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Spectral properties of the Laplacian of multiplex networks
Albert Sole-Ribalta, Manlio De Domenico, Nikos E. Kouvaris, Albert Diaz-Guilera, Sergio Gomez, Alex Arenas
TL;DR
The paper addresses how multiplex topology shapes dynamical processes by analyzing the Laplacian spectrum of multilayer networks. It decomposes the supra-Laplacian into intra- and interlayer contributions, derives asymptotic spectral expressions, and applies them to diffusion and synchronization. The results identify super-diffusion and an optimal interlayer coupling for synchronization stability.
Problem
The paper investigates how to capture the role of multiplex topology in dynamical processes beyond frameworks limited to particular network configurations.
Method
The paper decomposes the multiplex Laplacian into intralayer and interlayer structures, then uses perturbation and asymptotic analysis to characterize its spectrum.
Results
The derived spectrum characterizes diffusion and synchronization, revealing shorter multiplex diffusion times than in any individual layer and an optimal interlayer coupling for synchronization stability.
Takeaways & Limitations
The multiplex structure has dynamical implications without monoplex counterparts, including super-diffusive behavior and a most-stable synchronization regime at a particular interlayer coupling.
Abstract
from arXiv · showhide
One of the more challenging tasks in the understanding of dynamical properties of models on top of complex networks is to capture the precise role of multiplex topologies. In a recent paper, Gomez et al. [Phys. Rev. Lett. 101, 028701 (2013)] proposed a framework for the study of diffusion processes in such networks. Here, we extend the previous framework to deal with general configurations in several layers of networks, and analyze the behavior of the spectrum of the Laplacian of the full multiplex. We derive an interesting decoupling of the problem that allow us to unravel the role played by the interconnections of the multiplex in the dynamical processes on top of them. Capitalizing on this decoupling we perform an asymptotic analysis that allow us to derive analytical expressions for the full spectrum of eigenvalues. This spectrum is used to gain insight into physical phenomena on top of multiplex, specifically, diffusion processes and synchronizability.
I. INTRODUCTION
Multiplex networks represent the same objects across distinct layers with layer-specific connectivity and one-to-one interlayer connections. The paper decomposes the multiplex Laplacian into intralayer and interlayer contributions to analyze dynamical processes.
- I. INTRODUCTION: Multiplex networks encode the same objects in each layer while allowing different connectivity patterns and one-to-one interlayer connections.Examples include social relationships, transportation systems, and brain organization.
- I. INTRODUCTION: The Laplacian spectrum governs linearized dynamics, with λ2 linked to diffusion and synchronization times and the largest eigenvalue linked to synchronization stability.The decomposition supports spectral and perturbative analysis of diffusion timescales and synchronizability.
- I. INTRODUCTION: The interlayer network is modeled by an M × M Laplacian when interlayer connectivity is identical for all nodes.The analysis assumes undirected, globally connected intralayer and interlayer networks without self-loops.
- I. INTRODUCTION: The supra-Laplacian is separated into the direct sum of intralayer Laplacians and an interlayer contribution.The interlayer contribution is expressed using the interlayer Laplacian and an N × N identity matrix.
- I. INTRODUCTION: The supra-Laplacian decomposition is presented as the basis for discovering spectral properties of multiplex networks.The paper’s structure proceeds from decomposition and interlayer analysis to perturbation theory and dynamical implications.
III. THE ROLE OF THE INTERLAYER NETWORK
The interlayer Laplacian contributes directly to the supra-Laplacian spectrum: its eigenvalues are inherited, with multiplicities scaled by the number of nodes. Small-layer cases permit explicit formulas, while larger cases face algebraic limitations.
- III. THE ROLE OF THE INTERLAYER NETWORK: Each interlayer-Laplacian eigenvalue appears in the supra-Laplacian with multiplicity multiplied by N.Eigenvectors can be constructed as xI ⊗ 1, combining an interlayer eigenvector with the all-ones node vector.
- III. THE ROLE OF THE INTERLAYER NETWORK: The supra-Laplacian inherits every eigenvalue of the interlayer Laplacian.These eigenvalues generally require numerical computation unless the interlayer structure has a tractable special form.
- A. Solvable case: small number of layers: For two layers, the interlayer spectrum is {0, 2Dx}, where Dx controls the relative strength of interlayer and intralayer contributions.This gives an explicit example of how interlayer coupling enters the full spectrum.
