Average Jaccard index of random graphs

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 $n^2(1-p)\to\infty$ .

Root nodules of red alder (Alnus rubra) and Sitka alder (Alnus viridis ssp. sinuata) are inhabited by taxonomically diverse cultivable microbial endophytes

“…Homogeneity of dispersion among tree species was verified using vegdist, betadisper (bias. adjust=T), and permutest in vegan before proceeding to PERMA-NOVA with Jaccard distances (Feng et al, 2024) and 999 permutations to determine statistical differences in community composition (Llames et al, 2017). For phylogenetics, representative sequences of each OTU were aligned with downloaded sequences from closely related species (as determined through BLAST searching of the NCBI database) using MAFFT v7.45 (Katoh & Standley, 2013) as implemented in Geneious v10.2.6 (Kearse et al, 2012), followed by manual improvement.…”