Local Eigenvector Centrality (LEC)

• Local Eigenvector Centrality is a spectral measure that interpolates between global and community-scale influences using data-driven eigengap analysis.
• It aggregates multiple top eigenvectors from the adjacency matrix to expose modular hubs and bridging nodes while mitigating localization issues.
• LEC leverages efficient spectral algorithms and Krylov-subspace methods for sparse networks, with applications in social, transportation, and epidemiological studies.

Local Eigenvector Centrality (LEC) is a spectral graph centrality measure that captures both local and global node influence by combining information across multiple prominent spectral modes, as opposed to relying solely on the principal eigenvector. LEC quantitatively interpolates between global eigenvector centrality and cluster-scale or community-scale centrality through an automatic, data-driven selection of the number of spectral directions included. The metric is constructed from the eigendecomposition of the adjacency matrix (and, in certain variants, Laplacian matrix), using prominent eigengaps to identify community structure. LEC has been shown to reveal modular hubs, discover bridging nodes, and mitigate localization issues, with empirical validation across diverse network types including social and transportation networks (Clark et al., 5 Nov 2025).

1. Theoretical Motivation

Classical eigenvector centrality (EC) assigns centrality proportional to the corresponding node’s entry in the principal eigenvector of the adjacency matrix $A$. This approach is effective in globally homogeneous graphs but breaks down in networks with strong community or modular structure. In real-world systems—such as transmission in schools and urban road networks—the influence flows are not monopolized by a single global hub; multiple local hubs play crucial roles, and the standard EC either collapses centrality onto the single largest module (masking smaller clusters) or suffers from localization effects, concentrating values on dense subgraphs (Clark et al., 5 Nov 2025).

LEC addresses these limitations by identifying dominant scales of organization in a graph via analysis of the adjacency spectrum. It retains the set of eigenvectors associated with all eigenvalues prior to the largest eigengap and aggregates them in an isotropic ($\ell_2$-norm) fashion. This approach yields a centrality that is sensitive to the network’s modular structure and is capable of revealing both bridge nodes and community scale hubs (Clark et al., 5 Nov 2025).

2. Mathematical Formalism

Let $A \in \mathbb{R}^{n \times n}$ be the adjacency matrix. Compute its eigenvalues $\lambda_1, ..., \lambda_n \in \mathbb{C}$, ordered by real part:

$$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$

Define the eigengap sequence

$$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$

Select

$$k = \underset{i: \mathrm{Re}(\lambda_i)>0}{\arg\max} g_i.$$

If there are ties, select the smallest such $i$.

Extract a set of $k$ real, unit-norm eigenvectors associated with these top eigenvalues, forming $V \in \mathbb{R}^{n \times k}$. For complex conjugate eigenvalues, use the real and imaginary components as independent directions. The Local Eigenvector Centrality $\ell_2$0 is then given by:

$\ell_2$1

where $\ell_2$2 is the $\ell_2$3 column of $\ell_2$4 and $\ell_2$5 is the standard basis vector. This yields the Euclidean norm over the selected spectral features at each node. The construction presupposes $\ell_2$6 has no defective eigenvalues among the selected top eigenvalues; otherwise, use all available eigenvectors in the eigenspace (Clark et al., 5 Nov 2025).

LEC may also be constructed using the Laplacian $\ell_2$7 or normalized Laplacian $\ell_2$8, but on empirical examples, the adjacency-based LEC is superior for flow-centric centrality, whereas Laplacian-based variants highlight consensus or isolation (Clark et al., 5 Nov 2025, Shimono et al., 19 Jan 2025).

3. Algorithmic Aspects and Computational Complexity

The LEC construction proceeds as follows:

• Input: Adjacency matrix $\ell_2$9
• 1. Compute the leading $A \in \mathbb{R}^{n \times n}$0 eigenvalues and eigenvectors of $A \in \mathbb{R}^{n \times n}$1 ($A \in \mathbb{R}^{n \times n}$2)
• 2. Identify $A \in \mathbb{R}^{n \times n}$3 as the index of the largest eigengap with $A \in \mathbb{R}^{n \times n}$4
• 3. Extract $A \in \mathbb{R}^{n \times n}$5 real, normalized eigenvectors (or real/imaginary parts if complex)
• 4. Assemble $A \in \mathbb{R}^{n \times n}$6, then for each node $A \in \mathbb{R}^{n \times n}$7, compute $A \in \mathbb{R}^{n \times n}$8
• 5. Return $A \in \mathbb{R}^{n \times n}$9

Dense spectral decomposition requires $\lambda_1, ..., \lambda_n \in \mathbb{C}$0 time and $\lambda_1, ..., \lambda_n \in \mathbb{C}$1 storage. When $\lambda_1, ..., \lambda_n \in \mathbb{C}$2 is sparse and $\lambda_1, ..., \lambda_n \in \mathbb{C}$3, Krylov-subspace methods enable computation in $\lambda_1, ..., \lambda_n \in \mathbb{C}$4 time and $\lambda_1, ..., \lambda_n \in \mathbb{C}$5 space. Forming $\lambda_1, ..., \lambda_n \in \mathbb{C}$6 and $\lambda_1, ..., \lambda_n \in \mathbb{C}$7 is $\lambda_1, ..., \lambda_n \in \mathbb{C}$8 (Clark et al., 5 Nov 2025).