- A. Solvable case: small number of layers: For three layers, interlayer-link weights Dαβ determine the interlayer network Laplacian.Analytical expressions also exist for four and five layers but become too involved to be useful.
B. Solvable case: Uniform interlayer weights
Uniform interlayer weights make the interlayer spectrum analytically tractable in several topologies and embed its scaled eigenvalues in the full supra-Laplacian spectrum.
- Uniform fully connected networks: For a fully connected interlayer network with equal weight Dx, the spectrum is {0, MDx, . . . , MDx}, with MDx having multiplicity M −1.The nonzero interlayer eigenvalue scales with the number of layers and the common interlayer weight.
- Other uniform topologies: Uniform but non-fully-connected interlayer networks have eigenvalues proportional to Dx, with coefficients determined by the chosen topology.Regular graphs, cycles, and paths are among the cases that can be solved analytically.
- Spectral implications: Perturbation analysis shows that interlayer eigenvalues constrain the shape of the complete supra-Laplacian spectrum and help analyze multiplex dynamical phenomena.The paper uses these restrictions in subsequent analyses of physical processes on multiplex networks.
- Spectral decomposition: The supra-Laplacian separates into intralayer and interlayer contributions, with Dx encoding their relative strength.The interlayer Laplacian and weights are written with Dx factored out, while the remaining contributions are represented with hats.
- Spectral decomposition: The scaled interlayer eigenvalues are contained in the spectrum of the supra-Laplacian, while the intralayer contribution has the union of layer spectra.This provides a direct spectral link between interlayer structure and the full multiplex Laplacian.
A. Weak interlayer networks
When interlayer coupling is weak, the supra-Laplacian spectrum separates into small interlayer-controlled eigenvalues and intralayer eigenvalues perturbed at first order.
- Perturbative construction: Each selected intralayer eigenvector is lifted to the supra-Laplacian as v(γ) = eγ ⊗x(γ).Here eγ is the canonical layer vector selecting layer γ.
- Perturbative shifts: The first-order perturbation of intralayer eigenvalues is determined by the diagonal element ˆℓI γγ of the interlayer Laplacian.The perturbative calculation uses the symmetry of the supra-Laplacians and projection onto the unperturbed eigenvector.
- Perturbative shifts: For small Dx, every non-zero intralayer eigenvalue is shifted by ˆsI γγDx, a quantity depending only on the layers.The zero eigenvalues are not treated perturbatively because their exact values are already known from the interlayer spectrum.
B. Strong interlayer networks
When interlayer coupling is strong, perturbation around the interlayer supra-Laplacian yields average-network eigenvalues for slow modes and linearly growing eigenvalues for transverse modes.
- Perturbative construction: The strong-coupling analysis uses ǫ = 1/Dx and perturbs eigenpairs of the interlayer supra-Laplacian.The Kronecker-product structure supplies eigenvectors xI ⊗u for the unperturbed problem.
- Perturbative construction: The perturbation equation combines the interlayer supra-Laplacian with the intralayer supra-Laplacian to determine the eigenvalue corrections.Projection with the interlayer eigenvector is then used to obtain the correction equation.
- Strong interlayer networks: One shift is exactly zero for the uniform interlayer eigenvector, recovering the exact eigenvalues previously identified from the interlayer contribution.The other associated eigenvalues diverge linearly with Dx, with shifts determined by the perturbative equation.
- Strong interlayer networks: For Dx ≫1, the O(1) modes and the smallest N −1 non-zero eigenvalues approach those of the Laplacian of the average network.This extends the corresponding result previously established for the two-layer multiplex.
C. Global structure of the supra-Laplacian spectrum
The supra-Laplacian spectrum has a general asymptotic structure: weak coupling produces linear interlayer modes, while strong coupling separates average-network modes from modes scaling with interlayer strength.
- One eigenvalue remains exactly λ = 0 for every value of Dx.
- Weak interlayer coupling: For Dx ≪1, the smallest non-zero eigenvalues are O(Dx), while the remaining M(N − 1) eigenvalues are O(1).
- Strong interlayer coupling: For Dx ≫1, the smallest N −1 non-zero eigenvalues approach those of the average-network Laplacian, while the remaining N(M −1) eigenvalues are O(Dx).
- All eigenvalues vary continuously and non-decreasingly with Dx.
- Scope: The spectral structure is general for undirected, connected, loop-free layers and interlayer networks; directed networks may instead produce complex eigenvalues.
