Analysis of Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence

Kim, Woojin; Mémoli, Facundo; Smith, Zane

doi:10.1007/978-3-030-43408-3_14

Cited by 15 publications

(23 citation statements)

References 50 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recording both coordinates of these points (birth-death pairs) in the persistence diagram allows us to connect through time pairs that are close in this space and to identify statistically significant features and properties of a timechanging point cloud. As discussed in the introduction, this goal is much closer to the concept of vines and vineyards developed by Cohen-Steiner et al (2006) and more recently addressed by Kim et al (2020).…”

Section: Persistence Diagrams and Previous Analysis Of Time-series Datamentioning

confidence: 92%

“…These methods are applied to protein folding trajectories in Cohen-Steiner et al (2006). The study of Kim et al (2020) views dynamic data sets as time-varying graphs and extracts summaries of their clustering features, with potential applications to swarming behaviors. Our work is an approxima-tion of such theoretical approaches that is accurate for extracting the most significant feature and its emergence through time.…”

Section: Topological Data Analysis For Time-dependent Datamentioning

confidence: 99%

See 1 more Smart Citation

Topological Data Analysis Approaches to Uncovering the Timing of Ring Structure Onset in Filamentous Networks

et al. 2021

View full text Add to dashboard Cite

In developmental biology as well as in other biological systems, emerging structure and organization can be captured using time-series data of protein locations. In analyzing this time-dependent data, it is a common challenge not only to determine whether topological features emerge, but also to identify the timing of their formation. For instance, in most cells, actin filaments interact with myosin motor proteins and organize into polymer networks and higher-order structures. Ring channels are examples of such structures that maintain constant diameters over time and play key roles in processes such as cell division, development, and wound healing. Given the limitations in studying interactions of actin with myosin in vivo, we generate time-series data of protein polymer interactions in cells using complex agent-based models. Since the data has a filamentous structure, we propose sampling along the actin filaments and analyzing the topological structure of the resulting point cloud at each time. Building on existing tools from persistent homology, we develop a topological data analysis (TDA) method that assesses effective ring generation in this dynamic data. This method connects topological features through time in a path that corresponds to emergence of organization in the data. In this work, we also propose methods for assessing whether the topological features of interest are significant and thus whether they contribute to the formation of an emerging hole (ring channel) in the simulated protein interactions. In particular, we use the MEDYAN simulation platform to show that this technique can distinguish between the actin cytoskeleton organization resulting from distinct motor protein binding parameters.

show abstract

Section: Persistence Diagrams and Previous Analysis Of Time-series Datamentioning

confidence: 92%

Section: Topological Data Analysis For Time-dependent Datamentioning

confidence: 99%

Topological Data Analysis Approaches to Uncovering the Timing of Ring Structure Onset in Filamentous Networks

et al. 2021

View full text Add to dashboard Cite

show abstract

“…In the TDA literature it is often the case that the indexing set of a zigzag module is a subset of R [16,15,23,24,27]. This motivates us to construct a rescaled-version of the poset ZZ.…”

Section: Quadrants In Umentioning

confidence: 99%

“…From a different perspective, our decision to focus the presentation first on the setting of zigzag modules is motivated by extensive theoretical and algorithmic study on zigzag persistence [10,11,17,14,15,23,26,34,36], and the large number of concrete applications in mobile sensor networks, image processing, analysis of time-varying metric spaces/graphs that give rise to zigzag persistence [1,21,27,28,32]. In particular, in [27,28], zigzag persistence of graphs, partitions, or homology groups are induced as signatures of time-varying metric data. This motivates us to find stable invariants for zigzag persistence valued in other than the category of vector spaces.…”

Section: Introductionmentioning

confidence: 99%

Generalized Persistence Diagrams for Persistence Modules over Posets

Kim,

Memoli

2018

Preprint

Self Cite

View full text Add to dashboard Cite

When a category C satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors F : P → C from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel's recent extension. Specifically, the barcode of any interval decomposable persistence modules F : P → vec of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing poset P of F : P → C in defining Patel's generalized persistence diagram of F .By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel's semicontinuity theorem about type A persistence diagram to Lipschitz continuity theorem for the category of sets.

show abstract

“…One drawback of using standard non-zigzag persistence [12] is that it only allows addition of vertices and edges during the change, whereas deletion may also happen in practice. For example, many complex systems such as social networks, food webs, or disease spreading are modeled by the so-called "dynamic networks" [17,18,24], where vertices and edges can appear and disappear at different time. A variant of the standard persistence called zigzag persistence [3] is thus a more natural tool in such scenarios because simplices can be both added and deleted.…”

Section: Introductionmentioning

confidence: 99%

Computing Zigzag Persistence on Graphs in Near-Linear Time

Dey¹,

Hou²

2021

Preprint

View full text Add to dashboard Cite

Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one instrument for analyzing such changing graph data. However, standard persistent homology defined over a growing space cannot always capture such a dynamic process unless shrinking with deletions is also allowed. Hence, zigzag persistence which incorporates both insertions and deletions of simplices is more appropriate in such a setting. Unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general O(m ω ) time complexity are not known, where ω < 2.37286 is the matrix multiplication exponent. In this paper, we propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration with m additions and deletions on a graph with n vertices and edges, the algorithm for 0-dimension runs in O(m log 2 n + m log m) time and the algorithm for 1-dimension runs in O(m log 4 n) time. The algorithm for 0-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on 2-manifolds. The algorithm for 1-dimension pairs a negative edge with the earliest positive edge so that a 1-cycle containing both edges resides in all intermediate graphs. Both algorithms achieve the claimed time complexity via dynamic graph data structures proposed by Holm et al. In the end, using Alexander duality, we extend the algorithm for 0-dimension to compute the (p − 1)-dimensional zigzag persistence for R p -embedded complexes in O(m log 2 n + m log m + n log n) time. * This research is supported by NSF grants CCF 1839252 and 2049010. (a) (b) (c) (d) Figure 1: A sequence of graphs with four prominent clusters each colored differently. Black edges connect different clusters and forward (resp. backward) arrows indicate additions (resp. deletions) of vertices and edges. From (a) to (b), two clusters split; from (b) to (c), two clusters merge; from (c) to (d), one cluster disappears.

show abstract

Analysis of Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence

Cited by 15 publications

References 50 publications

Topological Data Analysis Approaches to Uncovering the Timing of Ring Structure Onset in Filamentous Networks

Topological Data Analysis Approaches to Uncovering the Timing of Ring Structure Onset in Filamentous Networks

Generalized Persistence Diagrams for Persistence Modules over Posets

Computing Zigzag Persistence on Graphs in Near-Linear Time

Contact Info

Product

Resources

About