Spectral Fluctuations in Multilayer Networks

• Spectral fluctuations in multilayer networks are characterized by the behavior of supra-Laplacian eigenvalues that govern interlayer and intralayer dynamics.
• Analytical methods, including perturbative expansions in weak and strong coupling regimes, yield explicit formulas linking spectral properties to diffusion and synchronization rates.
• This approach provides practical insights for designing networks by tuning interlayer coupling to optimize diffusion efficiency and enhance synchronization stability.

Spectral fluctuations in multilayer networks concern the detailed behavior of eigenvalues and eigenfunctions associated with operators—typically the adjacency or Laplacian matrices—constructed on networked systems composed of multiple interdependent layers. These spectral properties govern, in analytically tractable fashion, key dynamical features such as diffusion speed, synchronization stability, and critical phenomena in complex systems with layered structures.

1. Mathematical Formulation and Decoupling of the Supra-Laplacian

A general multilayer (specifically, multiplex) network with $M$ layers and $N$ nodes per layer is mathematically represented by the so-called supra-Laplacian, defined as

$$L = \left( \bigoplus_{\alpha=1}^M L^{(\alpha)} \right) + (I \otimes I_N)$$

where $L^{(\alpha)} = S^{(\alpha)} - W^{(\alpha)}$ is the Laplacian of layer $\alpha$, $W^{(\alpha)}$ its weighted adjacency, $S^{(\alpha)}$ the associated diagonal strength matrix, $I$ is the Laplacian of the (weighted) interlayer connectivity network, and $I_N$ the $N \times N$ identity matrix (Sole-Ribalta et al., 2013).

The direct sum, $\bigoplus$, gives a block-diagonal matrix for the isolated layers, while the Kronecker product, $I \otimes I_N$, encodes uniform interlayer coupling for each node across layers. The decoupling structure is exploited so that the spectrum of $L$ is explicitly determined by the spectra of the block-diagonal (intralayer) and Kronecker (interlayer) terms: every eigenvalue of $I$ appears in the spectrum of $L$ with (at least) $N$-fold multiplicity.

Introducing a control parameter, $D_x$, rescaling the interlayer versus intralayer connectivities, yields

$$L = L^{(\text{intra})} + D_x\, \hat{L}^{(\text{inter})}$$

where $D_x$ tunes the relative interlayer coupling strength and hatted matrices denote normalized versions.

2. Asymptotic Regimes and Spectral Fluctuation Structure

The intricate dependence of spectral fluctuations on $D_x$ is analytically revealed by perturbative expansion in two asymptotic regimes:

Weak Interlayer Coupling ($D_x \ll 1$)

• The spectrum is only slightly perturbed from the direct sum of layer spectra.
• For an intralayer eigenpair $$L^{(\gamma)}x^{(\gamma)} = \lambda^{(\gamma)} x^{(\gamma)}$$, the corresponding supra-Laplacian eigenvalue is

$$\lambda \approx \lambda^{(\gamma)} + \hat{s}^I_\gamma D_x$$

where $\hat{s}^I_\gamma$ is the (normalized) interlayer strength for layer $\gamma$.

• The smallest nonzero eigenvalues of the full Laplacian are

$\lambda \approx D_x \hat{\lambda}^I$

with $\hat{\lambda}^I$ nonzero eigenvalues of $I$.

Strong Interlayer Coupling ($D_x \gg 1$)

• The spectrum bifurcates:
  • A set scales as $D_x \hat{\lambda}^I + O(1)$ ("pinned" by interlayer coupling).
  • The other set remains $\mathcal{O}(1)$ and approximates the spectrum of the Laplacian of the "averaged network"

  $$W^{(\mathrm{AV})} = \frac{1}{M} \sum_{\alpha=1}^M W^{(\alpha)}$$

These asymptotic results provide explicit analytic formulas for essentially the entire spectrum under physically relevant limits, directly linking spectral fluctuations to network construction (Sole-Ribalta et al., 2013).

3. Dynamical Implications: Diffusion and Synchronizability

Spectral fluctuations have direct dynamical significance:

• Diffusion: The slowest relaxation time, $\tau \propto 1/\lambda_2$ ($\lambda_2$ the first nontrivial Laplacian eigenvalue), controls the convergence rate of diffusion. For $D_x \ll 1$, $\lambda_2 \propto D_x$ leads to slow diffusion across layers, while for $D_x \gg 1$, $\lambda_2$ approaches the gap of the average network, enabling possible "superdiffusion."
• Synchronization: The stability and robustness of synchronization is quantified by the eigenratio $R = \lambda_N / \lambda_2$. The ability to tune $D_x$ to minimize $R$ follows from analytic expressions for both eigenvalues in the two regimes. For intermediate coupling, optimal synchronizability may be achieved.

Such analytic dependency on $D_x$ permits design guidelines for desired dynamical states.

