HIGHER-ORDER STRUCTURE PREDICTION IN EVOLV-ING GRAPH SIMPLICIAL COMPLEXES Anonymous

Abstract

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 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 higherorder prediction methods. As a theoretical achievement, we prove the consistency and asymptotic normality in terms of Wasserstein distance of our estimator using Stein's method.

1. INTRODUCTION

Numerous types of networks like social (Liben-Nowell & Kleinberg, 2007a) , biological (Airoldi et al., 2006) , and chemical reaction networks (Wegscheider, 1911) are highly dynamic, as they evolve and grow rapidly via the appearance of new interactions, represented as the introduction of new links / edges between the nodes of a network. Identifying the underlying mechanisms by which such networks evolve over time is a fundamental question that is not yet fully understood. Typically, insight into the temporal evolution of networks has been obtained via a classical inferential problem called link prediction, where given a snapshot of the network at time t along with its linkage pattern, the task is to assess whether a pair of nodes will be linked at a later time t > t. While inferring pairwise links is an important problem, it is oftentimes observed that most of the real-world graphs exhibit higher-order group-wise interactions that involve more than two nodes at once. Examples illustrating human group behavior involve a co-author relationship on a single paper and a network of e-mails to multiple recipients. In nature too, one can observe several proteins interacting together in a biological network simultaneously. In spite of their significance, in comparison to single edge inference, relatively fewer works have studied the problem of predicting higher-order group-wise interactions. Benson et al. (2018) originally introduced a simplex to model group-wise interactions between nodes in a graph. They proposed predicting a simplicial closure event, whereby an open simplex (with just pairwise interactions between member vertices) transitions to a closed simplex (where all member vertices participate in the higher-order relationship simultaneously), in the near future. Figure 1 (Middle) shows an example of such a transition from an open triangle to a closed one. Recently, several works have proposed modeling higher-order interactions as hyperedges in a hypergraph (Xu et al., 2013; Zhang et al., 2018; Yoon et al., 2020; Patil et al., 2020) . Given a hyperedge h t at time t, the inference task is to predict the future arrival of a new hyperedge h t , which covers a larger set of vertices than h t and contains all the vertices in h t . Figure 1 (Right) illustrates this hyperedge prediction task.

