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Average Jaccard index of random graphs

Part of: Limit theorems Graph theory Combinatorial probability

Published online by Cambridge University Press:  26 February 2024

Qunqiang Feng Open the ORCID record for Qunqiang Feng [Opens in a new window] ,
Shuai Guo  and
Zhishui Hu Open the ORCID record for Zhishui Hu [Opens in a new window]
Qunqiang Feng*
Affiliation:
University of Science and Technology of China
Shuai Guo*
Affiliation:
University of Science and Technology of China
Zhishui Hu*
Affiliation:
University of Science and Technology of China
*
*Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China.
*Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China.
*Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China.
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Abstract

The asymptotic behavior of the Jaccard index in G(n, p), the classical Erdös–Rényi random graph model, is studied as n goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in G(n, p) is shown to be asymptotically normal under an additional mild condition that $np\to\infty$ and $n^2(1-p)\to\infty$.

Keywords

Erdös–Rényi Random graphJaccard similarityasymptotic distributioninverse moment

MSC classification

Primary: 05C80: Random graphs 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 14
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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