Topological strata of weighted complex networks

Abstract

The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and --more recently-- correlations between neighboring nodes. However, statistical methods quickly become cumbersome when dealing with many-body properties and do not capture the precise mesoscopic structure of complex networks. Here we introduce a novel method, based on persistent homology, to detect particular non-local structures, akin to weighted holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchically nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasilocal network properties, because of the intrinsic non-locality of homological properties, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. Moreover, this new method creates the first bridge between network theory and algebraic topology, which will allow to import the toolset of algebraic methods to complex systems.

26 pages, 19 figures, 1 table

References in corpus (18)

• Critical phenomena in complex networks
• Scale-free brain functional networks
• A tool for filtering information in complex systems
• Extracting the multiscale backbone of complex weighted networks
• High-resolution measurements of face-to-face contact patterns in a primary school
• Sustaining the Internet with Hyperbolic Mapping
• Prominence and control: The weighted rich-club effect
• Persistent Homology of Complex Networks
• Robust classification of salient links in complex networks
• Markov dynamics as a zooming lens for multiscale community detection: non clique-like communities and the field-of-view limit
• Spectral coarse-graining of complex networks
• Systematic Topology Analysis and Generation Using Degree Correlations
• Hierarchical information clustering by means of topologically embedded graphs
• Directed network modules
• The Organization of Strong Links in Complex Networks
• Degree Correlations in Random Geometric Graphs
• Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web
• Clear and Compress: Computing Persistent Homology in Chunks

Cited by in corpus (60)

• Networks beyond pairwise interactions: structure and dynamics
• Simplicial models of social contagion
• A roadmap for the computation of persistent homology
• Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes
• Network Analysis of Particles and Grains
• Emergent Hyperbolic Network Geometry
• Simplicial complexes and complex systems
• Topological data analysis of contagion maps for examining spreading processes on networks
• Persistent homology of time-dependent functional networks constructed from coupled time series
• Higher-order simplicial synchronization of coupled topological signals
• Interdisciplinary and physics challenges of Network Theory
• Hypergraph reconstruction from network data
• Emergent Complex Network Geometry
• Unveiling the higher-order organization of multivariate time series
• Weighted Growing Simplicial Complexes
• Beyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networks
• Construction of and efficient sampling from the simplicial configuration model
• Weighted simplicial complexes and their representation power of higher-order network data and topology
• Growing scale-free simplices
• Complex Quantum Network Geometries: Evolution and Phase Transitions
• Network Geometry and Complexity
• Efficient modeling of higher-order dependencies in networks: from algorithm to application for anomaly detection
• Demonstration of Topological Data Analysis on a Quantum Processor
• Persistent Homology Guided Force-Directed Graph Layouts
• Persistent Dirac for molecular representation
• Complex Quantum Network Manifolds in Dimension $d>2$ are Scale-Free
• A Short Survey of Topological Data Analysis in Time Series and Systems Analysis
• Topology and dynamics of higher-order multiplex networks
• Persistent homology of unweighted complex networks via discrete Morse theory
• Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guide
• Architecture and evolution of semantic networks in mathematics texts
• HONEM: Learning Embedding for Higher Order Networks
• Simplex2Vec embeddings for community detection in simplicial complexes
• On the reorderability of node-filtered order complexes
• Topological Data Analysis of Human Brain Networks Through Order Statistics
• Fast computation of persistent homology representatives with involuted persistent homology
• Understanding Complex Systems: From Networks to Optimal Higher-Order Models
• Measuring nodes centrality when local and global measures overlap
• Knowledge gaps in the early growth of semantic networks
• Topological data analysis of C. elegans locomotion and behavior
• Weighted Persistent Homology
• Characterizing Reaction Route Map of Realistic Molecular Reactions based on Weight Rank Clique Filtration of Persistent Homology
• Network science Ising states of matter
• Topological estimation of the latent geometry of a complex network
• Computational Tools in Weighted Persistent Homology
• Forman-Ricci curvature and Persistent homology of unweighted complex networks
• Weighted persistent homology for biomolecular data analysis
• $P$-persistent homology of finite topological spaces
• Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning
• Reviews: Topological Distances and Losses for Brain Networks
• Weighted Persistent Homology
• Community Detection in the Hyperbolic Space
• Persistent Path Homology of Directed Networks
• Weighted persistent homology for osmolyte molecular aggregation and hydrogen-bonding network analysis
• Persistent Homology and Graphs Representation Learning
• Structure of a media co-occurrence network
• Computational Human Dynamics
• weg2vec: Event embedding for temporal networks
• Topological characterization of S[B] systems: From data to models of complexity
• Topological Run-time Monitoring for Complex Systems