3,700 research outputs found
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
Topological Anomaly Detection in Dynamic Multilayer Blockchain Networks
Motivated by the recent surge of criminal activities with
cross-cryptocurrency trades, we introduce a new topological perspective to
structural anomaly detection in dynamic multilayer networks. We postulate that
anomalies in the underlying blockchain transaction graph that are composed of
multiple layers are likely to also be manifested in anomalous patterns of the
network shape properties. As such, we invoke the machinery of clique persistent
homology on graphs to systematically and efficiently track evolution of the
network shape and, as a result, to detect changes in the underlying network
topology and geometry. We develop a new persistence summary for multilayer
networks, called stacked persistence diagram, and prove its stability under
input data perturbations. We validate our new topological anomaly detection
framework in application to dynamic multilayer networks from the Ethereum
Blockchain and the Ripple Credit Network, and demonstrate that our stacked PD
approach substantially outperforms state-of-art techniques.Comment: 26 pages, 6 figures, 7 table
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
Mapper on Graphs for Network Visualization
Networks are an exceedingly popular type of data for representing
relationships between individuals, businesses, proteins, brain regions,
telecommunication endpoints, etc. Network or graph visualization provides an
intuitive way to explore the node-link structures of network data for instant
sense-making. However, naive node-link diagrams can fail to convey insights
regarding network structures, even for moderately sized data of a few hundred
nodes. We propose to apply the mapper construction--a popular tool in
topological data analysis--to graph visualization, which provides a strong
theoretical basis for summarizing network data while preserving their core
structures. We develop a variation of the mapper construction targeting
weighted, undirected graphs, called mapper on graphs, which generates
property-preserving summaries of graphs. We provide a software tool that
enables interactive explorations of such summaries and demonstrates the
effectiveness of our method for synthetic and real-world data. The mapper on
graphs approach we propose represents a new class of techniques that leverages
tools from topological data analysis in addressing challenges in graph
visualization
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