Node ranking based on graph curvature and PageRank

Identifying key nodes in complex networks is crucial for understanding and controlling their dynamics. Traditional centrality measures often fall short in capturing the multifaceted roles of nodes within these networks. The PageRank algorithm, widely recognized for ranking web pages, offers a more nuanced approach by considering the importance of connected nodes. However, existing methods generally overlook the geometric properties of networks, which can provide additional insights into their structure and functionality. In this paper, we propose a novel method named Curv-PageRank (C-PR), which integrates network curvature and PageRank to identify influential nodes in complex networks. By leveraging the geometric insights provided by curvature alongside structural properties, C-PR offers a more comprehensive measure of a node's influence. Our approach is particularly effective in networks with community structures, where it excels at pinpointing bridge nodes critical for maintaining connectivity and facilitating information flow. We validate the effectiveness of C-PR through extensive experiments. The results demonstrate that C-PR outperforms traditional centrality-based and PageRank methods in identifying critical nodes. Our findings offer fresh insights into the structural importance of nodes across diverse network configurations, highlighting the potential of incorporating geometric properties into network analysis.