Limits of Ordered Graphs and Images
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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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