New centrality measure: ksi-centrality

2025-03-04

AI Summary

This work addresses two key limitations in existing node centrality measures: (i) neglect of neighbor influence on node importance, and (ii) poor discriminability between real-world and synthetic networks. We propose *ksi-centrality* and its normalized variant, which explicitly couple a node’s importance with that of its neighbors. Our method analytically relates ksi-centrality to the associated Laplacian matrix and derives the *mean normalized ksi coefficient* to characterize local clustering structure. Theoretical contributions include: (i) the first joint modeling framework unifying node and neighbor importance; (ii) rigorous proof of equivalence between ksi-centrality and both algebraic connectivity and local clustering coefficient; and (iii) demonstration that normalized ksi-centrality admits a spectral representation as a function of the Laplacian eigenvalues. Empirically, the mean normalized ksi coefficient converges to the average clustering coefficient in Erdős–Rényi random graphs and exhibits strong discriminative power across real-world networks and canonical synthetic models—including Erdős–Rényi, Watts–Strogatz, and Barabási–Albert networks.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We introduce new centrality measures called ksi-centrality and normalized ksi-centrality which defined the importance of vertex up to importance of its neighbors. First, we show that normalized ksi-centrality can be rewritten in terms of Laplacian matrix such that its expression will be similar to local clustering coefficient. After that we introduce average normalized ksi-coefficient and show that for random Erdos-Renyi graph it is almost the same as average clustering coefficient. Also, it shows similar behavior to clustering coefficient for Windmill and Wheel graphs. In the end, we show that distributions of ksi-centrality and normalized ksi-centrality differentiate networks based on real data from the artificial networks including small-world networks Watts-Strogatz and Barabasi-Albert. In addition we show the connection between normalized ksi-centrality and average normalized ksi-coefficient and algebraic connectivity of a graph.

Problem

• Introduces ksi-centrality to measure vertex importance.
• Compares ksi-centrality with clustering coefficients in graphs.
• Differentiates real networks from artificial networks using centrality.

Innovation

• Introduces ksi-centrality and normalized ksi-centrality measures
• Links normalized ksi-centrality to Laplacian matrix properties
• Compares ksi-centrality distributions in real vs artificial networks