Barcode Embeddings for Metric Graphs

Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In this paper we study a rich homology-based invariant first defined by Dey, Shi, and Wang, which we think of as embedding a metric graph in the barcode space. We prove that this invariant is locally injective on the space of metric graphs and globally injective on a GH-dense subset. Moreover, we show that is globally injective on a full measure subset of metric graphs, in the appropriate sense.Comment: The newest draft clarifies the proofs in Sections 7 and 8, and provides improved figures therein. It also includes a results section in the introductio

Diffusion determines the recurrent graph

We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown by counterexamples this result is optimal. Without the recurrence assumption, the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each point has the same mass. These investigations provide discrete counterparts to studies of diffusion on Euclidean domains and manifolds initiated by Arendt and continued by Arendt/Biegert/ter Elst and Arendt/ter Elst. A crucial step in our considerations shows that order isomorphisms are actually unitary maps (up to a scaling) in our context.Comment: 30 page

Dual Feynman transform for modular operads

We introduce and study the notion of a dual Feynman transform of a modular operad. This generalizes and gives a conceptual explanation of Kontsevich's dual construction producing graph cohomology classes from a contractible differential graded Frobenius algebra. The dual Feynman transform of a modular operad is indeed linear dual to the Feynman transform introduced by Getzler and Kapranov when evaluated on vacuum graphs. In marked contrast to the Feynman transform, the dual notion admits an extremely simple presentation via generators and relations; this leads to an explicit and easy description of its algebras. We discuss a further generalization of the dual Feynman transform whose algebras are not necessarily contractible. This naturally gives rise to a two-colored graph complex analogous to the Boardman-Vogt topological tree complex.Comment: 27 pages. A few conceptual changes in the last section; in particular we prove that the two-colored graph complex is a resolution of the corresponding modular operad. It is now called 'BV-resolution' as suggested by Sasha Vorono