indicates a better similarity between the clusterings, with a maximum value of
1 indicating identical clusterings.

####  6.2.5. ARI (Adjusted Rand Index):

ARI is another metric for assessing the agreement between two clusterings. It
adjusts the Rand index to account for the expected value of random clustering.
ARI yields a score between -1 and 1, where higher values indicate a better
agreement between the clusterings, 0 represents a random agreement, and
negative values indicate worse than random chance.

####  6.2.6. NDCG (Normalized Discounted Cumulative Gain):

is a metric used to evaluate the quality of a ranked list of items, often in
the context of information retrieval or recommendation systems. This metric
considers both the predicted scores for the items and their graded relevance
values, typically real-valued and non-negative ground truth scores. NDCG
measures how well a ranking approach has performed in presenting the most
relevant items at the top of the list. It is crucial to acknowledge that the
metrics utilized in anomaly detection (AD) techniques for dynamic graphs
extend beyond those listed in Table 6. A more specialized analysis is
essential for comprehensive performance evaluation, as anomaly detection
entails diverse requirements specific to applications and network topologies,
as shown in Ma et al. (Ma et al., 2021).

Table 6. Commonly Used Evaluation Metrics in Literature Metrics |  Formula
---|---
Accuracy |  $\textit{Acc.}=\frac{(TN+TP)}{(TN+FN+FP+TP)}$
Precision |  $\textit{Prec.}=\frac{TP}{(TP+FP)}$
Recall/TPR |  $Rec.=\frac{TP}{(TP+FN)}$
F-1 score |  $F1=2\times\frac{\textit{Recall }\times\textit{Precision}}{(\textit{Recall }+\textit{Precision})}$
Specificity (TNR) |  TNR = 1 - FP
AUC |  Area Under ROC curve
Hits$@n$ |  $\frac{\textit{ \\# of detected anomalies at top n}}{n}$
MRR |  $\textit{MRR}=\frac{1}{|U_{\textit{all}}|}\sum_{u=1}^{|U_{\textit{all}}|}RR(u)$ $RR(u)=\sum_{i\leq L}\frac{\textit{relevance}}{\textit{rank}_{i}}$ $|U_{all}|$: total number of users
MAE |  $\textit{MAE}=\frac{1}{n}\sum_{i=1}^{n}\big{\lvert}y_{i}-y_{i}^{\textit{pred}}\big{\rvert}$
NMI |  $\textit{NMI (Y, C)}=\frac{2\times I(Y;C)}{[H(Y)+H(C)]}$ Y: labels, C: clusters, H(.): Entropy, I(Y;C): mutual information b/w Y and C
NDCG |  $\textit{DCG}@K=\sum_{i=1}^{k}\frac{r_{i}}{\log_{2}(i+1)}$
*$r_{i}$ in $\textit{DCG}@K$ is the graded relevance of the node at $i$
*RR (u): reciprocal rank of a user $u$ and the sum score for top $L$

##  7\. Challenges and Future Directions

The field of graph representation learning and knowledge graphing (KG) has
experienced rapid growth and attracted wide research attention over the past
two decades. The recent increase in publications in prestigious Artificial
Intelligence venues, as highlighted in Section 5, indicates that this trend is
apparent not only in the static graph representation but also in dynamic
graphs.

However, modeling graphs and detecting anomalous patterns in both discrete and
continuously evolving graph structures remains a prominent concern, shaping
potential future research directions. In this context, we examine common
challenges and pinpoint potential directions for future research.

###  7.1. Modeling Temporal Dynamics and Concept Drift

Despite the increasing volume of research in graph representation and
learning, anomaly detection in dynamic environments remains a challenging
task, particularly in the context of modeling temporal-evolving networks and
addressing concept drift.

Temporal dynamics: Researchers need to develop robust algorithms capable of
adapting to evolving graph structures over time, such as inductive learning
algorithms. For instance, detecting self-propagating malware, such as
polymorphic worms that independently replicate and spread across computer
networks, is a hot topic in cybersecurity and network analysis. Researchers
could focus on creating more robust heuristic approaches with adaptive
adjustment properties and incorporating behavioral analysis to detect and
defend against such sophisticated attacks.

