Weighted simplicial complexes and their representation power of higher-order network data and topology

Abstract

Hypergraphs and simplical complexes both capture the higher-order interactions of complex systems, ranging from higher-order collaboration networks to brain networks. One open problem in the field is what should drive the choice of the adopted mathematical framework to describe higher-order networks starting from data of higher-order interactions. Unweighted simplicial complexes typically involve a loss of information of the data, though having the benefit to capture the higher-order topology of the data. In this work we show that weighted simplicial complexes allow to circumvent all the limitations of unweighted simplicial complexes to represent higher-order interactions. In particular, weighted simplicial complexes can represent higher-order networks without loss of information, allowing at the same time to capture the weighted topology of the data. The higher-order topology is probed by studying the spectral properties of suitably defined weighted Hodge Laplacians displaying a normalized spectrum. The higher-order spectrum of (weighted) normalized Hodge Laplacians is here studied combining cohomology theory with information theory. In the proposed framework, we quantify and compare the information content of higher-order spectra of different dimension using higher-order spectral entropies and spectral relative entropies. The proposed methodology is tested on real higher-order collaboration networks and on the weighted version of the simplicial complex model "Network Geometry with Flavor".

arXiv comment20 pages, 11 figures

References in corpus (37)

• Fast unfolding of communities in large networks
• Who is the best connected scientist? A study of scientific coauthorship networks
• Networks beyond pairwise interactions: structure and dynamics
• The physics of higher-order interactions in complex systems
• A roadmap for the computation of persistent homology
• Dynamics on higher-order networks: A review
• Random Walks, Markov Processes and the Multiscale Modular Organization of Complex Networks
• Laplacian Dynamics and Multiscale Modular Structure in Networks
• Entropy measures for complex networks: Toward an information theory of complex topologies
• Damage and fluctuations induce loops in optimal transport networks
• Explosive higher-order Kuramoto dynamics on simplicial complexes
• Topological Signal Processing over Simplicial Complexes
• Topological strata of weighted complex networks
• Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian
• Spectral entropies as information-theoretic tools for complex network comparison
• Simplicial complexes and complex systems
• Forman curvature for complex networks
• Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond
• Network geometry with flavor: from complexity to quantum geometry
• The Shannon and the Von Neumann entropy of random networks with heterogeneous expected degree
• Coupled Dynamics on Hypergraphs: Master Stability of Steady States and Synchronization
• Higher-order simplicial synchronization of coupled topological signals
• Weighted Growing Simplicial Complexes
• Beyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networks
• Simplicial complexes: higher-order spectral dimension and dynamics
• Entropy of dynamical social networks
• Complex Quantum Network Geometries: Evolution and Phase Transitions
• Statistical physics of complex information dynamics
• Network Geometry and Complexity
• Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian
• The topological Dirac equation of networks and simplicial complexes
• Laplacian paths in complex networks: information core emerges from entropic transitions
• Spectral Detection of Simplicial Communities via Hodge Laplacians
• The hidden topological structure of flow network functionality
• Hysteresis and synchronization processes of Kuramoto oscillators on high-dimensional simplicial complexes with the competing simplex-encoded couplings
• Balanced Hodge Laplacians Optimize Consensus Dynamics over Simplicial Complexes
• The higher-order spectrum of simplicial complexes: a renormalization group approach

Cited by in corpus (25)

• Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes
• Complex Quantum Networks: a Topical Review
• The simpliciality of higher-order networks
• Persistent Dirac for molecular representation
• Dirac signal processing of higher-order topological signals
• Turing patterns on discrete topologies: from networks to higher-order structures
• Diffusion-driven instability of topological signals coupled by the Dirac operator
• Local Dirac Synchronization on Networks
• Topology and dynamics of higher-order multiplex networks
• The mass of simple and higher-order networks
• Dirac gauge theory for topological spinors in 3+1 dimensional networks
• A framework to generate hypergraphs with community structure
• Higher-order signal processing with the Dirac operator
• Quantum entropy couples matter with geometry
• Emergent information dynamics in many-body interconnected systems
• Synchronizability in randomized weighted simplicial complexes
• Global Topological Dirac Synchronization
• Persistent homology analysis of type 2 diabetes genome-wide association studies in protein-protein interaction networks
• Quantum walk on simplicial complexes for simplicial community detection
• Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning
• Topological Signal Processing on Quantum Computers for Higher-Order Network Analysis
• Provable quantum speedups for computing persistence in topological data analysis
• Designing topological cluster synchronization patterns with the Dirac operator
• HOI-Brain: a novel multi-channel transformers framework for brain disorder diagnosis by accurately extracting signed higher-order interactions from fMRI
• Towards a Quantitative Theory of Digraph-Based Complexes and its Applications in Brain Network Analysis