Temporal Network Analysis Using Zigzag Persistence

Abstract

This work presents a framework for studying temporal networks using zigzag persistence, a tool from the field of Topological Data Analysis (TDA). The resulting approach is general and applicable to a wide variety of time-varying graphs. For example, these graphs may correspond to a system modeled as a network with edges whose weights are functions of time, or they may represent a time series of a complex dynamical system. We use simplicial complexes to represent snapshots of the temporal networks that can then be analyzed using zigzag persistence. We show two applications of our method to dynamic networks: an analysis of commuting trends on multiple temporal scales, e.g., daily and weekly, in the Great Britain transportation network, and the detection of periodic/chaotic transitions due to intermittency in dynamical systems represented by temporal ordinal partition networks. Our findings show that the resulting zero- and one-dimensional zigzag persistence diagrams can detect changes in the networks' shapes that are missed by traditional connectivity and centrality graph statistics.

arXiv commentUpdated to include missing references

References in corpus (13)

• A roadmap for the computation of persistent homology
• Modern temporal network theory: A colloquium
• Dynamically induced cascading failures in power grids
• Ripser: efficient computation of Vietoris-Rips persistence barcodes
• The Multilayer Temporal Network of Public Transport in Great Britain
• Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems
• Persistent Homology of Complex Networks for Dynamic State Detection
• On the Automatic Parameter Selection for Permutation Entropy
• Spatio-temporal Persistent Homology for Dynamic Metric Spaces
• Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
• The Importance of Forgetting: Limiting Memory Improves Recovery of Topological Characteristics from Neural Data
• Detecting bifurcations in dynamical systems with CROCKER plots
• Interleaving by Parts: Join Decompositions of Interleavings and Join-Assemblage of Geodesics

Cited by in corpus (4)

• Topological data analysis and machine learning
• Detecting bifurcations in dynamical systems with CROCKER plots
• ZigzagNetVis: Suggesting temporal resolutions for graph visualization using zigzag persistence
• Bounding the Interleaving Distance for Mapper Graphs with a Loss Function