4,941 research outputs found
Persistent Homology Guided Force-Directed Graph Layouts
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
Multimapper: Data Density Sensitive Topological Visualization
Mapper is an algorithm that summarizes the topological information contained
in a dataset and provides an insightful visualization. It takes as input a
point cloud which is possibly high-dimensional, a filter function on it and an
open cover on the range of the function. It returns the nerve simplicial
complex of the pullback of the cover. Mapper can be considered a discrete
approximation of the topological construct called Reeb space, as analysed in
the -dimensional case by [Carriere et al.,2018]. Despite its success in
obtaining insights in various fields such as in [Kamruzzaman et al., 2016],
Mapper is an ad hoc technique requiring lots of parameter tuning. There is also
no measure to quantify goodness of the resulting visualization, which often
deviates from the Reeb space in practice. In this paper, we introduce a new
cover selection scheme for data that reduces the obscuration of topological
information at both the computation and visualisation steps. To achieve this,
we replace global scale selection of cover with a scale selection scheme
sensitive to local density of data points. We also propose a method to detect
some deviations in Mapper from Reeb space via computation of persistence
features on the Mapper graph.Comment: Accepted at ICDM
Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology
Topological data analysis is an emerging area in exploratory data analysis
and data mining. Its main tool, persistent homology, has become a popular
technique to study the structure of complex, high-dimensional data. In this
paper, we propose a novel method using persistent homology to quantify
structural changes in time-varying graphs. Specifically, we transform each
instance of the time-varying graph into metric spaces, extract topological
features using persistent homology, and compare those features over time. We
provide a visualization that assists in time-varying graph exploration and
helps to identify patterns of behavior within the data. To validate our
approach, we conduct several case studies on real world data sets and show how
our method can find cyclic patterns, deviations from those patterns, and
one-time events in time-varying graphs. We also examine whether
persistence-based similarity measure as a graph metric satisfies a set of
well-established, desirable properties for graph metrics
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