Seeded graph matching via large neighborhood statistics

We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic l...

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Bibliographic Details
Published in:Random structures & algorithms Vol. 57; no. 3; pp. 570 - 611
Main Authors: Mossel, Elchanan, Xu, Jiaming
Format: Journal Article
Language:English
Published: New York John Wiley & Sons, Inc 01.10.2020
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Subjects:
Algorithms
Apexes
branching process
graph isomorphism
Graph matching
Graph theory
Isomorphism
Polynomials
Seeds
subgraph count
ISSN:1042-9832, 1098-2418
Online Access:Get full text
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Summary:We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of vertices n. Moreover, we show the number of seeds needed for perfect recovery in polynomial‐time can be as low as nϵ in the sparse graph regime (with the average degree smaller than nϵ) and Ω(logn) in the dense graph regime, for a small positive constant ϵ. Unlike previous work on graph matching, which used small neighborhoods or small subgraphs with a logarithmic number of vertices in order to match vertices, our algorithms match vertices if their large neighborhoods have a significant overlap in the number of seeds.
Bibliography:Funding information
A conference (short) version of this paper is accepted to 2019 ACM‐SIAM Symposium on Discrete Algorithms (SODA).
This research was supported by the National Science Foundation, IIS‐1932630, CCF‐1850743, and CCF‐1856424.
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.20934