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Laplacian Eigenvector Centrality

Koya Shimono, Wataru Tamura

TL;DR

This paper introduces Laplacian Eigenvector Centrality (LEC), a spectral centrality based on the Laplacian eigenstructure with an adjustable order that modulates the scope of centrality considered. Framed as a dimensionality-reduction of the Laplacian, LEC aggregates squared eigenvector components up to order $r$, enabling transitions from core to broader network influence; the authors also propose a generalized form, gLEC, with weighted spectral components. They establish foundational properties, derive an economic model linking LEC to adaptation and coordination in networks, and compare LEC to Katz-Bonacich centrality, highlighting unique implications for shock diffusion and information design. The empirical diffusion application to microfinance demonstrates that LEC captures distinct aspects of network bottlenecks and coordination pressures, providing a practical tool for targeting interventions and resilience analysis in diffusion processes. Overall, LEC offers a scalable, robust spectral framework that complements existing centrality measures and informs policy design in networked economies.

Abstract

Networks significantly influence social, economic, and organizational outcomes, with centrality measures serving as crucial tools to capture the importance of individual nodes. This paper introduces Laplacian Eigenvector Centrality (LEC), a novel framework for network analysis based on spectral graph theory and the eigendecomposition of the Laplacian matrix. A distinctive feature of LEC is its adjustable parameter, the LEC order, which enables researchers to control and assess the scope of centrality measurement using the Laplacian spectrum. Using random graph models, LEC demonstrates robustness and scalability across diverse network structures. We connect LEC to equilibrium responses to external shocks in an economic model, showing how LEC quantifies agents' roles in attenuating shocks and facilitating coordinated responses through quadratic optimization. Finally, we apply LEC to the study of microfinance diffusion, illustrating how it complements classical centrality measures, such as eigenvector and Katz-Bonacich centralities, by capturing distinctive aspects of node positions within the network.

Laplacian Eigenvector Centrality

TL;DR

This paper introduces Laplacian Eigenvector Centrality (LEC), a spectral centrality based on the Laplacian eigenstructure with an adjustable order that modulates the scope of centrality considered. Framed as a dimensionality-reduction of the Laplacian, LEC aggregates squared eigenvector components up to order , enabling transitions from core to broader network influence; the authors also propose a generalized form, gLEC, with weighted spectral components. They establish foundational properties, derive an economic model linking LEC to adaptation and coordination in networks, and compare LEC to Katz-Bonacich centrality, highlighting unique implications for shock diffusion and information design. The empirical diffusion application to microfinance demonstrates that LEC captures distinct aspects of network bottlenecks and coordination pressures, providing a practical tool for targeting interventions and resilience analysis in diffusion processes. Overall, LEC offers a scalable, robust spectral framework that complements existing centrality measures and informs policy design in networked economies.

Abstract

Networks significantly influence social, economic, and organizational outcomes, with centrality measures serving as crucial tools to capture the importance of individual nodes. This paper introduces Laplacian Eigenvector Centrality (LEC), a novel framework for network analysis based on spectral graph theory and the eigendecomposition of the Laplacian matrix. A distinctive feature of LEC is its adjustable parameter, the LEC order, which enables researchers to control and assess the scope of centrality measurement using the Laplacian spectrum. Using random graph models, LEC demonstrates robustness and scalability across diverse network structures. We connect LEC to equilibrium responses to external shocks in an economic model, showing how LEC quantifies agents' roles in attenuating shocks and facilitating coordinated responses through quadratic optimization. Finally, we apply LEC to the study of microfinance diffusion, illustrating how it complements classical centrality measures, such as eigenvector and Katz-Bonacich centralities, by capturing distinctive aspects of node positions within the network.
Paper Structure (48 sections, 1 theorem, 38 equations, 18 figures, 8 tables)

This paper contains 48 sections, 1 theorem, 38 equations, 18 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related literature
  3. A non-technical introduction to LEC
  4. Laplacian eigenvector centrality: definition
  5. A formal introduction
  6. Basic properties
  7. Scoring interpretation
  8. Symmetry and positional properties
  9. Reflecting network positions
  10. LEC as dimensionality reduction
  11. Selecting the LEC order based on Laplacian spectrum
  12. Eigenvalue decay patterns
  13. Cumulative sum of eigenvalues as a threshold
  14. Statistical properties of LEC and pLEC
  15. Network size effects and scalability
  16. ...and 33 more sections

Key Result

Proposition 1

The government selects the target for intervention based on the generalized Laplacian eigenvector centrality (gLEC) of each agent, weighted by $\phi_j = 1 - \left( \frac{1}{1+\beta \lambda_j} \right)^2$ for $j=1, \dots, n-1$.

Figures (18)

  • Figure 1: Expansion of the scope of LEC in Florentine network.
  • Figure 2: LEC scores for different orders (Top 7 families).
  • Figure 3: Traditional centrality scores (Top 7 families).
  • Figure 4: Violation of neighborhood-inclusion principle.
  • Figure 5: Laplacian spectrum as a guide for the LEC order choice.
  • ...and 13 more figures

Theorems & Definitions (3)

  • Definition 1: LEC of order $r$
  • Definition 2: Generalized LEC with weight $\bm{w}$
  • Proposition 1