Understanding Higher-order Structures in Evolving Graphs: A Simplicial Complex based Kernel Estimation Approach
This work provides a theoretically grounded approach for predicting higher-order interactions in evolving graphs, which is an incremental improvement over existing heuristic-based methods for researchers working with complex network analysis.
This paper addresses the prediction of higher-order interactions in dynamic graphs, which are common in co-authorship and biological networks. The authors propose a nonparametric kernel estimator for simplices, which models their neighborhood using face-vectors and views the evolving graph as a time process. Their method significantly outperforms several baseline higher-order prediction methods.
Dynamic graphs are rife with higher-order interactions, such as co-authorship relationships and protein-protein interactions in biological networks, that naturally arise between more than two nodes at once. In spite of the ubiquitous presence of such higher-order interactions, limited attention has been paid to the higher-order counterpart of the popular pairwise link prediction problem. Existing higher-order structure prediction methods are mostly based on heuristic feature extraction procedures, which work well in practice but lack theoretical guarantees. Such heuristics are primarily focused on predicting links in a static snapshot of the graph. Moreover, these heuristic-based methods fail to effectively utilize and benefit from the knowledge of latent substructures already present within the higher-order structures. In this paper, we overcome these obstacles by capturing higher-order interactions succinctly as \textit{simplices}, model their neighborhood by face-vectors, and develop a nonparametric kernel estimator for simplices that views the evolving graph from the perspective of a time process (i.e., a sequence of graph snapshots). Our method substantially outperforms several baseline higher-order prediction methods. As a theoretical achievement, we prove the consistency and asymptotic normality in terms of the Wasserstein distance of our estimator using Stein's method.