Concept Drift: Researchers need to investigate adaptive and hybrid algorithms
for capturing dynamic graph structures by constantly adjusting to shifting
data patterns and demonstrating real-time responsiveness. Additionally, there
is a need to develop novel evaluation metrics that can measure the slight
drift in the evolving nature of anomalies.

###  7.2. Scalability

Dynamic network modeling faces challenges when dealing with large-scale graph
datasets that have a high volume of nodes and edges. To address these
challenges, there is a need for robust and scalable algorithms for real-time
anomaly detection. The influx of data in streaming graphs makes modeling slow,
less accurate, and computationally expensive. For example, approaches like
(Yuan et al., 2023; You et al., 2022; Huang et al., 2020; Wang et al., 2017;
Goyal et al., 2018; Liu et al., 2021) focus on learning graph patterns at each
snapshot. However, this could lead to a high computational space as the
network expands. Additionally, frequent snapshots could compromise the
accurate modeling of temporal networks (Skarding et al., 2021).

Key questions on scalability: How can dynamic graph models scale to adapt to
continuous input stream length? What is the processing time per input node or
edge compared to baseline approaches? Recent techniques like SedanSpot
(Eswaran and Faloutsos, 2018), which applied sub-processes like hashing,
random walk, and sampling algorithms in sublinear time $O(\log n)$, resulted
in a large computation time. PENminer (Belth et al., 2020) and F-FADE (Chang
et al., 2021), employing active pattern exploration and expensive frequency
factorization operations, respectively, also resulted in large computation
times. MIDAS-R (Bhatia et al., 2022), on the other hand, improved the
complexity of streaming graph algorithms by applying the CMS (count-min
sketch) (Ranshous et al., 2016) hashing data structure in constant time and
memory. However, it still struggles when the graph stream experiences
exponential growth, exhibiting suboptimal performance in subgraph and graph-
level anomaly detection tasks.

For future work, researchers could focus on improving continuous-time modeling
by exploring distributed and parallel algorithms to address scalability issues
associated with dynamic graphs and developing efficient data structures, as
well as exploring algorithms like Count-Min Sketch with Conservative Update
(CMSCU), FM Sketch (Flajolet-Martin Sketch), Lossy Counting, and other
lightweight data structures that allow for more memory-efficient
representations of streaming data.

###  7.3. Multi-view Graph Anomaly Detection

Multi-view anomaly detection refers to the task of identifying outliers in
data represented as a graph with multiple views. Each view provides a distinct
set of features associated with the nodes and edges of the graph. Researchers
could focus on key concepts such as graph heterogeneity (i.e., the presence of
diverse types of nodes, edges, or attributes within a graph), the integration
of diverse views, ensuring proper alignment, and adapting multi-view anomaly
detection algorithms into dynamic settings.

Exploring multi-view graph anomaly detection holds significant promise for
future research directions in dynamic graph anomaly detection. Researchers
could leverage multiple views of graphs to gain deeper insights into complex
systems and uncover hidden anomalies that may not be apparent in single-view
analyses (Huang et al., 2023a). One key example is addressing graph
heterogeneity, where the presence of different types of nodes, edges, or
attributes within a graph leads to multi-layer anomalies (Xie et al., 2023).
Furthermore, adapting these multi-view algorithms to dynamic settings could
open up a new research direction for understanding temporal anomaly patterns
and detecting evolving threats in real-time. Consequently, future research in
this direction could lead to the development of more robust and versatile
anomaly detection techniques capable of addressing the evolving challenges in
dynamic graph data analysis.

Multi-view detection algorithms are applied in various domains, including
social networks (Chen et al., 2023), time-series (Teng et al., 2017),
cybersecurity (Huang et al., 2023a), and fraud detection (Zhang et al.,
2024c), where a holistic understanding of complex relationships is crucial.
Recent works, such as MultiLAD (Huang et al., 2023a; Xie et al., 2023) and
MSTREAM (Bhatia et al., 2021), have targeted the detection of change point
anomalies in multi-view graphs and group anomalies in multi-aspect data,
respectively. However, this is an evolving research domain in dynamic graph
learning.

