Cluster Persistence for Weighted Graphs.

Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their 0-dimensional homology. While this area has been thoroughly studied, we present a new approach to constructing a filtration for cluster analysis via persistent homology. The key advantages of the new filtration is that (a) it provides richer signatures for connected components by introducing non-trivial birth times, and (b) it is robust to outliers. The key idea is that nodes are ignored until they belong to sufficiently large clusters. We demonstrate the computational efficiency of our filtration, its practical effectiveness, and explore into its properties when applied to random graphs

Homological percolation and the Euler characteristic

In this paper we study the connection between the phenomenon of homological percolation (the formation of "giant" cycles in persistent homology), and the zeros of the expected Euler characteristic curve. We perform an experimental study that covers four different models: site-percolation on the cubical and permutahedral lattices, the Poisson-Boolean model, and Gaussian random fields. All the models are generated on the flat torus $T^d$, for $d=2,3,4$. The simulation results strongly indicate that the zeros of the expected Euler characteristic curve approximate the critical values for homological-percolation. Our results also provide some insight about the approximation error. Further study of this connection could have powerful implications both in the study of percolation theory, and in the field of Topological Data Analysis

Randomly weighted d-complexes: Minimal spanning acycles and Persistence diagrams

A weighted d -complex is a simplicial complex of dimension d in which each face is assigned a real-valued weight. We derive three key results here concerning persistence diagrams and minimal spanning acycles (MSAs) of such complexes. First, we establish an equivalence between the MSA face-weights and death times in the persistence diagram. Next, we show a novel stability result for the MSA face-weights which, due to our first result, also holds true for the death and birth times, separately. Our final result concerns a perturbation of a mean-field model of randomly weighted d -complexes. The d -face weights here are perturbations of some i.i.d. distribution while all the lower-dimensional faces have a weight of 0 . If the perturbations decay sufficiently quickly, we show that suitably scaled extremal nearest face-weights, face-weights of the d -MSA, and the associated death times converge to an inhomogeneous Poisson point process. This result completely characterizes the extremal points of persistence diagrams and MSAs. The point process convergence and the asymptotic equivalence of three point processes are new for any weighted random complex model, including even the non-perturbed case. Lastly, as a consequence of our stability result, we show that Frieze's ζ ( 3 ) limit for random minimal spanning trees and the recent extension to random MSAs by Hino and Kanazawa also hold in suitable noisy settings

Dualities in persistent (co)homology

We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent information. We explain how one can use the existing algorithm for persistent homology to process any of the four modules, and relate it to a recently introduced persistent cohomology algorithm. We present experimental evidence for the practical efficiency of the latter algorithm.Comment: 16 pages, 3 figures, submitted to the Inverse Problems special issue on Topological Data Analysi

Interpreting Galilean Invariant Vector Field Analysis via Extended Robustness

The topological notion of robustness introduces mathematically rigorous approaches to interpret vector field data. Robustness quantifies the structural stability of critical points with respect to perturbations and has been shown to be useful for increasing the visual interpretability of vector fields. However, critical points, which are essential components of vector field topology, are defined with respect to a chosen frame of reference. The classical definition of robustness, therefore, depends also on the chosen frame of reference. We define a new Galilean invariant robustness framework that enables the simultaneous visualization of robust critical points across the dominating reference frames in different regions of the data. We also demonstrate a strong connection between such a robustness-based framework with the one recently proposed by Bujack et al., which is based on the determinant of the Jacobian. Our results include notable observations regarding the definition of stable features within the vector field data

Atropselective syntheses of (-) and (+) rugulotrosin A utilizing point-to-axial chirality transfer

Chiral, dimeric natural products containing complex structures and interesting biological properties have inspired chemists and biologists for decades. A seven-step total synthesis of the axially chiral, dimeric tetrahydroxanthone natural product rugulotrosin A is described. The synthesis employs a one-pot Suzuki coupling/dimerization to generate the requisite 2,2'-biaryl linkage. Highly selective point-to-axial chirality transfer was achieved using palladium catalysis with achiral phosphine ligands. Single X-ray crystal diffraction data were obtained to confirm both the atropisomeric configuration and absolute stereochemistry of rugulotrosin A. Computational studies are described to rationalize the atropselectivity observed in the key dimerization step. Comparison of the crude fungal extract with synthetic rugulotrosin A and its atropisomer verified that nature generates a single atropisomer of the natural product.P50 GM067041 - NIGMS NIH HHS; R01 GM099920 - NIGMS NIH HHS; GM-067041 - NIGMS NIH HHS; GM-099920 - NIGMS NIH HH

Persistent Homology in ℓ∞\ell_\infty Metric

Proximity complexes and filtrations are central constructions in topological data analysis. Built using distance functions, or more generally metrics, they are often used to infer connectivity information from point clouds. Here we investigate proximity complexes and filtrations built over the Chebyshev metric, also known as the maximum metric or $\ell_{\infty}$ metric, rather than the classical Euclidean metric. Somewhat surprisingly, the $\ell_{\infty}$ case has not been investigated thoroughly. In this paper, we examine a number of classical complexes under this metric, including the \v{C}ech, Vietoris-Rips, and Alpha complexes. We define two new families of flag complexes, which we call the Alpha flag and Minibox complexes, and prove their equivalence to \v{C}ech complexes in homological degrees zero and one. Moreover, we provide algorithms for finding Minibox edges of two, three, and higher-dimensional points. Finally, we present computational experiments on random points, which shows that Minibox filtrations can often be used to speed up persistent homology computations in homological degrees zero and one by reducing the number of simplices in the filtration.Comment: 36 pages, 15 figure