In this paper, we propose HLSAD, a novel method for detecting anomalies in time-evolving simplicial complexes. While traditional graph anomaly detection techniques have been extensively studied, they often fail to capture changes in higher-order interactions that are crucial for identifying complex structural anomalies. These higher-order interactions can arise either directly from the underlying data itself or through graph lifting techniques. Our approach leverages the spectral properties of Hodge Laplacians of simplicial complexes to effectively model multi-way interactions among data points. By incorporating higher-dimensional simplicial structures into our method, our method enhances both detection accuracy and computational efficiency. Through comprehensive experiments on both synthetic and real-world datasets, we demonstrate that our approach outperforms existing graph methods in detecting both events and change points.

In this paper, we extend the classical Color Refinement algorithm for static networks to temporal (undirected and directed) networks. This enables us to design an algorithm to sample synthetic networks that preserves the d-hop neighborhood structure of a given temporal network. The higher d is chosen, the better the temporal neighborhood structure of the original network is preserved. Specifically, we provide efficient algorithms that preserve time-respecting (”causal”) paths in the networks up to length d, and scale to real-world network sizes. We validate our approach theoretically (for Degree and Katz centrality) and experimentally (for edge persistence, causal triangles, and burstiness). An experimental comparison shows that our method retains these key temporal characteristics more effectively than existing randomization methods.

We develop a new method to efficiently sample synthetic networks that preserve the d-hop neighborhood structure of a given network for any given d. The proposed algorithm trades off the diversity in network samples against the depth of the neighborhood structure that is preserved. Our key innovation is to employ a colored Configuration Model with colors derived from iterations of the so-called Color Refinement algorithm. We prove that with increasing iterations the preserved structural information increases: the generated synthetic networks and the original network become more and more similar, and are eventually indistinguishable in terms of centrality measures such as PageRank, HITS, Katz centrality and eigenvector centrality. Our work enables to efficiently generate samples with a precisely controlled similarity to the original network, especially for large networks.

We bridge two research directions on graph neural networks (GNNs), by formalizing the relation between heterophily of node labels (i.e., connected nodes tend to have dissimilar labels) and the robustness of GNNs to adversarial attacks. Our theoretical and empirical analyses show that for homophilous graph data, impactful structural attacks always lead to reduced homophily, while for heterophilous graph data the change in the homophily level depends on the node degrees. These insights have practical implications for defending against attacks on real-world graphs: we deduce that separate aggregators for ego- and neighbor-embeddings, a design principle which has been identified to significantly improve prediction for heterophilous graph data, can also offer increased robustness to GNNs. Our comprehensive experiments show that GNNs merely adopting this design achieve improved empirical and certifiable robustness compared to the best-performing unvaccinated model. Additionally, combining this design with explicit defense mechanisms against adversarial attacks leads to an improved robustness with up to 18.33% performance increase under attacks compared to the best-performing vaccinated model.

We present a graph-based semi-supervised learning (SSL) method for learning edge flows defined on a graph. Specifically, given flow measurements on a subset of edges, we want to predict the flows on the remaining edges. To this end, we develop a computational framework that imposes certain constraints on the overall flows, such as (approximate) flow conservation. These constraints render our approach different from classical graph-based SSL for vertex labels, which posits that tightly connected nodes share similar labels and leverages the graph structure accordingly to extrapolate from a few vertex labels to the unlabeled vertices. We derive bounds for our method's reconstruction error and demonstrate its strong performance on synthetic and real-world flow networks from transportation, physical infrastructure, and the Web. Furthermore, we provide two active learning algorithms for selecting informative edges on which to measure flow, which has applications for optimal sensor deployment. The first strategy selects edges to minimize the reconstruction error bound and works well on flows that are approximately divergence-free. The second approach clusters the graph and selects bottleneck edges that cross cluster-boundaries, which works well on flows with global trends.