Bi-dimensional gravity-influence model: Quantitative assessment of node performance based on individual potential and geographic location

<sec>In complex networks, the accurate assessing of node importance is essential for understanding critical structures and optimizing dynamic processes. Traditional gravity-based methods often rely on local attributes or global shortest paths, which exhibit limitations in heterogeneous networks due to insufficient differentiation of node roles and their influences in different topologies. To address these challenges, we propose the bi-dimensional gravity influence model (BGIM) and its enhanced version (BGIM+). These models introduce a novel entropy-weighted gravity framework that integrates node information entropy, gravity correction factors, and asymmetric attraction factors. By replacing degree centrality with information entropy, BGIM captures nodes’ uncertainty and information richness, offering a more comprehensive view of their potential influence.</sec><sec>The gravity correction factor (NGCF) combines eigenvector centrality with network constraint coefficients to balance global feature and local feature, while the asymmetric attraction factor (AAF) consider gravitational asymmetry between core and peripheral nodes. This bi-dimensional method can evaluate the node importance in more detail and solve the problem of imbalanced influence distribution in different network structures. A normalization mechanism further enhances adaptability, thus ensuring robust performance in both sparse and dense networks.</sec><sec>Extensive experiments on real-world (e.g., Jazz, USAir, Email, Router) and synthetic (LFR-generated) networks validate the proposed models. The results demonstrate that BGIM and BGIM+ consistently outperform classical methods (such as Degree, Closeness, and Betweenness centralities) in identifying critical nodes and predicting their roles in propagation dynamics. In particular, BGIM+ exhibits superior performance in networks with complex topology, achieving high correlation with SIR (Susceptible-Infected-Recovered) model simulations under different propagation rates. Moreover, BGIM+ effectively balances the influences of local hubs and global bridges, thus it is particularly suitable for heterogeneous networks.</sec><sec>This study highlights the significance of incorporating multidimensional features into gravity models for accurate and robust node evaluation. The proposed model advances the development of complex network analysis by providing a universal tool for identifying influential nodes indifferent applications, including epidemic control, information dissemination, and infrastructure resilience. The applicability of BGIM in temporal and dynamic network contexts will be explored in future, so as to further expand its application scope.</sec>