Local Eigenvector Centrality

• Local eigenvector centrality is defined to overcome limitations of global eigenvector methods by focusing on localized supports within a network.
• Two approaches—regime-detection and spectral blending—extract key eigenvector information to robustly rank nodes even in the presence of hubs or K-core localization.
• LEC provides practical insights for applications like epidemiology and infrastructure analysis by isolating true influencer nodes in heterogeneous structures.

Local eigenvector centrality denotes a family of centrality measures designed to quantify the importance of nodes in a network based not only on global connectivity, but also on community- or locality-sensitive structural features. Conventional eigenvector centrality (EC) considers only the principal eigenvector of the adjacency matrix, which leads to well-known localization phenomena: under certain structural or spectral conditions, centrality mass may sharply concentrate on a small subset of nodes or communities, rendering traditional EC ineffective in ranking the remainder of the network. Local eigenvector centrality (LEC) addresses these pathological behaviors via algorithms that focus ranking on the supports where localization occurs, or by constructing blended centralities from the leading eigenspectrum, as in recent developments. These measures have substantial implications for analyzing and interpreting node influence in large-scale, heterogeneous, or modular complex networks.

1. Spectral Localization and Motivation

Eigenvector centrality is defined as the principal eigenvector $v$ of the adjacency matrix $A$ satisfying $A v = \lambda_1 v$, with $v_i \geq 0$ and $||v||_2=1$. Localization refers to the phenomenon where most of the squared weight $v_i^2$ is carried by a small number of nodes or subgraphs. This can be quantified using the inverse participation ratio (IPR):

$\mathrm{IPR} = Y = \sum_{i=1}^{N} v_i^4.$

For a fully delocalized $v$, $Y \sim N^{-1}$, but when $v$ is localized on $N_V \sim N^\beta$ nodes ($\beta<1$), $Y \sim N^{-\beta}$. In the extreme case, localization on a finite set yields $Y_0 > 0$ as $N\to\infty$ (Pastor-Satorras et al., 2015).

These effects arise in several contexts:

• In scale-free networks with degree distribution $P(q)\sim q^{-\gamma}$, for $\gamma > 5/2$ eigenvector mass localizes on the largest hub; for $\gamma < 5/2$, localization occurs on a mesoscopic $K$-core (Pastor-Satorras et al., 2015).
• In networks with a cut vertex (whose removal partitions the graph), EC can localize almost entirely onto a single dense subgraph connected via the cut-vertex, depending on spectral gaps and connection geometry (Sharkey, 2018).
• In the presence of high-degree hubs, such as random graph plus star models, centrality may concentrate exclusively on the hub and its immediate neighbors for sufficiently large hub degree, with a rigorously established localization threshold (Martin et al., 2014).

Localization renders EC unreliable for fine-grained ranking except on its support, motivating the development of local eigenvector centrality approaches.

2. Formal Definitions of Local Eigenvector Centrality

Two primary approaches to local eigenvector centrality are present in the literature:

Regime-Detection-Based Local Centrality

In (Pastor-Satorras et al., 2015), LEC focuses centrality on the identified support of the eigenvector localization:

1. Detect localization regime using spectral ratios:
   • Compute degrees $q_i$ and moments $\langle q\rangle, \langle q^2 \rangle$.
   • Compute principal eigenvalue $\lambda_1$.
   • Calculate $r_1 = \lambda_1/\sqrt{q_{\max}}$ and $$r_2 = \lambda_1/(\langle q^2\rangle/\langle q\rangle)$$.
   • $|r_1-1| < \epsilon$ implies hub-localization; $|r_2-1| < \epsilon$ implies K-core localization.
2. Extract localization support $V_{\rm loc}$ (the hub node or maximum $K$-core).
3. Compute the principal eigenvector $v^{(\rm loc)}$ of the induced subgraph adjacency $A_{\rm loc}$.
4. Define for each node $i$

$$c_i^{\rm (loc)} = \begin{cases} v_i^{(\rm loc)}, & i\in V_{\rm loc},\ 0, & i\notin V_{\rm loc}. \end{cases}$$

This isolates centrality only on the “true” support, preventing misleading background mass (Pastor-Satorras et al., 2015).

