Quantifying the Complexity of Nodes in Higher-Order Networks Using the Infomap Algorithm

Accurately quantifying the complexity of nodes in a network is crucial for revealing their roles and network complexity, as well as predicting network emergent phenomena. In this paper, we propose three novel complexity metrics for nodes to reflect the extent to which they participate in organized, structured interactions in higher-order networks. Our higher-order network is built using the BuildHON+ model, where communities are detected using the Infomap algorithm. Since a physical node may contain one or more higher-order nodes in higher-order networks, it may simultaneously exist in one or more communities. The complexity of a physical node is defined by the number and size of the communities to which it belongs, as well as the number of higher-order nodes it contains within the same community. Empirical flow datasets are used to evaluate the effectiveness of the proposed metrics, and the results demonstrate their efficacy in characterizing node complexity in higher-order networks.