Uniform generation of random regular graphs of moderate degree

McKay, Brendan D.; Wormald, Nicholas C.

doi:10.1016/0196-6774(90)90029-e

Cited by 133 publications

(199 citation statements)

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“…It has been shown [65] that the probability of such a graph being simple is exp(−k 2 /4), and the expected time to obtain a simple k-regular random graph with this algorithm is O(N ke k 2 /4 ). McKay and Wormald [51] improve this expected time to O(N 2 k 4 ) using a simple algorithm, and to O(N k 3 ) using a complicated and hard to implement algorithm.…”

Section: Centralized Constructions Of K-regular Random Graphsmentioning

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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Section: Centralized Constructions Of K-regular Random Graphsmentioning

confidence: 99%

Araneola: A scalable reliable multicast system for dynamic environments

Melamed

Keidar

2008

Journal of Parallel and Distributed Computing

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“…For a performing algorithm choosing one of this regular graphs at random see e.g. [10,23,5] and references therein.…”

Section: 4mentioning

confidence: 99%

A Note on Dynamical Models on Random Graphs and Fokker–Planck Equations

2016

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Abstract. We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e. a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erdős-Rényi graphs with edge probability pn, n is the number of vertices, such that limn→∞ pnn = ∞. The purpose of this note is twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker-Planck PDE (or, equivalently, by a nonlinear diffusion process) in the n = ∞ limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with n large but finite, for example the values of n that can be reached in simulations or that correspond to the typical number of interacting units in a biological system. 2010 Mathematics Subject Classification: 82C20, 60K35

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“…McKay and Wormald also introduced an algorithm based on a certain switching technique on the configuration model that achieves the best performance [32]. It produces random graphs with uniform distribution (better than FPRAS) and has a faster running time.…”

Section: Introductionmentioning

confidence: 99%

“…It has also become an important tool in a variety of real world applications including detecting motifs in biological networks [35] and simulating networking protocols on the Internet topology [39,18,30,13,2]. The best algorithm for this problem was given by McKay and Wormald [32] that uses certain switches on the configuration model and produces random graphs with uniform distribution in O(m 2 d 2 max ) time. However, this running time can be slow for the networks with millions of edges.…”

Section: Introductionmentioning

confidence: 99%