Graph MBO on Star Graphs and Regular Trees. With Corrections to DOI 10.1007/s00032-014-0216-8

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Abstract

The graph Merriman–Bence–Osher scheme produces, starting from an initial node subset, a sequence of node sets obtained by iteratively applying graph diffusion and thresholding to the characteristic (or indicator) function of the node subsets. One result in [14] gives sufficient conditions on the diffusion time to ensure that the set membership of a given node changes in one iteration of the scheme. In particular, these conditions only depend on local information at the node (information about neighbors and neighbors of neighbors of the node in question). In this paper we show that there does not exist any graph which satisfies these conditions. To make up for this negative result, this paper also presents positive results regarding the Merriman–Bence–Osher dynamics on star graphs and regular trees. In particular, we present sufficient (and in some cases necessary) results for the set membership of a given node to change in one iteration.

Original languageEnglish
Pages (from-to)141-168
Number of pages28
JournalMilan Journal of Mathematics
Volume87
Issue number1
DOIs
Publication statusPublished - 2019

Keywords

  • graph dynamics
  • Merriman–Bence–Osher scheme
  • regular tree graph
  • star graph
  • threshold dynamics

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