Abstract
Persistent homology enables fast and computable comparison of topological objects. We give some instances of a recent extension of the theory of persistence, guaranteeing robustness and computability for relevant data types, like simple graphs and digraphs. We focus on categorical persistence functions that allow us to study in full generality strong kinds of connectedness—clique communities, k-vertex, and k-edge connectedness—directly on simple graphs and strong connectedness in digraphs.
Highlights
Persistent homology allows for swift and robust comparison of topological objects.raw data are rarely endowed with a topological structure
Persistent homology and topological persistence are by their nature bound to topological spaces and simplicial complexes, so that, in persistent homology applications, data are mapped to topological spaces or simplicial complexes through auxiliary constructions (e.g., [1,2,3,4,5])
The blocks of a filtered graph do not form a simplicial complex because blocks lack the hereditariness that is required by simplicial constructions
Summary
Persistent homology allows for swift and robust comparison of topological objects. raw data are rarely endowed with a topological structure. We hope that the axiomatic foundation mentioned above will yield a tool agile enough to enable the usage of persistence in applications so far not liable to immediate, direct topological constructions ( a more elaborate one exists; see Remark 1) With this aim in mind, we introduce the definition of weakly directed properties as a way to build categorical persistence functions that describe graph-theoretical concepts of connectivity, e.g., clique communities, k-vertex and k-edge connectedness in graphs, and strong connectedness in digraphs. In the same figure we compute persistent Betti numbers of the example graph, seen as a simplicial complex
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