Regime-Detection Local Centrality

• Regime-detection-based local centrality is a method that identifies node importance by detecting structural regimes such as communities, core-periphery, or multiscale features.
• It integrates data-driven regime detection via spectral gaps, community inference, or diffusion processes to compute centrality that reflects local influence and information flow.
• By adapting to the underlying network structure, it overcomes limitations of global metrics and ensures robust, scalable node ranking in heterogeneous networks.

Regime-detection-based local centrality encompasses a class of network centrality measures that adaptively distinguish node importance according to structural regimes—such as communities, core-periphery divisions, or mesoscale geometric features—in a graph. These measures proceed via two key steps: (i) the data-driven detection or specification of local regimes (via spectral gaps, community inference, or timescale segmentation), and (ii) the computation of node centrality so as to capture influence, information flow, or structural dominance relevant within (and sometimes across) the detected regimes. This approach systematically overcomes the limitations of canonical global or purely local metrics, enabling robust node ranking in networks with pronounced community, core-periphery, or multiscale organization (Clark et al., 5 Nov 2025, Barucca et al., 2015, Blöcker et al., 2022, Arnaudon et al., 2019).

1. Mathematical Foundations and Definitions

Regime-detection-based local centrality measures are grounded in several mathematical concepts:

• Regime: A region or substructure of the network that is internally cohesive and can be discovered by spectral analysis (e.g., prominent eigengaps), statistical inference (e.g., community detection), or dynamical signatures (e.g., diffusion timescales).
• Adjacency and Spectral Data: Given $G=(V,E)$, the adjacency matrix $A \in \mathbb{R}^{n \times n}$ encodes connectivity. Spectral decompositions (eigenvalues $\lambda_1,\dots,\lambda_n$ of $A$ or Laplacian $L=K-A$) reveal structural scales.
• Locality: Centrality is computed using only information available within, or strongly related to, the structural regime in question. This can involve subspaces of eigenvectors, module-specific flow information, or spatially truncated diffusion kernels.
• Centrality Vector: For nodes $i \in V$, centrality scores $c(i)$ reflect their structural importance within a given regime, often combining influence on both local and global connectivity.

This foundation enables instantiations ranging from local eigenvector centrality (using prominent eigengaps) to module-aware information-theoretic measures, scale-dependent diffusion metrics, and belief-propagation marginals.

2. Principal Methodologies

Local Eigenvector Centrality (LEC) via Spectral Regimes

The approach of Clark et al. defines local eigenvector centrality by identifying the largest positive eigengap $$g_k = \mathrm{Re}(\lambda_k) - \mathrm{Re}(\lambda_{k+1})$$ in the spectrum of $A$, restricting to $\mathrm{Re}(\lambda_k) > 0$. This selects $k$ spectral modes, interpreted as encoding the dominant community scale:

$$c_j = \left\| \left( v_1(j), \ldots, v_k(j) \right) \right\|_2 .$$

Here, $v_1, \ldots, v_k$ are real orthonormal eigenvectors obtained from $A$ as detailed in the paper, and $c_j$ is the resulting centrality of node $j$ (Clark et al., 5 Nov 2025).

Map Equation Centrality: Information-Theoretic Module Locality

Map equation centrality quantifies the compressibility gain ($\Delta L_i$) from excluding a node $i$ in a two-level coding scheme where regimes are communities detected by an information-theoretic method (Infomap):

$$\Delta L_i = - (p_{m_i} - p_i) \log_2 \left( \frac{p_{m_i} - p_i}{p_{m_i}} \right)$$

where $p_{m_i}$ is the visit rate of module $m_i$ and $p_i$ is that of node $i$ (Blöcker et al., 2022). This measure is strictly local to the module and is fast to compute on large networks.

Regime Detection in Core–Periphery Networks

Within block-structured networks, regime-detection-based centralities use stochastic block models (SBM, degree-corrected SBM) to define core and periphery. Node importance is inferred either by maximum-likelihood assignment (belief-propagation marginals), degree, or PageRank. Spectral centralities such as EC and NBT exhibit localization in the presence of hubs, failing to detect the mesoscopic regime (Barucca et al., 2015).

Scale-Dependent Centrality from Diffusion Dynamics

Arnaudon–Peach–Barahona introduce a multiscale centrality $C_i(t)$ for each timescale $t$ of diffusion:

• Define $D_t(i,j)$, a diffusion-based distance between nodes.
• For each node $i$, $C_i(t)$ measures the fraction of node pairs $(j,k)$ for which $D_t(i,j) + D_t(i,k) - D_t(j,k) \leq 0$.

This quantifies "centrality at scale $t$" and exposes regime transitions as $t$ varies (Arnaudon et al., 2019).