###  7.4. Multi-task Anomaly Detection

Multi-task algorithms refer to models that simultaneously identify anomalous
patterns across multiple related tasks or domains. Many approaches have
focused on specific graph tasks, such as community detection (Yuan et al.,
2023; Alsentzer et al., 2020), node detection (Huang et al., 2020; Tian et
al., 2023; Goodge et al., 2022), and edge- or link-level prediction (Bhatia et
al., 2022; Chang et al., 2021; Bhatia et al., 2023). However, in a complex and
dynamic network, there may be two or more types of anomalies, posing a
significant threat to critical infrastructure in cybersecurity. Attackers
could potentially exploit the system with multiple kinds of attacks to bypass
detection models.

To address this challenge, researchers could focus on developing dynamic
fusion models that adaptively integrate information from multiple anomaly
detection tasks. Additionally, future work could design algorithms that are
task-aware, considering the unique properties within the network. Also, it’s a
good idea to encourage experts from different fields to work together on
anomaly detection, dynamic graph theory, and improving multi-task anomaly
detection systems.

###  7.5. Graph Theoretical Foundation and Explainability

Most existing anomaly detection algorithms for dynamic graphs are designed and
evaluated through empirical experiments, lacking sufficient theoretical
foundations to verify their reliability. Consequently, there is a tendency to
overlook the explainability of the learned representations and detection
results. For example, understanding which nodes, edge features, and adjacency
matrices are most crucial in the graph or which edges or links predominantly
influence the drift in network changes. Relying solely on anomaly scores may
not be sufficient to conclusively determine if network traffic is anomalous or
not.

Therefore, we believe that future work should focus on exploring the
foundational knowledge of graph theory for modeling dynamic graph
relationships and identifying patterns. This emphasis on the theoretical
aspect of graphs will pave the way for new directions in model
interpretability and advanced visual analytics.

Additionally, future models could incorporate human-in-the-loop approaches to
enhance the explainability and interpretability of dynamic graph algorithms.
This holistic approach aims to bridge the gap between empirical evaluations
and theoretical foundations, fostering a more comprehensive and reliable
understanding of anomaly detection in dynamic graphs.

###  7.6. Adversarial of Graph Models and Data Privacy

The increase in graph-based research has also attracted a wide range of
sophisticated attacks on graph-based models due to the availability and mining
of big data from real-world networks (Kyle et al., 2023; Ahmed et al., 2022).
Although several adversarial-resistant models have been developed recently,
graph models are highly susceptible to structural adversarial attacks that can
manipulate node or edge features to deceive the model into making incorrect
detections (Ma et al., 2021; Kim et al., 2022). Graph-based models are
particularly prone to evasion attacks involving subtle data modifications,
data poisoning attacks, and data privacy concerns.

Future work could explore the development of techniques to mitigate such
attacks and implement privacy-preserving techniques, such as differential
privacy, to safeguard sensitive information in graph datasets.

###  7.7. Fairness in Graph Anomaly Detection

Fairness refers to the equal and impartial treatment of individuals or
societal groups by an AI system. Addressing fairness in anomaly detection is
crucial for ensuring equitable and effective models, especially in real-world
applications. For instance, consider a fraud detection model that relies on
historical data to predict anomalous transactions. If this data is biased,
reflecting systemic discrimination against a specific demographic group, the
model may flag potential transactions from that group. This could result in
the unfair denial of legitimate transactions for individuals within that
demographic or geographical segment.

Recent articles have explored these aspects in various contexts. Notable works
include SRGNN by Zhang et al. (Zhang et al., 2024a) in 2024. This study
addresses fairness issues related to sensitive node attributes by considering
the impact of both low-degree and high-degree graph nodes in the GNN model for
learning fair representations in decision-making. Furthermore, SRGNN employs
adversarial learning to acquire fair representations through gradient
normalization, ensuring the separation of each node’s representation from
sensitive attribute information. Likewise, in 2024, Ling et al. (Ling et al.,
2024) addressed the problem of fair feature selection for classification
decision tasks in static graphs. The authors propose a fair causal feature
selection algorithm called FairCFS. Specifically, FairCFS constructs a
localized causal graph that identifies the Markov blankets of class and
sensitive variables to block the transmission of sensitive information for
selecting fair causal features.

