Capability centrality: the next step from scale-free property

In a new preprint on arXiv, researcher Mikhail Tuzhilin proposes a game-changing network centrality measure called ksi-centrality that goes beyond traditional scale-free analysis. While degree centrality has long been the gold standard for distinguishing real networks from random ones, ksi-centrality offers an independent and complementary lens. Tuzhilin demonstrates that the ksi-centrality distribution is right-skewed for real-world networks (e.g., social, biological) but centered for random Erdos-Renyi graphs — a pattern reminiscent of degree centrality. Crucially, this property holds even for synthetic models designed to mimic real networks, such as Barabasi-Albert, Watts-Strogatz, and Boccaletti-Hwang-Latora, confirming that ksi-centrality captures a distinct structural signature not reducible to scale-freeness.

The paper also introduces a normalized version of ksi-centrality and establishes its deep connections to algebraic connectivity and Cheeger's value — key concepts in spectral graph theory. Most notably, the average normalized ksi-centrality of a network has a bijective correspondence with the parameter m in the Barabasi-Albert preferential attachment model, which controls how many edges a new node connects to existing nodes. This gives practitioners a concrete, data-driven method to choose m when modeling a specific real-world network, rather than relying on guesswork. The findings have immediate implications for network science, AI graph neural networks, and any domain where understanding the true structure of connectivity matters — from epidemiology to recommendation systems.