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The Spectral Gap of Sparse Random Digraphs

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Simon Coste

Probability · Submitted 1 Aug 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper, we give an asymptotic upper bound for the second eigenvalue when $P$ is the transition matrix of the simple random walk over a random directed graph with given degree sequence.

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arXiv ID1708.00530 (version 2)
Pith IDSNBR562H, Pith's permanent record of this paper and its review
CitationSimon Coste. "The Spectral Gap of Sparse Random Digraphs." arXiv:1708.00530 (2017). https://pith.science/paper/SNBR562H
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,255 input tokens.

What the abstract itself states, as probabilities: claims first 0.94 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.02 · states quantitative result 0.42 · compares with prior results 0.47 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

The second largest eigenvalue of a transition matrix $P$ has connections with many properties of the underlying Markov chain, and especially its convergence rate towards the stationary distribution. In this paper, we give an asymptotic upper bound for the second eigenvalue when $P$ is the transition matrix of the simple random walk over a random directed graph with given degree sequence. This is the first result concerning the asymptotic behavior of the spectral gap for sparse non-reversible Markov chains with an unknown stationary distribution. An immediate consequence of our result is a proof of the Alon conjecture for directed regular graphs. Our result is based on a variation of the trace method introduced by Bordenave (2015).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The second largest eigenvalue of a transition matrix $P$ has connections with many properties of the underlying Markov chain, and especially its convergence rate towards the stationary distribution. background 1.00 0.04 0.03 asserted 0.99
s1 In this paper, we give an asymptotic upper bound for the second eigenvalue when $P$ is the transition matrix of the simple random walk over a random directed graph with given degree sequence. contribution 0.79 0.96 0.75 asserted 1.00
s2 This is the first result concerning the asymptotic behavior of the spectral gap for sparse non-reversible Markov chains with an unknown stationary distribution. contribution 0.47 0.86 0.14 asserted 1.00
s3 An immediate consequence of our result is a proof of the Alon conjecture for directed regular graphs. interpretation 0.75 0.88 0.12 asserted 1.00
s4 Our result is based on a variation of the trace method introduced by Bordenave (2015). method 1.00 0.53 0.03 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

The second largest eigenvalue of a transition matrix $P$ has connections with many properties of the underlying Markov chain, and especially its convergence rate towards the stationary distribution. In this paper, we give an asymptotic upper bound for the second eigenvalue when $P$ is the transition matrix of the simple random walk over a random directed graph with given degree sequence. This is the first result concerning the asymptotic behavior of the spectral gap for sparse non-reversible Markov chains with an unknown stationary distribution. An immediate consequence of our result is a proof of the Alon conjecture for directed regular graphs. Our result is based on a variation of the trace method introduced by Bordenave (2015).

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8 arXiv papers cite this in 12 sentences

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Papers citing it · 8 sentences without a confident label

Compares with it 2 papers
Cites it as background 1 paper
  • In the analysis of Markov chains, the spectral gap is a valuable metric to consider because it provides insights into the speed at which the chain reaches its stationary state [ref] .

    Linear Log-Normal Attention with Unbiased Concentration · Spectral Gap and Attention Concentration · 0.85

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