- Illustration: Figure 2 compares exact eigenvalues with analytical approximations for weak and strong coupling in a three-layer, five-node toy multiplex.
A. On the timescale of diffusion dynamics
For linear diffusion on a multiplex, the convergence timescale is governed by the supra-Laplacian’s second eigenvalue, whose controlling structure changes between weak and strong interlayer coupling.
- Diffusion model: The multiplex diffusion model uses linear intralayer and interlayer couplings, with one interlayer coupling constant shared across corresponding nodes.
- Diffusion model: The diffusion solution consists of normal modes decaying as φi(t) = φ(0)e−λit, where λi are supra-Laplacian eigenvalues.
- Validation: Figure 3 evaluates the approximation for small and large Dx in a four-layer multiplex whose layers have Barabási–Albert power-law networks of 200 nodes.
- Timescale: The second eigenvalue governs convergence, giving a diffusion timescale τ ∝ λ2^-1.
- Weak interlayer coupling: For Dx ≪1, λ2(L) corresponds to the second eigenvalue of the interlayer network, so the timescale is controlled by interlayer connectivity.
- Strong interlayer coupling: For Dx ≫1, λ2(L) is approximated by the second eigenvalue of the average network, yielding τ ∝ (λ2(W AV))^-1.
B. On the stability of the synchronization manifold of coupled phase oscillators
Multiplex synchronizability is analyzed through the eigenratio R = λN/λ2, with weak- and strong-coupling approximations used to characterize its dependence on interlayer coupling.
- Synchronizability measure: Synchronizability is reduced to computing the multiplex eigenratio R = λN/λ2, where λN is the largest Laplacian eigenvalue.
- Validation: Figure 4 compares exact eigenratio values with weak- and strong-coupling approximations for 200-node Erdős–Rényi layers with edge probability 0.5.
- Asymptotic regimes: The analysis considers asymptotic eigenratio behavior for weak and strong interlayer coupling, Dx ≪1 and Dx ≫1.
- Weak interlayer coupling: For weak coupling, λ2(L) ≈ λ2(LI), while the largest eigenvalue equals the largest intralayer Laplacian eigenvalue across layers.
- Strong interlayer coupling: For strong coupling, λ2 is approximated by the average-network second eigenvalue, while λN depends on the interlayer-network maximum eigenvalue plus a layer-dependent offset.
VI. CONCLUSIONS
The paper derives asymptotic spectral expressions for multiplex Laplacians and applies them to diffusion and synchronization, revealing effects absent from single-layer networks.
- The analytical spectrum supports inference about diffusion and synchronization dynamics on multiplex networks.
- Diffusion can be super-diffusive, with multiplex timescales shorter than those of every individual layer network.
- The synchronization eigenratio has an optimal Dx at which the full multiplex structure is most stable.
- The spectral results may apply to other processes governed by Laplacian or related-matrix spectra.
Appendix A: Supplemental material
The supplemental material compares eigen-ratio approximations and second Laplacian eigenvalues across multiplexes with varied layer structures, community overlap, and interlayer connectivity.
- Eigen-ratio approximations: Eigen-ratio plots compare the multiplex ratio with proposed approximations for weak and strong interlayer networks.The plots use multiplexes of three layers with either scale-free or strongly overlapping community-based layer networks.
- Second-eigenvalue comparisons: For four-layer multiplexes, the supplemental comparisons examine the second eigenvalue across different Laplacians for homogeneous Erdős–Rényi layers.Each layer contains 200 nodes with edge probability 0.5.
- Second-eigenvalue comparisons: Additional four-layer comparisons use strongly overlapping community networks with low inter-community edge probability.The community-based layers have within-community edge probability 0.5 and inter-community probability 0.05.
- Second-eigenvalue comparisons: Mixed four-layer multiplexes combine two strongly overlapping community layers with two homogeneous Erdős–Rényi layers.All layers contain 200 nodes; the Erdős–Rényi layers use edge probability 0.5.
- Second-eigenvalue comparisons: The appendix also compares multiplexes combining three Barabási–Albert scale-free layers with one Erdős–Rényi layer.Each layer contains 200 nodes.
- Second-eigenvalue comparisons: Finally, it varies scale-free layer heterogeneity by using attachment to 3 existing nodes in two layers and 7 in the other two.This comparison concerns four-layer multiplexes whose layers each contain 200-node Barabási–Albert networks.