and practical contributions.", "weaknesses": "Theoretical Justification for EDBD: While the energy-aggregation scheme is introduced as a key component of the EDBD, the paper would benefit from a more rigorous theoretical foundation explaining why this method effectively mitigates the mixing of OOD scores. Providing mathematical justifications or theoretical comparisons with other aggregation methods could strengthen the credibility of this contribution.\n\nSensitivity Analysis: Conducting a sensitivity analysis on the parameters of the EDBD model could provide valuable insights into how different configurations affect performance. Understanding how the model behaves under various conditions would help in fine-tuning the method for practical applications and could guide users in optimizing the approach for specific contexts.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IzFJorKQ7p", "reviewer": "Reviewer_xogV", "summary": "This paper introduces a more realistic benchmark for node-level out-of-distribution (OOD) detection on graphs, called “spreading OOD detection”, where  OOD characteristics spread across connected nodes, similar to real-world diffusion processes like disease spreading. The paper dedicates one part to defining this benchmark in detail, using the “Spreading COVID-19” dataset to show its practical value. This benchmark is different from traditional approaches that treat OOD nodes as isolated and do not consider interactions. Another part of the paper is focused on a new method, the Energy Distribution-Based Detector (EDBD), designed to address this benchmark. EDBD uses an energy-aggregation scheme to keep OOD and in-distribution (ID) scores from mixing, helping to identify OOD nodes more accurately.", "questions": "Q1: Since the dataset is central to the benchmark, it is crucial to understand how well it reflects realistic contact-driven networks for virus spreading. Information on whether the dataset’s assumptions have been validated against more realistic epidemic scenarios would also help clarify its suitability for simulating disease dynamics. Could the authors provide detailed information on the construction and structure of the LastFM dataset, including the main mechanisms of edge formation, the presence of high-degree nodes that could act as super-spreaders, and the dataset’s time span in relation to episode durations required to reach 75% node infection? \n\nQ2: Could the authors discuss the sensitivity of the proposed method to variations in the SIS model parameters and its adaptability to different diffusion patterns? Understanding this sensitivity is important for evaluating the method’s robustness in various scenarios. Additionally, could the authors clarify why OOD detection is not approached as a binary classification between ID and OOD, or as a one-class classification in cases where only ID labels are available? Addressing these points would provide valuable insights into the generalizability of the benchmark and the chosen approach for OOD detection.\n\nQ3: Given the high error bars in Tables 2 and 3, how conclusive are the results? Could the authors explain if the similar performance of EDBD and the Energy method suggests that data features, rather than graph structure, drive the performance? Also, since Eq. 4 of the paper is quite similar to Eq. 7 of GNNSAFE, could the authors clarify the exact differences between EDBD and GNNSAFE beyond adding the energy similarity and consistency matrices? For example, does the first row in Table 4 represent GNNSAFE?\n\nQ4: Since hyperparameter tuning can greatly impact performance, could the authors explain how it was done for both EDBD and baseline methods? Was the search for competitors restricted compared to EDBD? For GNNSAFE, was only the supervised loss used, or did the authors include GNNSAFE++ with energy regularization? If only the supervised loss was used, what motivated this choice?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "Although the benchmark is presented as realistic, it relies on assumptions in simulations that may be hard to verify, particularly given the use of the LastFM dataset, which may not accurately represent human contact patterns essential for modeling disease spread. The suitability of LastFM’s social connections as a proxy for physical contact is uncertain, as are other factors like spatial constraints and the presence of high-degree nodes that could act as unrealistic “super-spreaders.” To address this concern and enhance the benchmark’s realism, the authors could validate these assumptions against datasets explicitly designed for human contact networks (such as those used in epidemiology studies) or test the model on multiple datasets with diverse structural properties to better reflect real-world contact interactions.