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Euclidean vs. Graph Metric

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Erdős Centennial

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 25))

Abstract

The theory of sparse graph limits concerns itself with versions of local convergence and global convergence, see e.g. [44]. Informally, in local convergence we look at a large neighborhood around a random uniformly chosen vertex in a graph and in global convergence we observe the whole graph from afar. In this note rather than surveying the general theory we will consider some concrete examples and problems of global and local convergence, with a geometric viewpoint. We will discuss how well large graphs approximate continuous spaces such as the Euclidean space. Or how properties of Euclidean space such as scale invariance and rotational invariance can appear in large graphs.

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Author information

Authors and Affiliations

  1. Weizmann Institute of Science, 234 Herzl St., Rehovot, 76100, Israel

    Itai Benjamini

Authors
  1. Itai Benjamini
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Eötvös Lóránd University, Pázmány P. sétány 1/c, Budapest, 1117, Hungary

    László Lovász

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, Budapest, 1053, Hungary

    Imre Z. Ruzsa  & Vera T. Sós  & 

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© 2013 János Bolyai Mathematical Society and Springer-Verlag

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Benjamini, I. (2013). Euclidean vs. Graph Metric. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds) Erdős Centennial. Bolyai Society Mathematical Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39286-3_2

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