9 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
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
A Sparse Stress Model
Force-directed layout methods constitute the most common approach to draw
general graphs. Among them, stress minimization produces layouts of
comparatively high quality but also imposes comparatively high computational
demands. We propose a speed-up method based on the aggregation of terms in the
objective function. It is akin to aggregate repulsion from far-away nodes
during spring embedding but transfers the idea from the layout space into a
preprocessing phase. An initial experimental study informs a method to select
representatives, and subsequent more extensive experiments indicate that our
method yields better approximations of minimum-stress layouts in less time than
related methods.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
New Algorithm for Drawings of 3-Planar Graphs
Graphs arise in a natural way in many applications, together with the need to be drawn. Except for very small instances, drawing a graph by hand becomes a very complex task, which must be performed by automatic tools. The field of graph drawing is concerned with finding algorithms to draw graph in an aesthetically pleasant way, based upon a certain number of aesthetic criteria that define what a good drawing, (synonyms: diagrams, pictures, layouts), of a graph should be. This problem can be found in many such as in the computer networks, data networks, class inter-relationship diagrams in object oriented databases and object oriented programs, visual programming interfaces, database design systems, software engineering…etc. Given a plane graph G, we wish to find a drawing of G in the plane such that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints without any edge-intersection. Such drawings are called planar straight-line drawings of G. An additional objective is to minimize the area of the rectangular grid in which G is drawn. In this paper we introduce a new algorithms that finds an embedding of 3-planar graph. Keywords: 3- Planar Graph; Graph Drawing; drawing on grid
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