Spectral Blending via Eigengaps

Clark et al. (Clark et al., 5 Nov 2025) define LEC using the leading $k$ eigenvectors associated with the most prominent positive eigengap:

• Let $A$ be the (possibly weighted, directed) adjacency matrix with eigenvalues $\lambda_1 \geq \Re(\lambda_2) \geq \cdots$.
• Define $g_i = \Re(\lambda_i) - \Re(\lambda_{i+1})$.
• Choose $k = \arg\max_{\Re(\lambda_i)>0} g_i$ (largest positive eigengap).
• Assemble $V \in \mathbb{R}^{n \times k}$ from the first $k$ eigenvectors (real or, for complex pairs, by splitting into real and imaginary parts).
• LEC for node $i$ is the Euclidean norm of the $i$th row of $V$:

$$c(i) = \sqrt{\sum_{j=1}^k v_j(i)^2} = \|e_i^T V\|_2,$$

where $v_j(i)$ is the $i$th entry of the $j$th eigenvector (Clark et al., 5 Nov 2025).

This definition provides a multiscale notion of centrality that reflects both global (hub) and local (community) influence.

3. Algorithmic Procedures

Pseudocode for Regime-Detection-Based LEC (Pastor-Satorras et al., 2015)

Input: Graph G(V, E) Output: c_loc[1..N] 1. Compute degrees q[i], qmax = max_i q[i] 2. <q> = (1/N) sum_i q[i], <q2> = (1/N) sum_i q[i]^2 3. Compute (lambda1, v) principal eigenpair of A 4. r1 = lambda1 / sqrt(qmax) r2 = lambda1 / ((<q2>)/(<q>)) 5. if |r1 - 1| < eps: regime = "hub" else if |r2 - 1| < eps: regime = "kcore" else: regime = "ambiguous" 6. if regime == "hub": Vloc = {argmax_i q[i]} else: compute k-core indices core[i], Kmax = max_i core[i] Vloc = {i : core[i] == Kmax} 7. Build subgraph G_loc induced by Vloc 8. Compute principal eigenvector v_loc of G_loc 9. for i in V: if i in Vloc: c_loc[i] = v_loc[i]; else: c_loc[i] = 0 10. return c_loc

Pseudocode for Spectral-Blending LEC (Clark et al., 5 Nov 2025)

Input: Adjacency A (n x n) Output: c (size n) 1. Compute top part of spectrum {lambda_i, x_i} of A 2. Sort so that Re(lambda_1) >= Re(lambda_2) >= ... 3. Compute eigengaps g_i = Re(lambda_i) - Re(lambda_{i+1}) 4. k = argmax_{Re(lambda_i)>0} g_i 5. Build V (n x k): for i = 1 to k if lambda_i real: v_i = x_i / ||x_i|| else if complex: v_i = Re(x_i)/||Re(x_i)||; v_{i+1} = Im(x_i)/||Im(x_i)|| 6. For each i, c(i) = sqrt(sum_{j=1}^k v_j(i)^2) 7. Return c

The computational bottleneck is in steps 1 and 5 (partial eigendecomposition). For sparse large networks, iterative methods such as Lanczos/Arnoldi are necessary.

4. Relations to Alternative Centralities and Localization Mitigation

Eigenvector centrality is susceptible to spurious localization, with nearly all centrality weight absorbed by a single high-degree hub or dense subgraph. Katz centrality $x = (I - \alpha A)^{-1}\mathbf{1}$ blends information from the full spectrum and avoids the pathological blow-up observed in EC. As $\alpha \to 1/\lambda_1$, normalized Katz centrality converges to EC, but for any $\alpha < 1/\lambda_1$, sub-leading eigenvector contributions are retained, enhancing robustness (Sharkey, 2018). Damping, as utilized in PageRank (with teleportation parameter $\alpha=0.85$), similarly mitigates localization.

