The higher-order spectrum of simplicial complexes: a renormalization group approach

Abstract

Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are investigated in network topology are simplicial complexes. Simplicial complexes generalize networks by not only taking pairwise interactions into account, but also taking into account many-body interactions between more than two nodes. Higher-order Laplacians are topological operators that describe higher-order diffusion on simplicial complexes and constitute the natural mathematical objects that capture the interplay between network topology and dynamics. We show that higher-order up and down Laplacians can have a finite spectral dimension, characterizing the long time behaviour of the diffusion process on simplicial complexes that depends on their order $m$. We provide a renormalization group theory for the calculation of the higher-order spectral dimension of two deterministic models of simplicial complexes: the Apollonian and the pseudo-fractal simplicial complexes. We show that the RG flow is affected by the fixed point at zero mass, which determines the higher-order spectral dimension $d_S$ of the up-Laplacians of order $m$ with $m\ge 0$.

arXiv comment(39 pages,6 figures)

References in corpus (36)

• A tool for filtering information in complex systems
• Simplicial models of social contagion
• Pseudofractal Scale-free Web
• Spectral Dimension of the Universe
• Reconstructing the Universe
• Critical behavior of colored tensor models in the large N limit
• Explosive higher-order Kuramoto dynamics on simplicial complexes
• Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplices
• Spectra of complex networks
• Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes
• Fractal and Transfractal Recursive Scale-Free Nets
• Fractal properties of quantum spacetime
• Emergent Hyperbolic Network Geometry
• Network geometry with flavor: from complexity to quantum geometry
• Spectral geometry as a probe of quantum spacetime
• Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?
• Interdisciplinary and physics challenges of Network Theory
• Ordinary Percolation with Discontinuous Transitions
• Emergent Complex Network Geometry
• Percolation in Hierarchical Scale-Free Nets
• Simplicial complexes: spectrum, homology and random walks
• PCA meets RG
• High dimensional random Apollonian networks
• Synchronization in Network Geometries with Finite Spectral Dimension
• The spectral dimension of the branched polymers phase of two-dimensional quantum gravity
• The spectral dimension of generic trees
• Spectral dimensions of hierarchical scale-free networks with shortcuts
• Topological Percolation on Hyperbolic Simplicial Complexes
• The spectral dimension of simplicial complexes: a renormalization group theory
• Small-World Bonds and Patchy Percolation on the Hanoi Network
• Network Geometry and Complexity
• Critical fluctuations in spatial complex networks
• The inverse Mermin-Wagner theorem for classical spin models on graphs
• Percolation on branching simplicial and cell complexes and its relation to interdependent percolation
• Renormalization group for link percolation on planar hyperbolic manifolds
• Spectral renormalization group theory on networks

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• What are higher-order networks?
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• Complex Quantum Networks: a Topical Review
• The topological Dirac equation of networks and simplicial complexes
• Turing patterns on discrete topologies: from networks to higher-order structures
• Diffusion-driven instability of topological signals coupled by the Dirac operator
• Complex networks with tuneable dimensions as a universality playground
• Predicting hyperlinks via hypernetwork loop structure
• The mass of simple and higher-order networks
• Geometry, Topology and Simplicial Synchronization
• Dirac gauge theory for topological spinors in 3+1 dimensional networks
• Global topological synchronization of weighted simplicial complexes
• Random matrices with row constraints and eigenvalue distributions of graph Laplacians
• Network Renormalization
• Generalised spectral dimensions in non-perturbative quantum gravity
• Hyper-diffusion on multiplex networks
• Hammerstein equations for sparse random matrices
• The Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution
• Deterministic simplicial complexes