Persistent homology in two-dimensional atomic networks

Persistent Homology of Collaboration Networks

We apply persistent homology to four collaboration networks. We show that the intervals for the zeroth and first Betti numbers correspond to tangible features of the structure of these networks. Finally, we use persistent homology to distinguish collaboration networks from similar random networks.

Coverage in Sensor Networks via Persistent Homology

We consider coverage problems in sensor networks with minimal sensing capabilities. In particular, we demonstrate that a stationary collection of sensor nodes with no localization can verify coverage in a bounded domain of unknown topological type, so long as the boundary is not too pinched. The only sensing capabilities required by the nodes are a binary form of distance estimation between nod...

One-Dimensional Reduction of Multidimensional Persistent Homology

A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional persistent homology. Some reflections on i-essentiality of homological critical values conclude the paper.

Persistent Path Homology of Directed Networks

Two-dimensional atomic crystals.

We report free-standing atomic crystals that are strictly 2D and can be viewed as individual atomic planes pulled out of bulk crystals or as unrolled single-wall nanotubes. By using micromechanical cleavage, we have prepared and studied a variety of 2D crystals including single layers of boron nitride, graphite, several dichalcogenides, and complex oxides. These atomically thin sheets (essentia...