Limits of Ordered Graphs and Images
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by
Omri Ben-Eliezer, Eldar Fischer, Amit Levi, Yuichi Yoshida
2018
Abstract
The emerging theory of graph limits exhibits an interesting analytic
perspective on graphs, showing that many important concepts and tools in graph
theory and its applications can be described naturally in analytic language. We
extend the theory of graph limits to the ordered setting, presenting a limit
object for dense vertex-ordered graphs, which we call an orderon. Images are an
example of dense ordered bipartite graphs, where the rows and the columns
constitute the vertices, and pixel colors are represented by row-column edges;
thus, as a special case, we obtain a limit object for images.
Along the way, we devise an ordered locality-preserving variant of the cut
distance between ordered graphs, showing that two graphs are close with respect
to this distance if and only if they are similar in terms of their ordered
subgraph frequencies. We show that the space of orderons is compact with
respect to this distance notion, which is key to a successful analysis of
combinatorial objects through their limits. For the proof we combine techniques
used in the unordered setting with several new techniques specifically designed
to overcome the challenges arising in the ordered setting. We derive several
results related to sampling and property testing on ordered graphs and images;
For example, we describe how one can use the analytic machinery to obtain a new
proof of the ordered graph removal lemma [Alon et al., FOCS 2017].
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