Decimation of Fast States and Weak Nodes: Topological Variation via Persistent Homology

Donato, Irene; Petri, Giovanni; Scolamiero, Martina; Rondoni, Lamberto; Vaccarino, Francesco

doi:10.1007/978-3-319-00395-5_39

Cited by 7 publications

(5 citation statements)

References 7 publications

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“…[72], while Harer and Edelsbrunner [25] detailed the concept and history of persistent homology. Persistent homology has shown great promise to assist in the analysis of complex graphs due to the types of features it extracts and its ability to differentiate signal from noise [18,43,54,55,71]. For example, Rieck et al [58] used persistent homology to track the evolution of clique communities across different edge weight thresholds.…”

Section: Persistent Homologymentioning

confidence: 99%

Untangling Force-Directed Layouts Using Persistent Homology

Doppalapudi¹,

Wang²,

Rosen³

2022

Preprint

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Static Layout MethodsFig. 1: Our approach in using persistent homology to untangle force-directed layouts has two main functionalities. First, as demonstrated via the HIC 1K NET dataset, we use persistent homology to formulate an initial graph layout (column 1) that improves both the convergence rate of the graph layout and the final layout quality. Second, our approach comes with interactive capabilities, as illustrated in Fig. 2. We use the local continuity meta criterion (Q LCMC ) for layout evaluation (larger is better). In terms of convergence, our approach (column 2, rows 2 and 3) shows the formation of major graph structures as early as 5 iterations, whereas the standard approach (column 4, row 1) takes 25+ iterations. In terms of final layout quality, the Q LCMC scores for the final layout (column 5) show that our approach significantly exceeds the standard one. Three static layout methods, neato, fdp, and sfdp, are also included for comparison (column 6), with only sfdp (column 6, row 3) showing a comparable Q LCMC score.

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Section: Persistent Homologymentioning

confidence: 99%

Untangling Force-Directed Layouts Using Persistent Homology

Doppalapudi¹,

Wang²,

Rosen³

2022

Preprint

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“…Persistent Homology and Graphs. PH is an emerging tool in studying complex graphs [22,26,46,74,75], including collaboration networks [7,12] and brain networks [13,17,[58][59][60][61]76]. PH has recently been used in the visualization community for graph analysis targeting clique communities [78] and time-varying graphs [41].…”

Section: Prior Workmentioning

confidence: 99%

Persistent Homology Guided Force-Directed Graph Layouts

Suh

Hajij

Wang

et al. 2019

IEEE Trans. Visual. Comput. Graphics

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Graphs are commonly used to encode relationships among entities, yet their abstractness makes them difficult to analyze. Node-link diagrams are popular for drawing graphs, and force-directed layouts provide a flexible method for node arrangements that use local relationships in an attempt to reveal the global shape of the graph. However, clutter and overlap of unrelated structures can lead to confusing graph visualizations. This paper leverages the persistent homology features of an undirected graph as derived information for interactive manipulation of force-directed layouts. We first discuss how to efficiently extract 0-dimensional persistent homology features from both weighted and unweighted undirected graphs. We then introduce the interactive persistence barcode used to manipulate the force-directed graph layout. In particular, the user adds and removes contracting and repulsing forces generated by the persistent homology features, eventually selecting the set of persistent homology features that most improve the layout. Finally, we demonstrate the utility of our approach across a variety of synthetic and real datasets.

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“…Such complexes are computed from point data or complex networks, including sensor networks [dSG07, SHPW17], brain networks [DMFC12, CRS15], social networks [BG14] and others [DPS*13]. Examples of high‐dimensional simplicial complexes that can be computed on these type of data are alpha shapes [EM94], tidy sets [Zom10b] or Vietoris‐Rips (VR) complexes [Zom10a].…”

Section: Introductionmentioning

confidence: 99%

Efficient Homology‐Preserving Simplification of High‐Dimensional Simplicial Shapes

Fellegara

Iuricich

Floriani

et al. 2019

Computer Graphics Forum

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Simplicial complexes are widely used to discretize shapes. In low dimensions, a 3D shape is represented by discretizing its boundary surface, encoded as a triangle mesh, or by discretizing the enclosed volume, encoded as a tetrahedral mesh. High‐dimensional simplicial complexes have recently found their application in topological data analysis. Topological data analysis aims at studying a point cloud P, possibly embedded in a high‐dimensional metric space, by investigating the topological characteristics of the simplicial complexes built on P. Analysing such complexes is not feasible due to their size and dimensions. To this aim, the idea of simplifying a complex while preserving its topological features has been proposed in the literature. Here, we consider the problem of efficiently simplifying simplicial complexes in arbitrary dimensions. We provide a new definition for the edge contraction operator, based on a top‐based data structure, with the objective of preserving structural aspects of a simplicial shape (i.e., its homology), and a new algorithm for verifying the link condition on a top‐based representation. We implement the simplification algorithm obtained by coupling the new edge contraction and the link condition on a specific top‐based data structure, that we use to demonstrate the scalability of our approach.

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Decimation of Fast States and Weak Nodes: Topological Variation via Persistent Homology

Cited by 7 publications

References 7 publications

Untangling Force-Directed Layouts Using Persistent Homology

Untangling Force-Directed Layouts Using Persistent Homology

Persistent Homology Guided Force-Directed Graph Layouts

Efficient Homology‐Preserving Simplification of High‐Dimensional Simplicial Shapes

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