Source-linked AI summary
Diffusion dynamics on multiplex networks
Sergio Gomez, Albert Diaz-Guilera, Jesus Gomez-Gardeñes, Conrad J. Perez-Vicente, Yamir Moreno, Alex Arenas
TL;DR
Diffusion on interconnected network layers requires models that capture multiple structural and temporal scales. The paper constructs a supra-Laplacian and uses perturbative spectral analysis to derive asymptotic eigenvalue behavior, showing how interlayer coupling controls diffusion timescales and can accelerate diffusion.
Problem
Characterizing the topology and dynamics of systems composed of interconnected networks remains difficult because standard models treat all links at one temporal and structural scale.
Method
The paper dimensionally lifts individual layer Laplacians into a supra-Laplacian and applies perturbative spectral analysis to a two-layer multiplex with conserved nodes.
Results
For weak interlayer diffusion, the global diffusion timescale is controlled by 1/(2D_x), while strong coupling yields divergent and finite spectral branches with smallest nonzero eigenvalue λ_s/2.
Takeaways & Limitations
Multiplex structure can accelerate the less diffusive layer and may produce super-diffusion when interlayer diffusion exceeds intralayer diffusion.
Takeaways & Limitations
The analysis assumes connected, undirected layers and is specifically developed for two-layer multiplexes with nodes preserved across layers.
Abstract
from arXiv · showhide
We study the time scales associated to diffusion processes that take place on multiplex networks, i.e. on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-Laplacian matrix, which consists of a dimensional lifting of the Laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-Laplacian allows us to understand the physics of diffusion-like processes on top of multiplex networks.
Supplemental material
The supplemental material compares the second-smallest eigenvalues λ2 of different Laplacians across multiplex networks with varied layer structures. One configuration shows an absence of super-diffusion, attributed to semi-superposition forming a weighted spanning graph.
- Scale-free layers: λ2 comparisons use multiplexes with two 1000-node layers, including scale-free layers with degree exponents −2.5 and −3.The first layer has P(k) ∼k−2.5, while the second has P(k) ∼k−3.
- Scale-free and Erd¨os-R´enyi layers: λ2 is also compared when a P(k) ∼ k−2.5 scale-free layer is paired with an Erd¨os-R´enyi layer of average degree ⟨k⟩= 8.Both layers contain 1000 nodes.
- Scale-free and small-world layers: Another λ2 comparison pairs the P(k) ∼k−2.5 scale-free layer with a small-world layer having ⟨k⟩= 8 and replacement probability r = 0.3.The multiplex again consists of two layers with 1000 nodes each.
- Scale-free and lattice layers: A lattice configuration compares λ2 for a P(k) ∼k−2.5 scale-free layer and a 40×25 lattice with eight neighbors per node and periodic boundary conditions.The two layers each contain 1000 nodes.
- Absence of super-diffusion: The scale-free network with 400 extra random links shows absence of super-diffusion because semi-superposition (W1 + W2)/2 is a weighted spanning graph.This configuration compares the second-smallest eigenvalues λ2 of different Laplacians in a two-layer, 1000-node-per-layer multiplex.
- Community-structured layers: Further λ2 comparisons vary community structure, including four-community layers and a pairing of four and 17 communities with specified internal, external, and clustering characteristics.The four-community configurations use 250-node communities; the latter pairing has ⟨kint⟩= 50 and ⟨kext⟩= 8 in the 17-community layer.