An improved gravity centrality for finding important nodes in multi-layer networks based on multi-PageRank

How to identify important nodes in multi-layer networks is still an unresolved issue in network science, which has aroused the interest of many researchers. In addition, the relationships between entities in many real-world systems are diverse and can be modeled as multi-layer networks. In the past few decades, scholars have defined various centrality methods from different perspectives to find influential nodes in multi-layer networks, but they only utilize the local or global topology information. Recently, various gravity centralities that utilize both the local and global topological structure information have been defined for identifying key nodes in single-layer networks. In the gravity model, the interaction between two nodes is related to their mass and distance. In consideration of the advantages of gravity model, in this paper, we define an improved gravity centrality for identifying key nodes in multi-layer networks based on multi-PageRank centrality, referred to as the PRGC. Unlike the existing gravity centralities that treat each node degree as its mass, our proposed centrality views the multi-PageRank centrality value of each node as its mass. Furthermore, PRGC weights the shortest path distance between any two nodes across all network layers to define their distance in multi-layer networks. Fianlly, to illustrate the effectiveness and superiority of the proposed centrality approach, numerical experiments are conducted on six real-world multi-layer networks show that our proposed centrality method outperforms state-of-the-art centralities.