In future research, it is important to integrate fairness into dynamic graph
anomaly detection models. This involves not only considering the technical
aspects of anomaly detection but also examining how these models may impact
diverse communities and addressing potential biases. The development of
algorithms that are fair, transparent, and unbiased will contribute to the
responsible and ethical deployment of anomaly detection systems in various
domains.

###  7.8. Cost Sensitivity in Graph Anomaly Detection

Cost sensitivity in anomaly detection involves accounting for varying costs
associated with misclassifying anomalies or normal instances. This is
particularly relevant in scenarios where the consequences of false positives
or false negatives differ significantly. For instance, consider a credit card
fraud detection system. The cost associated with a false negative (allowing
fraud) is typically much higher than the cost of a false positive (blocking a
legitimate transaction). Therefore, the anomaly detection models need to be
cost-sensitive to prioritize minimizing false negatives, even if it means
accepting a higher rate of false positives.

Several works have explored the concept of cost sensitivity, including the
works of Zhang et al. (Zhang, 2018). The authors propose a general target-
resource framework involving multiple kinds of cost scales that minimize one
kind of cost scale (called target cost scale) while controlling the others
(called resource cost scales) in given resource budgets. Similarly, Huang et
al. (Huang et al., 2023c) introduce a multivariate fusion prediction system
that tackles the extraction of predictive information from multi-scale
information systems. These approaches helped in assessing the data features
more comprehensively and globally and highlighted the superiority degree
between different samples.

In future research, it is important to integrate cost sensitivity into dynamic
graph anomaly detection models. This involves designing algorithms that are
not only accurate but also consider the economic implications of
misclassifications. Developing models that are cost-sensitive will contribute
to the practical and efficient deployment of anomaly detection systems in
various domains.

###  7.9. Further Research Directions

Other open challenges and future work include, but are not limited to, the
following:

  * •

Designing faster data streaming graph models, considering the tradeoff between
speed and memory size.

  * •

Diversifying graph models for new applications, including environmental
monitoring, medical data, and dynamic data streams.

  * •

Exploring hybrid approaches (integration of deep learning models with
streaming data structures)

  * •

Addressing the imbalance problem in graph datasets

  * •

Tackling other emerging research topics in graph representation learning

##  8\. Conclusion

Due to the growing interest in research on graph representation learning and
anomaly detection (AD) in dynamic graphs, we have conducted a comprehensive
survey of existing AD methods in dynamic graphs. To the best of our knowledge,
this is the most recent and holistic survey dedicated to anomaly detection in
dynamic graphs, covering a wide range of modern techniques.

In Section 2, we provided a concrete mathematical background and explained the
different types of anomalies that can occur in both static and dynamic graphs.
Section 3 discussed classical graph representation learning, specifically the
GNN architecture, while Section 4 delved into dynamic graph representations.
This set the stage for our survey on anomaly detection techniques in dynamic
graphs. In Section 5, we reviewed and categorized current AD techniques,
including (1) traditional machine learning (tree-based, density-based, and
distance-based); (2) matrix factorization approaches; (3) probabilistic
approaches; and (4) deep learning approaches. We presented a detailed summary
and comparison of different anomaly detection techniques, current trends, and
limitations. Furthermore, to aid future research advancement in this field, we
presented a systematic timeline illustrating the chronological progression of
all reviewed techniques. In Section 6, we also conducted a structured
benchmarking of commonly used datasets (both real-world and synthetic data)
and provided commonly used evaluation metrics for dynamic graph models.
Finally, in Section 7, we highlighted potential research directions for future
work based on the survey results.

We are optimistic that the rapid increase in research associated with dynamic
graph learning will benefit numerous applications from diverse domains, and
this survey provides a valuable contribution.

###### Acknowledgements.

The authors wish to thank the College of Engineering, the Machine Intelligence
and Data Science (MInDS) Center, and the Department of Computer Science at
Tennessee Tech University for providing resources and funding to work on this
project.

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