6 papers
Discrete homology computations by reduction to zero differentials
Sterling Ebel, Chris Kapulkin, Nathan Kershaw
We develop a new algorithm for computing (persistent) discrete homology of graphs using reduction to zero differentials and active enumeration. This allows us to compute the fourth…
RedZeD: Computing persistent homology by Reduction to Zero Differentials
Chris Kapulkin, Nathan Kershaw
We introduce a new algorithm for computing persistent homology of Vietoris-Rips filtrations, which offers a considerable improvement both in terms of time and memory over the exist…
Topological data analysis using persistent discrete homology
Chris Kapulkin, Nathan Kershaw
We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods. In particular, we provide empirical evidence t…
Categorical foundations of discrete dynamical systems
Daniel Carranza, Chris Kapulkin, Nathan Kershaw +2
We develop categorical foundations of discrete dynamical systems, aimed at understanding how the structure of the system affects its dynamics. We introduce the notion of cycle sets…
Faster computations of discrete homology
Chris Kapulkin, Nathan Kershaw
Machine computation of the discrete homology of graphs has stopped at degree two. We present an algorithm that reaches degree four. It generates the singular cubes inductively, pai…
Closed symmetric monoidal structures on the category of graphs
Chris Kapulkin, Nathan Kershaw
We show that the category of (reflexive) graphs and graph maps carries exactly two closed symmetric monoidal products: the box product and the categorical product.