GAD methods when they are adapted to the semi-supervised setting.

Despite the importance of graph OD and many algorithms being developed for it in recent years, there is no comprehensive benchmark on graph outlier detection, which we believe has hindered the development and understanding of graph OD algorithms.

Structured data, such as graphs, are vital in machine learning due to their capacity to capture complex relationships and interactions. In recent years, the Fused Gromov-Wasserstein (FGW) distance has attracted growing interest because it enables the comparison of structured data by jointly accounting for feature similarity and geometric structure. However, as a variant of optimal transport (OT), classical FGW assumes an equal mass constraint on the compared data. In this work, we relax this mass constraint and propose the Fused Partial Gromov-Wasserstein (FPGW) framework, which extends FGW to accommodate unbalanced data. Theoretically, we establish the relationship between FPGW and FGW and prove the metric properties of FPGW. Numerically, we introduce Frank-Wolfe solvers for the proposed FPGW framework and provide a convergence analysis. Finally, we evaluate the FPGW distance through graph classification and clustering experiments, demonstrating its robust performance, especially when data is corrupted by outlier noise.

This work considers a practical semi-supervised graph anomaly detection (GAD) scenario, where part of the nodes in a graph are known to be normal, contrasting to the extensively explored unsupervised setting with a fully unlabeled graph. We reveal that having access to the normal nodes, even just a small percentage of normal nodes, helps enhance the detection performance of existing unsupervised GAD methods when they are adapted to the semi-supervised setting. However, their utilization of these normal nodes is limited. In this paper, we propose a novel Generative GAD approach (namely GGAD) for the semi-supervised scenario to better exploit the normal nodes. The key idea is to generate pseudo anomaly nodes, referred to as 'outlier nodes', for providing effective negative node samples in training a discriminative one-class classifier. The main challenge here lies in the lack of ground truth information about real anomaly nodes. To address this challenge, GGAD is designed to leverage two important priors about the anomaly nodes -- asymmetric local affinity and egocentric closeness -- to generate reliable outlier nodes that assimilate anomaly nodes in both graph structure and feature representations. Comprehensive experiments on six real-world GAD datasets are performed to establish a benchmark for semi-supervised GAD and show that GGAD substantially outperforms state-of-the-art unsupervised and semi-supervised GAD methods with varying numbers of training normal nodes. Code will be made available at https://github.com/mala-lab/GGAD.

Detecting which nodes in graphs are outliers is a relatively new machine learning task with numerous applications. Despite the proliferation of algorithms developed in recent years for this task, there has been no standard comprehensive setting for performance evaluation. Consequently, it has been difficult to understand which methods work well and when under a broad range of settings. To bridge this gap, we present--to the best of our knowledge--the first comprehensive benchmark for unsupervised outlier node detection on static attributed graphs called BOND, with the following highlights. (1) We benchmark the outlier detection performance of 14 methods ranging from classical matrix factorization to the latest graph neural networks. (2) Using nine real datasets, our benchmark assesses how the different detection methods respond to two major types of synthetic outliers and separately to "organic" (real non-synthetic) outliers. (3) Using an existing random graph generation technique, we produce a family of synthetically generated datasets of different graph sizes that enable us to compare the running time and memory usage of the different outlier detection algorithms. Based on our experimental results, we discuss the pros and cons of existing graph outlier detection algorithms, and we highlight opportunities for future research. Importantly, our code is freely available and meant to be easily extendable: https://github.com/pygod-team/pygod/tree/main/benchmark

Network (or graph) embedding is the task to map the nodes of a graph to a lower dimensional vector space, such that it preserves the graph properties and facilitates the downstream network mining tasks. Real world networks often come with (community) outlier nodes, which behave differently from the regular nodes of the community. These outlier nodes can affect the embedding of the regular nodes, if not handled carefully. In this paper, we propose a novel unsupervised graph embedding approach (called DMGD) which integrates outlier and community detection with node embedding. We extend the idea of deep support vector data description to the framework of graph embedding when there are multiple communities present in the given network, and an outlier is characterized relative to its community. We also show the theoretical bounds on the number of outliers detected by DMGD. Our formulation boils down to an interesting minimax game between the outliers, community assignments and the node embedding function. We also propose an efficient algorithm to solve this optimization framework. Experimental results on both synthetic and real world networks show the merit of our approach compared to state-of-the-arts.