New centrality measure: ksi-centrality
Mikhail Tuzhilin
TL;DR
The paper introduces ksi-centrality, a new node centrality measuring a node's importance through the power of its neighbors, with both the unnormalized $\xi_i$ and normalized $\hat{\xi}_i$ versions. It shows a Laplacian-based representation for $\hat{\xi}_i$ analogous to the local clustering coefficient, and defines the average coefficient $\hat{\Xi}(G)$ with Erdős–Rényi graphs yielding $\mathbb{E}[\hat{\xi}_i] \approx p$, connecting to the algebraic connectivity via $\hat{\Xi}(G) \ge \lambda_2/n$ and to the Cheeger number. It analyzes various graphs (star, windmill, wheel, nested triangles) and real vs artificial networks, showing distributions of $\xi_i$ and $\hat{\xi}_i$ distinguish real data from models and relate to spectral properties. The results indicate practical potential for identifying influential nodes and discriminating network types, while noting limitations and directions for future work to refine average coefficients and applications.
Abstract
We introduce new centrality measures, called ksi-centrality and normalized ksi-centrality measure the importance of a node up to the importance of its neighbors. First, we show that normalized ksi-centrality can be rewritten in terms of the Laplacian matrix such that its expression is similar to the local clustering coefficient. After that we introduce average normalized ksi-coefficient and show that for a random Erdos-Renyi graph it is almost the same as average clustering coefficient. It also shows behavior similar to the clustering coefficient for the Windmill and Wheel graphs. Finally, we show that the distributions of ksi centrality and normalized ksi centrality distinguish networks based on real data from artificial networks, including the Watts-Strogatz, Barabasi-Albert and Boccaletti-Hwang-Latora small-world networks. Furthermore, we show the relationship between normalized ksi centrality and the average normalized ksi coefficient and the algebraic connectivity of the graph and the Chegeer number.