4. Spectral Fluctuations Across Structural Transitions

Spectral fluctuations undergo qualitative rearrangement as the system is tuned through a structural transition in the interlayer coupling:

• As $D_x$ crosses a critical threshold, the slope of $\lambda_2$ as a function of $D_x$ exhibits a discontinuity, marking a switch from interlayer-dominant to intralayer-coherent diffusive modes.
• The corresponding eigenvectors reorganize, with the components flipping from antiphase between layers (for $D_x < D^*_x$) to alignment (for $D_x > D^*_x$), revealing how structural coherence emerges from entanglement of the layers.
• This crossover reflects the fundamental fluctuation regime change and is central for the emergence of "superdiffusion" or abrupt changes in system-wide timescales (Boccaletti et al., 2014).

5. Rigorous Bounds and Dimensionality Reduction via Quotient Graphs

Eigenvalue interlacing theorems link the spectral fluctuations of the full multilayer network to those of reduced representations (quotient graphs):

• For any partition into subnetworks (layers, aggregates, or "network of layers"), the eigenvalues of the quotient Laplacian or adjacency matrix interlace those of the full supra-matrix:

$$\lambda_i \leq \mu_i \leq \lambda_{i + (n-m)},\quad i=1,\dots,m$$

where $(\lambda_\bullet)$ are eigenvalues of the full network, $(\mu_\bullet)$ those of the $m \times m$ quotient (Sánchez-García et al., 2013, Cozzo et al., 2015).

• Consequently, spectral fluctuations in the multilayer structure are bounded and can sometimes be approximated by analysis on much lower-dimensional objects (aggregate or network-of-layers graphs).
• This enables rigorous, tractable assessment of dynamical critical points using only simplified models, with epidemics, synchronization, and diffusive timescales bounded between those of the full and aggregated system.

6. Sensitivity, Noise, and the Impact of Structural Perturbations

Spectral fluctuations exhibit sensitive dependence on both microscopic and macroscopic network modifications:

• The addition or alteration of a single interlayer edge can induce discontinuous "jumps" in $\lambda_2$ (algebraic connectivity), abruptly changing relaxation times and dynamical regimes (Diakonova et al., 2016).
• The propagation of noise and fluctuations between layers is fundamentally governed by the overlap of slow eigenmodes (e.g., Fiedler vectors), with amplification factors scaling as inverse powers of the algebraic connectivity. High overlap and small $\lambda_2$ in either layer can lead to large variance amplification, even when only one layer is directly driven by noise (Tyloo, 2022).
• These properties underscore the role of spectral fluctuations in system robustness and vulnerability, as small topological changes yield outsized impacts on dynamical behavior.

7. Broader Methodological, Structural, and Application Context

The analytical framework for spectral fluctuations in multilayer networks extends beyond simple eigenvalue counting:

• Scaling laws derived from random matrix theory show that localization properties of eigenfunctions (e.g., normalized localization length $\beta = x^*/(1+x^*)$) as well as eigenvalue spacing statistics universally collapse onto curves parameterized by effective network "bandwidth" and system size. This unifies spectral fluctuations across a wide variety of multilayer architectures and topologies (Méndez-Bermúdez et al., 2016, Tapia-Labra et al., 26 Sep 2024).
• Graph product multilayer networks allow explicit construction of networks with predictable spectral composition, as the spectra of products (Cartesian, direct, strong) are algebraically related to those of factor graphs—permitting direct control and insight into spectral fluctuation structure (Sayama, 2017).
• Core-periphery structure and nonlinear spectral optimization generalize classical spectral approaches to account for dual node-layer organization, with spectral fluctuation phenomena emerging as variations in coreness ranking across layers and the spectrum (Bergermann et al., 5 Dec 2024).
• Data-driven and inference applications: Spectral properties are used to inform community detection, segmentation, and inference in various domains (hyperspectral imaging, protein structure, airline transport, social graphs), often leveraging the sensitivity and universality of spectral fluctuations as a diagnostic and analytic tool (Zhang et al., 2021, Zhao et al., 12 Mar 2025, Shekhar et al., 18 Aug 2025).

These methods collectively establish spectral fluctuation analysis as a central approach for diagnosing, predicting, and engineering the dynamics and structure of complex multilayer networks. The universal features elucidated via scaling laws and RMT, as well as the analytic tractability afforded by decoupling and perturbative expansions, provide rigorous underpinnings for advances in theory and application.

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Table 1: Key Regimes and Spectral Properties in Multilayer Networks

Regime	Sup. Laplacian Eigenvalues	Dynamical Consequence
$D_x \ll 1$ (weak interlayer)	$\lambda \approx D_x \hat\lambda^I$ (for smallest eig.)	Slow interlayer processes
	$$\lambda \approx \lambda^{(\gamma)} + \hat{s}_\gamma^I D_x$$ (others)	Layer-dominated dynamics
$D_x \gg 1$ (strong interlayer)	$\lambda \approx D_x \hat\lambda^I +O(1)$	Modes "pinned" by interlayer
	$\lambda \approx \lambda_{\mathrm{AV}}$	Averaged-layer behavior

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Spectral fluctuations in multilayer networks thus hinge on explicit analytic relationships derived from the network construction, admit rigorous bounds and perturbative expansions, sharply characterize dynamical regimes, and connect directly to universal behaviors established in contemporary random matrix theory. These properties position spectral analysis as both a foundational and powerful methodology in complex network science.