For Laplacian Eigenvector Centrality (as in (Shimono et al., 19 Jan 2025)), similar algorithms apply, with spectral analysis on $\lambda_1, ..., \lambda_n \in \mathbb{C}$9 and accumulation of squared components up to the desired order $$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$0.

4. Empirical Validation and Case Studies

LEC has been empirically validated on several benchmark datasets:

• Primary-School Contact Network: With $$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$1, eigengaps at $$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$2 correspond to “whole-school”, “year-group”, and “class” levels. LEC with $$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$3 or $$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$4 recovers centralities reflective of known structural modules, outperforming global EC and showing close similarity to per-class EC (Clark et al., 5 Nov 2025).
• High-School Contact Network: Largest eigengap occurs at $$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$5, revealing five local hubs not aligned with official classes. LEC identifies bridging nodes and modular influence, with deviations from per-class EC highlighting individuals with extensive inter-class connections (Clark et al., 5 Nov 2025).
• Localization and De-localization: LEC can exhibit strong localization on dense hubs. Comparison to PageRank (with teleportation) reveals that PageRank distributes centrality more evenly. A nonlinear Hadamard rescaling ($$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$6) with $$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$7 can diminish this localization effect, matching PageRank more closely while preserving LEC’s spectral interpretability (Clark et al., 5 Nov 2025).
• Road Networks: In large-scale urban networks (Glasgow, Chicago), adjacency-based LEC reveals multiple traffic centers (“hubs”) corresponding to urban subcenters, while EC only identifies global maxima. Laplacian-based LEC predominantly highlights isolated bottlenecks, confirming the importance of adjacency-based variants for flow (Clark et al., 5 Nov 2025).

5. Relationship to Other Centralities and Localization

Comparison among Centralities

Measure	Matrix	spectrum used	Community sensitivity	Localization
Eigenvector Centrality	$$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$8	principal eigenvec.	low	high
LEC (Adjacency-based)	$$\mathrm{Re}(\lambda_1) \ge \mathrm{Re}(\lambda_2) \ge \cdots \ge \mathrm{Re}(\lambda_n).$$9	top-$$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$0 eigenspace	adaptive	mitigated
Laplacian Eigenvector Cent.	$$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$1	top-$$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$2 eigenspace	spectral, diffusion	low
PageRank	$$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$3	stochastic, all modes	mitigates modularity	lowest

Classical EC is globally oriented and neglects multi-community structure. Adjacency-based LEC, using multiple leading eigenvectors up to the dominant eigengap, interpolates between global and local perspectives, discovering multi-scale hubs and bridge nodes. Laplacian-based LEC, as shown in (Shimono et al., 19 Jan 2025), is sensitive to diffusion structure and attenuates localization, focusing on consensus and isolation.

PageRank’s random-walk with teleportation delivers uniformization, serving as an effective de-localization tool. A nonlinear transformation of LEC ($$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$4) narrows the gap between LEC and PageRank, but the mapping is ad hoc, as highlighted by the use of $$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$5 yielding the minimal Euclidean distance between the two centrality vectors in empirical studies (Clark et al., 5 Nov 2025).

6. Advantages, Limitations, and Extensions

Strengths

• Adaptive detection of prominent community scales via eigengap analysis.
• Isotropic aggregation of multiple spectral directions, avoiding single-eigenvector bias.
• Identification of hubs and bridging nodes without requiring prior community labels.
• Direct comparability with classical EC ($$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$6) and with per-community EC for larger $$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$7 (Clark et al., 5 Nov 2025).

Limitations

• Requires at least partial spectral decomposition, which is computationally demanding for large-scale graphs.
• Effectiveness depends on the presence of clear eigengaps; smoothly decaying spectra render $$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$8 unstable.
• Localization can persist in highly dense subgraphs; ad hoc nonlinear rescaling is required to mitigate (Clark et al., 5 Nov 2025).

Extensions

• Replacement of $$g_i = \mathrm{Re}(\lambda_i) - \mathrm{Re}(\lambda_{i+1}), \quad i=1,...,n-1.$$9 with non-backtracking or other matrices can further reduce localization.
• Incorporation of exogenous community labels allows multi-gap selection to capture hierarchical structures.
• Alternative norms ($$k = \underset{i: \mathrm{Re}(\lambda_i)>0}{\arg\max} g_i.$$0, $$k = \underset{i: \mathrm{Re}(\lambda_i)>0}{\arg\max} g_i.$$1) for aggregation could modulate emphasis on multicommunity membership.
• Development of streaming or incremental approximations for dynamic or massive networks (Clark et al., 5 Nov 2025).

7. Applications and Outlook

LEC’s spectrum-informed construction equips it to serve a wide range of analytic purposes, from epidemiological modeling in contact networks to urban planning in transportation networks. Its adaptability to the intrinsic modular scale, discovery of both intra- and inter-community roles, and compatibility with modern spectral clustering and graph signal processing frameworks position it as a flexible and principled centrality measure for complex networks. For Laplacian-based centralities, their grounding in diffusion and consensus is instrumental in economic modeling (e.g., equilibrium response to shocks), social coordination, and infrastructure robustness analysis (Shimono et al., 19 Jan 2025). Open challenges remain in extending LEC to directed, weighted, or time-evolving networks and in integrating filter functions for tunable sensitivity to spectral bands.

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References:

Clark et al., “A local eigenvector centrality” (Clark et al., 5 Nov 2025); Shimono and Tamura, “Laplacian Eigenvector Centrality” (Shimono et al., 19 Jan 2025).