A note on the random triadic process

TL;DR

This paper analyzes the size of the random r-generalized triadic process, showing it behaves like a Erdős–Rényi graph under certain conditions and providing bounds on the number of edges added.

Contribution

It extends previous work on the triadic process by examining the process for general r and establishing its asymptotic behavior in a new probabilistic regime.

Findings

Graph size approximates n^2p with high probability

Process behaves like G(n,p) under specified p conditions

Final edge count is n^2p(1  o(1)) when p=n^{-2}

Abstract

For a fixed integer $r\geqslant 3$, let $\mathbb{H}_r(n,p)$ be a random $r$-uniform hypergraph on the vertex set $[n]$, where each $r$-set is an edge randomly and independently with probability $p$. The random $r$-generalized triadic process starts with a complete bipartite graph $K_{r-2,n-r+2}$ on the same vertex set, chooses two distinct vertices $x$ and $y$ uniformly at random and iteratively adds $\{x,y\}$ as an edge if there is a subset $Z$ with size $r-2$, denoted as $Z=\{z_1,\cdots,z_{r-2}\}$, such that $\{x,z_i\}$ and $\{y,z_i\}$ for $1\leqslant i\leqslant r-2$ are already edges in the graph and $\{x,y, z_1,\cdots,z_{r-2}\}$ is an edge in $\mathbb{H}_r(n,p)$. The random triadic process is an abbreviation for the random $3$-generalized triadic process. Kor\'{a}ndi et al. proved a sharp threshold probability for the propagation of the random triadic process, that is, if $p= cn^{ - \frac 12}$ for some positive constant $c$, with high probability, the triadic process reaches the complete graph when $c> \frac 12$ and stops at $O(n^{\frac 32})$ edges when $c< \frac 12$. In this note, we consider the final size of the random $r$-generalized triadic process when $p=o( n^{- \frac 12}\log^{ \alpha(3-r)} n)$ with a constant $\alpha> \frac 12$. We show that the generated graph of the process essentially behaves like $\mathbb{G}(n,p)$. The final number of added edges in the process, with high probability, equals $ \frac {1}{2}n^{2}p(1\pm o(1))$ provided that $p=\omega(n^{-2})$. The results partially complement the ones on the case of $r=3$.