It is demonstrated that higher-order patterns beyond pairwise relations can significantly enhance the learning capability of existing graph-based models, and simplex is one of the primary form for graphically representing higher-order patterns. Predicting unknown (disappeared) simplices in real-world complex networks can provide us with deeper insights, thereby assisting us in making better decisions. Nevertheless, previous efforts to predict simplices suffer from two issues: (i) they mainly focus on 2- or 3-simplices, and there are few models available for predicting simplices of arbitrary orders, and (ii) they lack the ability to analyze and learn the features of simplices from the perspective of dynamics. In this paper, we present a Higher-order Neurodynamical Equation for Simplex Prediction of arbitrary order (HNESP), which is a framework that combines neural networks and neurodynamics. Specifically, HNESP simulates the dynamical coupling process of nodes in simplicial complexes through different relations (i.e., strong pairwise relation, weak pairwise relation, and simplex) to learn node-level representations, while explaining the learning mechanism of neural networks from neurodynamics. To enrich the higher-order information contained in simplices, we exploit the entropy and normalized multivariate mutual information of different sub-structures of simplices to acquire simplex-level representations. Furthermore, simplex-level representations and multi-layer perceptron are used to quantify the existence probability of simplices. The effectiveness of HNESP is demonstrated by extensive simulations on seven higher-order benchmarks. Experimental results show that HNESP improves the AUC values of the state-of-the-art baselines by an average of 8.32%. Our implementations will be publicly available at: https://github.com/jianruichen/HNESP.
Keywords: Higher-order information; Mutual information; Neurodynamical equation; Representation learning; Simplex prediction.
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