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arXiv · 1804.09758

Analysis of a Canonical Labeling Algorithm for the Alignment of Correlated Erd\H{o}s-R\'enyi Graphs

Osman Emre Dai·Daniel Cullina·Negar Kiyavash·Matthias Grossglauser
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Abstract

Graph alignment in two correlated random graphs refers to the task of identifying the correspondence between vertex sets of the graphs. Recent results have characterized the exact information-theoretic threshold for graph alignment in correlated Erd\H{o}s-R\'enyi graphs. However, very little is known about the existence of efficient algorithms to achieve graph alignment without seeds. In this work we identify a region in which a straightforward $O(n^{11/5} \log n )$-time canonical labeling algorithm, initially introduced in the context of graph isomorphism, succeeds in aligning correlated Erd\H{o}s-R\'enyi graphs. The algorithm has two steps. In the first step, all vertices are labeled by their degrees and a trivial minimum distance alignment (i.e., sorting vertices according to their degrees) matches a fixed number of highest degree vertices in the two graphs. Having identified this subset of vertices, the remaining vertices are matched using a alignment algorithm for bipartite graphs.

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Osman Emre Dai, Daniel Cullina, Negar Kiyavash, Matthias Grossglauser. 2018-04-25. Analysis of a Canonical Labeling Algorithm for the Alignment of Correlated Erd\H{o}s-R\'enyi Graphs. https://arxiv.org/abs/1804.09758

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