\n\nThe EDBD method’s performance appears closely tied to assumptions within the benchmark setup, making it difficult to assess its independent technical relevance. The method seems to build on GNNSAFE with the addition of energy similarity and consistency matrices, yet these connections and differences are not fully discussed. To clarify the contribution, the authors could provide a more detailed comparison between EDBD and GNNSAFE, explaining how the added matrices influence performance and to what extent the results depend on benchmark-specific assumptions.", "weaknesses": "Although the benchmark is presented as realistic, it relies on assumptions in simulations that may be hard to verify, particularly given the use of the LastFM dataset, which may not accurately represent human contact patterns essential for modeling disease spread. The literature in this area includes foundational works from over a decade ago, which should be discussed, e.g. [1,2,3].\nThe suitability of LastFM’s social connections as a proxy for physical contact is uncertain, as are other factors like spatial constraints and the presence of high-degree nodes that could act as unrealistic “super-spreaders.” To address this concern and enhance the benchmark’s realism, the authors could validate these assumptions against datasets explicitly designed for human contact networks (such as those used in epidemiology studies) or test the model on multiple datasets with diverse structural properties to better reflect real-world interactions.\n\nThe EDBD method’s performance appears closely tied to assumptions within the benchmark setup, making it difficult to assess its technical relevance. The method seems to build on GNNSAFE with the addition of energy similarity and consistency matrices, yet these connections and differences are not fully discussed. To clarify the contribution, the authors could provide a more detailed comparison between EDBD and GNNSAFE, explaining how the added matrices influence performance and to what extent the results depend on benchmark-specific assumptions.\n\n[1] Mossong, J., Hens, N., Jit, M., Beutels, P., Auranen, K., Mikolajczyk, R., ... & Edmunds, W. J. (2008). Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS medicine, 5(3), e74.   \n[2] Balcan, D., Colizza, V., Gonçalves, B., Hu, H., Ramasco, J. J., & Vespignani, A. (2009). Multiscale mobility networks and the spatial spreading of infectious diseases. Proceedings of the national academy of sciences, 106(51), 21484-21489.   \n[3] Pastor-Satorras, R., Castellano, C., Van Mieghem, P., & Vespignani, A. (2015). Epidemic processes in complex networks. Reviews of modern physics, 87(3), 925-979.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IBSIumW7ZM", "reviewer": "Reviewer_8nsJ", "summary": "This paper introduces \"spreading OOD detection,\" for node-level out-of-distribution (OOD) detection on graphs. It addresses limitations in existing methods by modeling the spread of OOD samples through graph structures. The authors propose a new dataset, \"Spreading COVID-19,\" and present a novel approach called Energy Distribution-Based Detector (EDBD) that outperforms existing methods in both spreading and conventional OOD detection tasks.", "questions": "1. Are there any specific graph structures or spreading patterns where EDBD might underperform compared to other methods?\n2. How sensitive is the performance of EDBD to the choice of hyperparameters in the energy-aggregation scheme?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The concept of spreading OOD detection is novel and addresses a gap in existing methods.\n2. A new dataset is introduced which will help future research\n3. The figures used in the paper manage to illustrate the problem and approach well.", "weaknesses": "1. It would be interesting to see comparison of EDBD with SOTA in terms of computational complexity and scalability.\n2. Some discussion on how the proposed dataset compares to real-world datasets and its characteristics would benefit the paper", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7xuQCTYjoV", "reviewer": "Reviewer_t4Yw", "summary": "This paper introduces a novel task called spreading out-of-distribution (OOD) detection, which focuses on identifying OOD nodes within graph structures, particularly emphasizing the interactions and influence these nodes have on their neighbors. The authors leverage epidemic spreading models to create realistic benchmarks that reflect the dynamics of OOD node propagation in real-world scenarios. The authors propose an innovative approach known as the Energy Distribution-Based Detector (EDBD), which utilizes energy scores to enhance the aggregation process of node information. This method aims to mitigate the mixing of OOD scores between in-distribution (ID) and OOD nodes, thereby improving the accuracy of OOD detection.", "questions": "1. What is the sensitivity of EDBD to different hyperparameter settings? How do variations in these parameters affect the model's performance?\n2. How does EDBD perform on larger graphs or more complex networks? Are there any scalability issues that arise when applying the method to real-world scenarios with extensive node interactions?