2020/03/31 by Marcus Reitz, Ginestra Bianconi · 1 citation
Computer Science · Mathematics · Neuroscience · Physics and Astronomy · #Abstract simplicial complex #Betti number #Dimension (graph theory) #Functional Brain Connectivity Studies #Simplicial approximation theorem #Simplicial complex #Simplicial homology #Simplicial manifold #Spectrum (functional analysis) #Topological and Geometric Data Analysis #Topology (electrical circuits) #advanced mathematical theories #cond-mat.dis-nn #cond-mat.stat-mech #gr-qc #hep-lat #physics.soc-ph
paper · pdf · doi:10.1088/1751-8121/ab9338
published as J. Phys. A: Math. Theor. 53 295001 (2020) · (39 pages,6 figures)
openalex created_date 2020/03/27 · arxiv created 2020/05/14 · openalex publication_date 2020/05/14 · arxiv updated 2020/07/15 · openalex updated_date 2026/08/05
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 ⩾ 0.