Discrete Morse theory and persistent homology

2019 ◽  
pp. 117-148
Keyword(s):  
Morse Theory ◽  
Persistent Homology ◽  
Discrete Morse Theory

Efficient computation of 3D Morse–Smale complexes and persistent homology using discrete Morse theory

2012 ◽  
Vol 28 (10) ◽  
pp. 959-969 ◽  
Author(s):  
David Günther ◽  
Jan Reininghaus ◽  
Hubert Wagner ◽  
Ingrid Hotz
Keyword(s):  
Morse Theory ◽  
Persistent Homology ◽  
Discrete Morse Theory ◽  
Efficient Computation

scholarly journals Persistent homology of unweighted complex networks via discrete Morse theory

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Harish Kannan ◽  
Emil Saucan ◽  
Indrava Roy ◽  
Areejit Samal
Keyword(s):  
Complex Networks ◽  
Morse Theory ◽  
Persistent Homology ◽  
Higher Order ◽  
Topological Data Analysis ◽  
Discrete Morse Theory ◽  
Order Structure ◽  
Global Topology ◽  
Minimum Number ◽  
Theoretical Minimum

Abstract 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.

scholarly journals Counting and Discrete Morse Theory

2019 ◽  
Vol 21 (1) ◽  
Author(s):  
Andrew Sack
Keyword(s):  
Morse Theory ◽  
Vector Fields ◽  
Discrete Morse Theory ◽  
Gradient Vector ◽  
Morse Functions

We examine enumerating discrete Morse functions on graphs up to equivalence by gradient vector fields and by restrictions on the codomain.  We give formulae for the number of discrete Morse functions on specific classes of graphs (line, cycle, and bouquet of circles).

scholarly journals Strong Discrete Morse Theory and Simplicial L–S Category: A Discrete Version of the Lusternik–Schnirelmann Theorem

2019 ◽  
Vol 63 (3) ◽  
pp. 607-623
Author(s):  
Desamparados Fernández-Ternero ◽  
Enrique Macías-Virgós ◽  
Nicholas A. Scoville ◽  
José Antonio Vilches
Keyword(s):  
Morse Theory ◽  
Discrete Morse Theory ◽  
Discrete Version ◽  
Lusternik Schnirelmann

scholarly journals Discrete Morse theory for totally non-negative flag varieties

2010 ◽  
Vol 223 (6) ◽  
pp. 1855-1884 ◽  
Author(s):  
Konstanze Rietsch ◽  
Lauren Williams
Keyword(s):  
Morse Theory ◽  
Discrete Morse Theory ◽  
Flag Varieties
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