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When Is Degree Enough? Bounds on Degree-Eigenvector Misalignment in Assortative Structured Networks

Sreerag Puravankara, Vipin P. Veetil

Abstract

A tight alignment between the degree vector and the leading eigenvector arises naturally in networks with neutral degree mixing and the absence of local structures. Many real-world networks, however, violate both conditions. We derive bounds on the divergence between the degree vector and the eigenvector in networks with degree assortativity and local mesoscopic structures such as communities, core-peripheries, and cycles. Our approach is constructive. We design sufficiently general degree-preserving rewiring algorithms that start from a neutral benchmark and monotonically increase assortativity and the strength of local structures, with each step inducing a perturbation of the adjacency matrix. Using the Stewart--Sun Perturbation Bound, together with explicit spectral-norm control of the rewiring steps, we derive upper bounds on the angle between the eigenvector and the degree vector for modest levels of assortativity and local structures. Our analytical bounds delineate regions of `spectral safety' in which a node's degree can be used as a reliable measure of its systemic importance in real-world networks. We also substantiate our analytical bounds with numerical simulations that compute the exact angles of deviation.

When Is Degree Enough? Bounds on Degree-Eigenvector Misalignment in Assortative Structured Networks

Abstract

A tight alignment between the degree vector and the leading eigenvector arises naturally in networks with neutral degree mixing and the absence of local structures. Many real-world networks, however, violate both conditions. We derive bounds on the divergence between the degree vector and the eigenvector in networks with degree assortativity and local mesoscopic structures such as communities, core-peripheries, and cycles. Our approach is constructive. We design sufficiently general degree-preserving rewiring algorithms that start from a neutral benchmark and monotonically increase assortativity and the strength of local structures, with each step inducing a perturbation of the adjacency matrix. Using the Stewart--Sun Perturbation Bound, together with explicit spectral-norm control of the rewiring steps, we derive upper bounds on the angle between the eigenvector and the degree vector for modest levels of assortativity and local structures. Our analytical bounds delineate regions of `spectral safety' in which a node's degree can be used as a reliable measure of its systemic importance in real-world networks. We also substantiate our analytical bounds with numerical simulations that compute the exact angles of deviation.
Paper Structure (31 sections, 7 theorems, 122 equations, 2 figures)

This paper contains 31 sections, 7 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spectral Preliminaries and the Stewart--Sun Perturbation Bound
  3. Basic notation
  4. Neutral matrix and its alignment with the degree vector
  5. Perturbation matrices and their spectral norms
  6. Stewart--Sun eigenvector perturbation bound
  7. Bounding moments of angle of deviation with moments of network statistic
  8. Assortativity
  9. Assortativity inducing rewiring
  10. Local structures
  11. Communities
  12. Community strengthening rewiring
  13. Extension to many communities
  14. Core-periphery
  15. Degree-based partition and a swap
  16. ...and 16 more sections

Key Result

Proposition 1

Let $\mathbf v$ denote the unit Perron eigenvector of $\mathbf A$ , and let $\mathbf d$ denote the corresponding unit degree vector. Then so that degree centrality coincides with eigenvector centrality in the neutral baseline Newman2010Networks. Note that when $\mathbf v$ is the right eigenvector, then $\mathbf d$ is the out-degree, equivalently for the left eigenvector and the in-degree. $\circ$

Figures (2)

  • Figure 1: Assortativity experiments : Each panel fixes a focus (in--in, in--out, out--in, out--out). The x-axis is the corresponding degree-endpoint correlation $r(\cdot)$ computed across directed edges; the dashed vertical line marks $r=0$. The y-axis shows the deviation angles $\theta_R(\mathrm{out})$ (right eigenvector vs out-degree; blue circles) and $\theta_L(\mathrm{in})$ (left eigenvector vs in-degree; orange squares), in degrees.
  • Figure 2: Local-structure experiments. Top: communities ($\phi_{\mathrm{com}}$). Middle: triangles ($\phi_{3\text{-}\mathrm{cyc}}$). Bottom: core--periphery ($\phi_{\mathrm{cp}}$). Scatter plots show logged snapshots: solid curves show empirical mean angles across the Monte Carlo sample.

Theorems & Definitions (23)

  • Definition 1: Distance between two vectors as an angle
  • Definition 2: Neutral network
  • Proposition 1: Degree vector as a proxy for the leading eigenvector in neutral networks
  • Definition 3: Spectral gap
  • Definition 4: Perturbation matrices
  • Proposition 2: Spectral norm bound for the cumulative rewiring perturbation
  • proof
  • Definition 5: Distortion factor
  • Proposition 3: Angle of deviation between the degree vector and the eigenvector
  • Conjecture 1: Wandering angle under strictly $\phi$-improving rewiring
  • ...and 13 more