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The $k$-core of a graph and its high-order spectra

Chunmeng Liu, Qing Xu, Changjiang Bu

TL;DR

This work establishes a spectral characterization of the graph $k$-core through the $k$-adjacency tensor, proving that a nonempty $k$-core exists iff the tensor's spectral radius $\rho_k(G) \ge 1$ and that Perron vector support localizes to the $k$-core. It shows that, when the core is connected, the core coincides with the support of the Perron vector, and introduces the $k$-th order eigenvector centrality as the Perron vector components, providing a natural, higher-order core-restricted measure of vertex influence. An accompanying algorithm computes both core existence and centrality, and extensive experiments on real networks validate the theory and reveal meaningful relationships between centrality, cycle structure, and core membership. The results offer a principled, scalable pathway to analyze core-periphery structure using higher-order spectral information. Practical implications include improved understanding of hierarchical structure and more nuanced centrality measures in complex networks.

Abstract

The $k$-core of a graph is its largest subgraph with minimum degree at least $k$, a fundamental concept for uncovering hierarchical structures. In this paper, we establish a connection between the $k$-core and the high-order spectra of graphs, a concept originally introduced by Cvetković, Doob, and Sachs. Specifically, we consider the high-order spectra defined via the $k$-adjacency tensor. Within this framework, we prove that a graph admits a non-empty $k$-core if and only if the spectral radius of the $k$-adjacency tensor is greater than or equal to $1$. Moreover, when the $k$-core exists, vertices corresponding to positive entries in the Perron vector of the $k$-adjacency tensor belong to the $k$-core. We thus define the $k$-order eigenvector centrality via the Perron vector, which provides both membership identification and a measure of relative influence within the $k$-core. Numerical experiments confirm our theoretical findings and illustrate the properties of this centrality measure in some real-world networks.

The $k$-core of a graph and its high-order spectra

TL;DR

This work establishes a spectral characterization of the graph -core through the -adjacency tensor, proving that a nonempty -core exists iff the tensor's spectral radius and that Perron vector support localizes to the -core. It shows that, when the core is connected, the core coincides with the support of the Perron vector, and introduces the -th order eigenvector centrality as the Perron vector components, providing a natural, higher-order core-restricted measure of vertex influence. An accompanying algorithm computes both core existence and centrality, and extensive experiments on real networks validate the theory and reveal meaningful relationships between centrality, cycle structure, and core membership. The results offer a principled, scalable pathway to analyze core-periphery structure using higher-order spectral information. Practical implications include improved understanding of hierarchical structure and more nuanced centrality measures in complex networks.

Abstract

The -core of a graph is its largest subgraph with minimum degree at least , a fundamental concept for uncovering hierarchical structures. In this paper, we establish a connection between the -core and the high-order spectra of graphs, a concept originally introduced by Cvetković, Doob, and Sachs. Specifically, we consider the high-order spectra defined via the -adjacency tensor. Within this framework, we prove that a graph admits a non-empty -core if and only if the spectral radius of the -adjacency tensor is greater than or equal to . Moreover, when the -core exists, vertices corresponding to positive entries in the Perron vector of the -adjacency tensor belong to the -core. We thus define the -order eigenvector centrality via the Perron vector, which provides both membership identification and a measure of relative influence within the -core. Numerical experiments confirm our theoretical findings and illustrate the properties of this centrality measure in some real-world networks.

Paper Structure

This paper contains 9 sections, 5 theorems, 12 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. $k$-th Order Eigenvector Centrality: Definition and Algorithm
  5. Experimental results
  6. Case Study: Verification and Illustration
  7. Structural Correlates of the Centrality Measure
  8. Comparative Analysis with Classical Centrality Measures
  9. Concluding remarks

Key Result

Lemma 2.1

Perron Th. If $\mathcal{A}$ is a nonnegative weakly irreducible tensor, then its spectral radius $\rho(\mathcal{A})$ is the unique positive eigenvalue of $\mathcal{A}$, and there exists a unique positive eigenvector $\mathbf{x}$ (up to a positive scaling factor) corresponding to $\rho(\mathcal{A})$.

Figures (4)

  • Figure 1: An example graph $G$ with 26 vertices. The labels $k_s = 1, 2, 3$ mark the encircled regions representing the 1-core, 2-core, and 3-core, respectively.
  • Figure 2: Visualizations of vertices centrality (The darker the color, the higher the centrality)
  • Figure 3: Karate network
  • Figure 8: Karate network

Theorems & Definitions (9)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Definition 4.1