Traditional network analysis focuses on the representation of complex systems with only pairwise interactions between nodes. However, the higher-order structure, which is beyond pairwise interactions, has a great influence on both network dynamics and function. Ranking cliques could help understand more emergent dynamical phenomena in large-scale complex networks with higher-order structures, regarding important issues, such as behavioral synchronization, dynamical evolution, and epidemic spreading. In this paper, motivated by multi-node interactions in a topological simplex, several higher-order centralities are proposed, namely, higher-order cycle (HOC) ratio, higher-order degree, higher-order H-index, and higher-order PageRank (HOP), to quantify and rank the importance of cliques. Experiments on both synthetic and real-world networks support that, compared with other traditional network metrics, the proposed higher-order centralities effectively reduce the dimension of a large-scale network and are more accurate in finding a set of vital nodes. Moreover, since the critical cliques ranked by the HOP and the HOC are scattered over a complex network, the HOP and the HOC outperform other metrics in ranking cliques that are vital in maintaining the network connectivity, thereby facilitating network dynamical synchronization and virus spread control in applications.
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