TY - GEN

T1 - Random Walks on Rotating Expanders

AU - Cohen, Gil

AU - Maor, Gal

N1 - Publisher Copyright:
© 2023 ACM.

PY - 2023/6/2

Y1 - 2023/6/2

N2 - Random walks on expanders are a powerful tool which found applications in many areas of theoretical computer science, and beyond. However, they come with an inherent cost-the spectral expansion of the corresponding power graph deteriorates at a rate that is exponential in the length of the walk. As an example, when G is a d-regular Ramanujan graph, the power graph Gt has spectral expansion 2ω(t) D, where D = dt is the regularity of Gt, thus, Gt is 2ω(t) away from being Ramanujan. This exponential blowup manifests itself in many applications. In this work we bypass this barrier by permuting the vertices of the given graph after each random step. We prove that there exists a sequence of permutations for which the spectral expansion deteriorates by only a linear factor in t. In the Ramanujan case this yields an expansion of O(t D). We stress that the permutations are tailor-made to the graph at hand and require no randomness to generate. Our proof, which holds for all sufficiently high girth graphs, makes heavy use of the powerful framework of finite free probability and interlacing families that was developed in a seminal sequence of works by Marcus, Spielman and Srivastava.

AB - Random walks on expanders are a powerful tool which found applications in many areas of theoretical computer science, and beyond. However, they come with an inherent cost-the spectral expansion of the corresponding power graph deteriorates at a rate that is exponential in the length of the walk. As an example, when G is a d-regular Ramanujan graph, the power graph Gt has spectral expansion 2ω(t) D, where D = dt is the regularity of Gt, thus, Gt is 2ω(t) away from being Ramanujan. This exponential blowup manifests itself in many applications. In this work we bypass this barrier by permuting the vertices of the given graph after each random step. We prove that there exists a sequence of permutations for which the spectral expansion deteriorates by only a linear factor in t. In the Ramanujan case this yields an expansion of O(t D). We stress that the permutations are tailor-made to the graph at hand and require no randomness to generate. Our proof, which holds for all sufficiently high girth graphs, makes heavy use of the powerful framework of finite free probability and interlacing families that was developed in a seminal sequence of works by Marcus, Spielman and Srivastava.

KW - expander graphs

KW - finite free probability

KW - interlacing families

KW - random walks on graphs

KW - spectral graph theory

UR - http://www.scopus.com/inward/record.url?scp=85163083888&partnerID=8YFLogxK

U2 - 10.1145/3564246.3585133

DO - 10.1145/3564246.3585133

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AN - SCOPUS:85163083888

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 971

EP - 984

BT - STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing

A2 - Saha, Barna

A2 - Servedio, Rocco A.

PB - Association for Computing Machinery

T2 - 55th Annual ACM Symposium on Theory of Computing, STOC 2023

Y2 - 20 June 2023 through 23 June 2023

ER -