  In the node classification task, it is intuitively understood that densely
connected nodes tend to exhibit similar attributes. However, it is crucial to
first define what constitutes a dense connection and to develop a reliable
mathematical tool for assessing node cohesiveness. In this paper, we propose a
probability-based objective function for semi-supervised node classification
that takes advantage of higher-order networks' capabilities. The proposed
function embodies the philosophy most aligned with the intuition behind
classifying within higher-order networks, as it is designed to reduce the
likelihood of nodes interconnected through higher-order networks bearing
different labels. We evaluate the function using both balanced and imbalanced
datasets generated by the Planted Partition Model (PPM), as well as a
real-world political book dataset. According to the results, in challenging
classification contexts characterized by low homo-connection probability, high
hetero-connection probability, and limited prior information of nodes,
higher-order networks outperform pairwise interactions in terms of objective
function performance. Notably, the objective function exhibits elevated Recall
and F1-score relative to Precision in the imbalanced dataset, indicating its
potential applicability in many domains where detecting false negatives is
critical, even at the expense of some false positives.
