Stable Signatures for Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence

12/11/2017
by   Woojin Kim, et al.
0

When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. In a similar vein, studying the dynamics of social networks leads to the problem of characterizing groups/communities as they form and disperse throughout time. Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent data. Given a finite dynamic graph (DG), we first construct a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG. Based on standard results, we then obtain a persistence diagram or barcode from this zigzag persistence module. We prove that these barcodes are stable under perturbations in the input DG under a suitable distance between DGs that we identify. More precisely, our stability theorem can be interpreted as providing a lower bound for the distance between DGs. Since it relies on barcodes, and their bottleneck distance, this lower bound can be computed in polynomial time from the DG inputs. Since DGs can be given rise by applying the Rips functor (with a fixed threshold) to dynamic metric spaces, we are also able to derive related stable invariants for these richer class of dynamic objects. Along the way, we propose a summarization of dynamic graphs that captures their time-dependent clustering features which we call formigrams. These set-valued functions generalize the notion of dendrogram, a prevalent tool for hierarchical clustering. In order to elucidate the relationship between our distance between two DGs and the bottleneck distance between their associated barcodes, we exploit recent advances in the stability of zigzag persistence due to Botnan and Lesnick, and to Bjerkevik.

READ FULL TEXT

page 1

page 2

page 3

page 4

12/11/2017

Stable Signatures for Dynamic Metric Spaces via Zigzag Persistent Homology

When studying flocking/swarming behaviors in animals one is interested i...
12/03/2018

Stable Persistent Homology Features of Dynamic Metric Spaces

Characterizing the dynamics of time-evolving data within the framework o...
08/16/2008

Persistent Clustering and a Theorem of J. Kleinberg

We construct a framework for studying clustering algorithms, which inclu...
08/16/2017

Distances and Isomorphism between Networks and the Stability of Network Invariants

We develop the theoretical foundations of a network distance that has re...
07/30/2022

On the bottleneck stability of rank decompositions of multi-parameter persistence modules

The notion of rank decomposition of a multi-parameter persistence module...
02/12/2020

Graph Similarity Using PageRank and Persistent Homology

The PageRank of a graph is a scalar function defined on the node set of ...
12/13/2018

The Relationship Between the Intrinsic Cech and Persistence Distortion Distances for Metric Graphs

Metric graphs are meaningful objects for modeling complex structures tha...