112 research outputs found
Persistent homology of unweighted complex networks via discrete Morse theory
Topological data analysis can reveal higher-order structure beyond pairwise
connections between vertices in complex networks. We present a new method based
on discrete Morse theory to study topological properties of unweighted and
undirected networks using persistent homology. Leveraging on the features of
discrete Morse theory, our method not only captures the topology of the clique
complex of such graphs via the concept of critical simplices, but also achieves
close to the theoretical minimum number of critical simplices in several
analyzed model and real networks. This leads to a reduced filtration scheme
based on the subsequence of the corresponding critical weights, thereby leading
to a significant increase in computational efficiency. We have employed our
filtration scheme to explore the persistent homology of several model and
real-world networks. In particular, we show that our method can detect
differences in the higher-order structure of networks, and the corresponding
persistence diagrams can be used to distinguish between different model
networks. In summary, our method based on discrete Morse theory further
increases the applicability of persistent homology to investigate the global
topology of complex networks.Comment: 36 pages, 6 main figures, SI Appendix and SI Figures; SI Tables
available upon request from author
Forman-Ricci curvature and Persistent homology of unweighted complex networks
We present the application of topological data analysis (TDA) to study
unweighted complex networks via their persistent homology. By endowing
appropriate weights that capture the inherent topological characteristics of
such a network, we convert an unweighted network into a weighted one. Standard
TDA tools are then used to compute their persistent homology. To this end, we
use two main quantifiers: a local measure based on Forman's discretized version
of Ricci curvature, and a global measure based on edge betweenness centrality.
We have employed these methods to study various model and real-world networks.
Our results show that persistent homology can be used to distinguish between
model and real networks with different topological properties.Comment: 25 pages, 6 Main figures, 10 SI figure
Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunIT
Topological data analysis (TDA) delivers invaluable and complementary
information on the intrinsic properties of data inaccessible to conventional
methods. However, high computational costs remain the primary roadblock
hindering the successful application of TDA in real-world studies, particularly
with machine learning on large complex networks.
Indeed, most modern networks such as citation, blockchain, and online social
networks often have hundreds of thousands of vertices, making the application
of existing TDA methods infeasible. We develop two new, remarkably simple but
effective algorithms to compute the exact persistence diagrams of large graphs
to address this major TDA limitation. First, we prove that -core of a
graph suffices to compute its persistence diagram,
. Second, we introduce a pruning algorithm for graphs to
compute their persistence diagrams by removing the dominated vertices. Our
experiments on large networks show that our novel approach can achieve
computational gains up to 95%.
The developed framework provides the first bridge between the graph theory
and TDA, with applications in machine learning of large complex networks. Our
implementation is available at
https://github.com/cakcora/PersistentHomologyWithCoralPrunitComment: Spotlight paper at NeurIPS 202
Visualizing Topological Importance: A Class-Driven Approach
This paper presents the first approach to visualize the importance of
topological features that define classes of data. Topological features, with
their ability to abstract the fundamental structure of complex data, are an
integral component of visualization and analysis pipelines. Although not all
topological features present in data are of equal importance. To date, the
default definition of feature importance is often assumed and fixed. This work
shows how proven explainable deep learning approaches can be adapted for use in
topological classification. In doing so, it provides the first technique that
illuminates what topological structures are important in each dataset in
regards to their class label. In particular, the approach uses a learned metric
classifier with a density estimator of the points of a persistence diagram as
input. This metric learns how to reweigh this density such that classification
accuracy is high. By extracting this weight, an importance field on persistent
point density can be created. This provides an intuitive representation of
persistence point importance that can be used to drive new visualizations. This
work provides two examples: Visualization on each diagram directly and, in the
case of sublevel set filtrations on images, directly on the images themselves.
This work highlights real-world examples of this approach visualizing the
important topological features in graph, 3D shape, and medical image data.Comment: 11 pages, 11 figure
PHDP: Preserving Persistent Homology in Differentially Private Graph Publications
Online social networks (OSNs) routinely share and analyze user data. This requires protection of sensitive user information. Researchers have proposed several techniques to anonymize the data of OSNs. Some differential-privacy techniques claim to preserve graph utility under certain graph metrics, as well as guarantee strict privacy. However, each graph utility metric reveals the whole graph in specific aspects.We employ persistent homology to give a comprehensive description of the graph utility in OSNs. This paper proposes a novel anonymization scheme, called PHDP, which preserves persistent homology and satisfies differential privacy. To strengthen privacy protection, we add exponential noise to the adjacency matrix of the network and find the number of adding/deleting edges. To maintain persistent homology, we collect edges along persistent structures and avoid perturbation on these edges. Our regeneration algorithms balance persistent homology with differential privacy, publishing an anonymized graph with a guarantee of both. Evaluation result show that the PHDP-anonymized graph achieves high graph utility, both in graph metrics and application metrics
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