The spectral dimension of simplicial complexes: a renormalization group theory

Abstract

Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the spectral dimension of the graph Laplacian of two classes of non-amenable $d$ dimensional simplicial complexes: the Apollonian networks and the pseudo-fractal networks. We analyse the scaling of the spectral dimension with the topological dimension $d$ for $d\to \infty$ and we point out that randomness such as the one present in Network Geometry with Flavor can diminish the value of the spectral dimension of these structures.

(30 pages, 5 figures)

References in corpus (10)

• A tool for filtering information in complex systems
• Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplices
• Fractal and Transfractal Recursive Scale-Free Nets
• Fractal properties of quantum spacetime
• Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?
• Simplicial SIS model in scale-free uniform hypergraph
• Emergent Complex Network Geometry
• Percolation in Hierarchical Scale-Free Nets
• The spectral dimension of generic trees
• Small-World Bonds and Patchy Percolation on the Hanoi Network

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• Laplacian Renormalization Group for heterogeneous networks
• Does the brain behave like a (complex) network? I. Dynamics
• Random matrices with row constraints and eigenvalue distributions of graph Laplacians
• Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes
• Network Renormalization
• Hammerstein equations for sparse random matrices
• Clustering-induced localization of quantum walks on networks
• Anomalous finite-size scaling in higher-order processes with absorbing states
• Deterministic simplicial complexes