In (Clark et al., 5 Nov 2025), a nonlinear rescaling against PageRank is used to correct LEC in the presence of localization. Given LEC vector $x$ and PageRank $y$, find $p\in(0,1)$ to minimize $\|\mathcal N_p(x) - y\|_2$, where

$$\mathcal N_p(x) = \frac{x^{\circ p}}{\sum_i x_i^p}.$$

This calibration preserves the community-sensitive resolution of LEC while suppressing over-amplified scores on singular hubs.

The nonbacktracking (Hashimoto) matrix approach (Martin et al., 2014) provides node centralities less sensitive to localization: the leading eigenvector of the nonbacktracking matrix remains delocalized even as hubs emerge, because no eigenvalue crossing (and hence no localization transition) occurs in this spectrum. The computational complexity is $O(m)$ (up to logarithmic factors), only marginally higher than conventional EC.

5. Empirical Results and Applications

Empirical studies across synthetic and real networks consistently identify regime-dependent localization and draw distinctions between global and local centrality distributions:

• In primary school contact networks, LEC at $k=5$ corresponds to year-group communities, while $k=10$ recovers class-level hubs. Nodes with exceptional between-community connectivity are distinguished from purely local hubs (Clark et al., 5 Nov 2025).
• City-scale road networks exhibit a variety of eigengap profiles: in Glasgow, LEC identifies multiple principal junction clusters, whereas in Chicago, a sharp eigengap at $k=1$ indicates dominance by a single global hub, but secondary peaks (e.g., $k=10$) allow LEC to surface local hubs.
• In real-world large-scale networks (coauthorship, web graphs, online retailers), either hub or K-core localization is quantitatively detected, and local EC accordingly isolates centrality on the principal structural support (Pastor-Satorras et al., 2015).

Table: Summary of Localization Supports and LEC Algorithm (Pastor-Satorras et al., 2015, Clark et al., 5 Nov 2025)

Regime	Localization support	LEC type
Hub-localized	Max-degree node (hub)	Assign EC only on the hub
K-core-localized	Nodes in max $K$-core shell	Assign EC only within $K_M$-core
Multicommunity (eigengap)	Leading communities via $k$-eigenvectors	Euclidean norm of first $k$ EVs

6. Limitations and Open Questions

Local eigenvector centralities fundamentally depend on spectral features and may inherit certain degeneracies:

• If the adjacency matrix is defective, providing fewer than $k$ independent eigenvectors, LEC uses the eigenvectors available; this can affect ranking granularity (Clark et al., 5 Nov 2025).
• Structures such as directed paths with zero eigenvalues yield identically zero LEC, and simple cycles distribute centrality uniformly, aligning with expected structural uniformity.
• In ambiguous localization regimes (where neither hub nor K-core regime is detected), the regime-detection algorithm may fail to assign a unique localization support (Pastor-Satorras et al., 2015).
• Spectral-blending LEC is sensitive to numerical eigenvector computation, requiring careful treatment for large or nearly defective networks.

The selection of $k$ via the most prominent positive eigengap provides an adaptive resolution, but in highly multiscale or hierarchical networks, further algorithmic refinements may be necessary for optimal interpretability.

7. Theoretical and Practical Significance

LEC methodologies provide principled mechanisms for quantifying node importance in structurally heterogeneous networks—particularly those exhibiting prominent community divisions or singular hubs. By restricting centrality to true localization supports or by constructing norm-based blends across leading eigenvectors, LEC produces rankings more robust to the spectral artifacts that plague standard EC. This has direct consequences in epidemiology (identifying super-spreader clusters), infrastructure analysis (resilient hub detection), social systems (multiscale influencer ranking), and network science research. Alternatively, where a uniform ranking is demanded, recalibration strategies referencing PageRank or nonbacktracking centrality can be employed to suppress localization artifacts.

A plausible implication is that local eigenvector centrality (in both the regime-detection and eigengap-resolved forms) complements rather than replaces other centrality measures. Its effectiveness in revealing both local and global influencers in real data suggests substantial utility for theory-driven and application-oriented network analysis (Pastor-Satorras et al., 2015, Clark et al., 5 Nov 2025, Sharkey, 2018, Martin et al., 2014).