Structural analysis and the sum of nodes’ betweenness centrality in complex networks

Structural analysis in the field of network science aims to uncover the hidden information embedded within the topological structure of complex networks. Betweenness centrality is a measure that quantifies a node’s influence on the overall network, based on the shortest paths between different pairs of nodes. This unique property of betweenness centrality allows it to capture more structural information than the commonly employed degree centrality, which is a fundamental characteristic of network structure. In this study, we demonstrate that the sum of nodes’ betweenness centralities (SBC) can be utilized as a novel structural index to reveal the underlying rules governing the growth of a network. Additionally, we have developed a method that combines K-shell decomposition and SBC analysis to investigate the growth rules that have shaped the evolution of static networks in the past. Our findings indicate that the Barabási–Albert model guides the network’s SBC to grow in a logarithmic fashion, whereas the Erdős–Rényi model leads to the convergence of SBC values. Interestingly, we also discover that the convergence or divergence of SBC within the k-shell-like decomposition of a static network can be used to distinguish between developed networks (where growth rules have reached saturation) and developing networks (still undergoing expansion). These results highlight the utility of SBC as a reasonable and effective index for the structural analysis of complex networks, providing insights into the rules governing their evolution.