\n3. How interpretable are the results produced by EDBD? Can the authors provide insights into why certain nodes are classified as OOD based on the energy aggregation process?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1.The creation of the Spreading COVID-19 dataset provides a realistic and relevant context for evaluating OOD detection methods. This dataset simulates real-world scenarios, making the findings more applicable to practical situations.\n2. The authors conducted comprehensive experiments, comparing EDBD against several state-of-the-art methods across multiple datasets.\n3. The paper is well-structured and clearly presents the methodology, experimental setup, and results, making it accessible for readers and researchers interested in the topic.", "weaknesses": "1. While the results demonstrate superior performance, the paper may lack comprehensive statistical analysis (e.g., significance testing) to support the claims of superiority over existing methods.\n2.The energy-based aggregation approach may introduce additional complexity compared to simpler methods. This complexity could make the method less accessible for practitioners who may prefer more straightforward solutions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JvIOGZthoo", "reviewer": "Reviewer_peqA", "summary": "This paper formulates spreading OOD detection, a new benchmark setup for node-level OOD detection method that incorporates interactions among OOD samples and establishes a new dataset called Spreading COVID-19 to benchmark the spreading OOD detection scenario in real-world data.\n\nA node-level OOD detector called EDBD employs an attentive energy aggregation scheme that prevents the mixing of energies between ID nodes and OOD nodes. EDBD is applied to enhance the OOD detection.", "questions": "**Q1.** How might the proposed spreading OOD detection model adapt to more complex epidemic models, such as SEIR (Susceptible-Exposed-Infectious-Recovered) or multi-host transmission models? Are there specific limitations or computational challenges that prevent their integration?\n\n**Q2.** What steps could be taken to ensure the model’s robustness on graphs with missing or noisy edges and nodes? Would incorporating graph-denoising techniques or imputing missing links improve EDBD’s performance?\n\n**Q3.** Have alternative case studies, such as the spread of misinformation or emergent diseases, been considered to showcase the model’s versatility? How might these cases impact the model's applicability and reception in current contexts?\n\n**Q4.** What’s the relationship of this new problem with other problems, e.g., node classification or outlier detection on graphs?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "**S1.**  This paper proposes a new benchmark that models the realistic spread of OOD properties across graph nodes. This setup provides a more applicable framework for real-world scenarios.\n\n**S2.**  This paper leverages EDBD to prevent unwanted mixing of OOD and in-distribution (ID) nodes. By controlling the influence of energy distribution, EDBD achieves better OOD discrimination on proposed datasets and spreading situations.\n\n**S3.**  The paper's aggregation scheme in EDBD is versatile and adaptable to various scenarios on graph-structured data.", "weaknesses": "**W1.**  The paper’s reliance on COVID-19 to introduce and validate the spreading OOD detection model may feel outdated. As COVID-19 has become less urgent as a topic, using it as the central example could reduce the model's perceived relevance. A more contemporary application, such as misinformation spread or novel pathogens, might enhance the model's applicability and appeal\n\n**W2.** What’s the relationship of this new problem with other problems, e.g., node classification or outlier detection on graphs? I’m not sure what’s the main difference between this problem and the outlier detection problem. The outliers are not seen in the training set? Can these algorithms be used as the baselines? \n\n**W3.** The effectiveness of the EDBD approach heavily depends on the graph structure and quality of connectivity information. In cases where graph structures are incomplete or noisy, the method’s performance may degrade, potentially affecting its robustness in real-world scenarios.\n\n**W4.** While the paper employs SI and SIS models for simulating OOD spreading, it leaves the integration of more complex epidemic models for future work. This limits the current approach's applicability to more intricate real-world spreading phenomena that may not be accurately represented by simple models.\n\n**W5.** Although the \"Spreading COVID-19\" dataset simulates virus spread, the lack of training on actual epidemic or contagion data might reduce the generalizability of the model when faced with real-world data that includes nuanced spreading patterns not captured by synthetic simulations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)