On the second eigenvalue and random walks in randomd-regular graphs

Friedman, Joel

doi:10.1007/bf01275669

Cited by 140 publications

(287 citation statements)

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“…Say that X is ν-weakly Ramanujan if Spec(X) ⊂ [−ν, ν] ∪ {−λ 0 , λ 0 } (also we usually insist that ν < λ 0 to prevent a trivial situation). Then building a k-regular graph from k/2 permutations (assuming that k is even), a number of papers have shown that most graphs are ν-weakly Ramanujan (see [BS87,FKS89,Fri91]) for certain values of ν; for example, in [Fri91] it is shown that there is a constant C such that most k-regular graphs on a sufficiently large number of vertices are ν-weakly Ramanujan with ν = 2 √ k + 2 log k + C. Furthermore, numerical experiments (like those in [Fri93]) suggest that most random regular graphs on a large number of vertices are Ramanujan.…”

Section: Introductionmentioning

confidence: 99%

Relative expanders or weakly relatively Ramanujan graphs

Friedman¹

2003

Duke Math. J.

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110

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Let G be a fixed graph with largest (adjacency matrix) eigenvalue λ 0 and with its universal cover having spectral radius ρ. We show that a random cover of large degree over G has its "new" eigenvalues bounded in absolute value by roughly √ λ 0 ρ. This gives a positive result about finite quotients of certain trees having "small" eigenvalues, provided we ignore the "old" eigenvalues. This positive result contrasts with the negative result of LubotzkyNagnibeda that showed that there is a tree all of whose finite quotients are not "Ramanujan" in the sense of Lubotzky-Philips-Sarnak and Greenberg.Our main result is a "relative version" of the Broder-Shamir bound on eigenvalues of random regular graphs. Some of their combinatorial techniques are replaced by spectral techniques on the universal cover of G. For the choice of G that specializes our theorem to the BroderShamir setting, our result slightly improves theirs.MSC 2000 numbers: Primary: 05C50; Secondary: 05C80,

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Section: Introductionmentioning

confidence: 99%

Relative expanders or weakly relatively Ramanujan graphs

Friedman¹

2003

Duke Math. J.

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110

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“…The O(d 3/4 ) bound for ρ 2 (G) was improved to O( √ d) by Friedman (using the same model) [50]. A weaker version than Friedman's was obtained independently by Kahn and Szemerédi (cf.…”

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confidence: 94%

Eigenvalues in Combinatorial Optimization

Mohar

Poljak

1993

Combinatorial and Graph-Theoretical Problems in Linear Algebra

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“…A kregular random graph with N nodes is a graph chosen uniformly at random from the set of k-regular graphs with N nodes. In contrast to a normal random graph [11], where node degrees vary, in a k-regular graph, each node's degree is exactly k. For k ≥ 3, a k-regular random graph is almost always a good expander [25], which implies that (i) its diameter grows logarithmically with N [65]; and (ii) it remains connected after random failures of a linear subset of its nodes and/or edges [27]. In addition, such a graph is generally k-connected, i.e., there are k disjoint paths between every two nodes in the graph 2 .…”

Section: Introductionmentioning

confidence: 99%

Araneola: A scalable reliable multicast system for dynamic environments

Melamed

Keidar

2008

Journal of Parallel and Distributed Computing

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This paper presents Araneola 1 , a scalable reliable application-level multicast system for highly dynamic wide-area environments. Araneola supports multi-point to multi-point reliable communication in a fully distributed manner while incurring constant load (in terms of message and space complexity) on each node. For a tunable parameter k ≥ 3, Araneola constructs and dynamically maintains a basic overlay structure in which each node's degree is either k or k + 1, and roughly 90% of the nodes have degree k. Empirical evaluation shows that Araneola's basic overlay achieves three important mathematical properties of k-regular random graphs (i.e., random graphs in which each node has exactly k neighbors) with N nodes: (i) its diameter grows logarithmically with N ; (ii) it is generally k-connected; and (iii) it remains highly connected following random removal of linear-size subsets of edges or nodes. The overlay is constructed and maintained at a low cost: each join, leave, or failure is handled locally, and entails the sending of only about 3k messages in total, independent of N . Moreover, this cost decreases as the churn rate increases.The low degree of Araneola's basic overlay structure allows for allocating plenty of additional bandwidth for specific application needs. In this paper, we give an example for such a need -communicating with nearby nodes; we enhance the basic overlay with additional links chosen according to geographic proximity and available bandwidth. We show that this approach, i.e., a combination of random and nearby links, reduces the number of physical hops messages traverse without hurting the overlay's robustness as compared to completely random Araneola overlays (in which all the links are random) with the same average node degree.Given Araneola's overlay, we sketch out several message dissemination techniques that can be implemented on top of this overlay. We present a full implementation and evaluation of a gossip-based multicast scheme with up to 10,000 nodes. We show that compared to a (non-overlay-based) gossipbased multicast protocol, gossiping over Araneola achieves substantial improvements in load, reliability, and latency.

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On the second eigenvalue and random walks in randomd-regular graphs

Cited by 140 publications

References 8 publications

Relative expanders or weakly relatively Ramanujan graphs

Relative expanders or weakly relatively Ramanujan graphs

Eigenvalues in Combinatorial Optimization

Araneola: A scalable reliable multicast system for dynamic environments

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