3. Algorithmic Procedures and Computational Aspects

The practical computation of regime-detection local centralities varies by methodology:

Centrality	Main Steps	Complexity
LEC (Clark et al., 5 Nov 2025)	Eigendecomposition, eigengap selection, norm ops.	$O(n^3)$ (dense); $O(k\,\mathrm{nnz}(A))$ (sparse)
Map Equation (Blöcker et al., 2022)	Community detection via Infomap, per-module visit-rate summation, per-node $\Delta L_i$	$O(n \log n)$ (sparse)
BP Marginals (Barucca et al., 2015)	Iterative BP messages, marginal compute	$O(NkT)$ for $k$ groups, $T$ iters
Diffusion-Based (Arnaudon et al., 2019)	Matrix exponential/Krylov, $O(N^3)$ triangle counting	$O(N^3)$ worst case

LEC and diffusion-based centrality are dominated by spectral and pairwise operations, respectively. Map equation centrality achieves scalability by separating regime detection (community finding) and local computation, trivializing parallelization.

4. Comparative Analyses and Empirical Findings

Community and Contact Networks

• In school contact networks, the transition index $k$ (maximum eigengap) in LEC aligns with known community structure: entire school, year groups, individual classes. LEC ranks central pupils per regime scale; LEC matches community-restricted EC but diverges from global PageRank. Nonlinear renormalization bridges this gap (Clark et al., 5 Nov 2025).

Core–Periphery Networks

• BP marginals consistently achieve maximum overlap with true core sets under SBM; PageRank and degree centrality follow closely. EC and NBT degrade in the presence of heavy-tailed hubs due to localization. PageRank's teleportation mitigates this, sustaining global sensitivity (Barucca et al., 2015).

Road Networks and Multiscale Examples

• For urban road graphs, LEC uncovers both global and local connectivity hubs (e.g., sectional traffic nodes vs. motorway junctions), depending on the eigengap regime. Diffusion centrality $C_i(t)$ correlates with degree at short timescales and with closeness or bridge roles at large $t$ (Clark et al., 5 Nov 2025, Arnaudon et al., 2019).

Flow-Based and Information-Theoretic Networks

• Map equation centrality outperforms alternatives in predicting influential spreaders in threshold and SIR models across real and synthetic datasets, due to its module-local, flow-aware design (Blöcker et al., 2022).

Localization and Spectral Limitations

• The inverse participation ratio (IPR) serves as an indicator of localization. LEC and similar spectral centralities avoid excessive localization by incorporating multiple eigenvectors or by non-linear postprocessing, while PageRank's random walk basis imparts robustness against localization transitions in heterogeneous degree distributions (Clark et al., 5 Nov 2025, Barucca et al., 2015).

5. Theoretical and Practical Advantages

Regime-detection-based local centrality unifies the following benefits:

• Adaptivity: Automatically tunes the locality/globality of centrality to actual detected regime structure via eigengaps, flow partitions, or diffusion horizons.
• Delocalization: Mitigates hub domination and localization transitions, yielding more interpretable, regime-sensitive rankings.
• No Ground-Truth Labels Required: Recovers community-sensitive or core-aware centralities without a priori specification of regimes or labels (Clark et al., 5 Nov 2025, Blöcker et al., 2022).
• Robustness Across Regimes: Retains performance in both homogeneous and heavily heterogeneous scenarios by selecting suitable metrics (BP, PageRank) or postprocessing.
• Computational Feasibility: Exploits spectral, message-passing, or information-theoretic decompositions to enable efficient and scalable implementations.

A plausible implication is that such measures provide a canonical way to extract multi-scale backbone structure in complex networks, bridging classical local (degree), global (eigenvector), and mesoscopic (community-aware) node ranking approaches.

6. Applications and Open Directions

Regime-detection-based local centralities are applied to diverse domains: social contact tracing, urban infrastructure analysis, core-periphery financial systems, biological networks, and information flow modeling. Their success in detecting structurally important nodes at multiple scales or regimes supports their adoption in fields where classical centrality measures suffer from localization failure or inability to resolve intermediate structures.

Ongoing open problems include formal criteria for optimal regime selection in noisy or overlapping communities, computational improvements for large diffusion-based measures, and further understanding of the dynamic interplay between different regime-detection methods and application-specific centrality requirements.

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Key references:

• "A local eigenvector centrality" (Clark et al., 5 Nov 2025)
• "Centrality metrics and localization in core-periphery networks" (Barucca et al., 2015)
• "Map Equation Centrality: Community-aware Centrality based on the Map Equation" (Blöcker et al., 2022)
• "Scale-dependent measure of network centrality from diffusion dynamics" (Arnaudon et al., 2019)