On the largest strongly connected component of randomly oriented divisor graphs

TL;DR

This paper investigates the expected size of the largest strongly connected component in randomly oriented divisor graphs, revealing it is asymptotic to the number of vertices for fixed edge-reversal probability.

Contribution

It provides a lower bound for the largest strongly connected component size in randomly oriented divisor graphs and an effective version of Hardy-Ramanujan's theorem on divisor function distribution.

Findings

Expected size of the largest strongly connected component is asymptotic to N for fixed ρ.

Established a lower bound for the component size based on divisor function distribution.

Proved an effective version of Hardy and Ramanujan's theorem on log τ(n).

Abstract

We introduce the study of \textit{randomly oriented divisor graphs}. For each $\rho \in [0,1]$, the randomly oriented divisor graph $\mathcal{D}_\rho(N)$ is obtained from the divisor graph on $\{1, 2, \ldots, N\}$ by directing each edge according to divisibility and independently reversing the direction of each edge with probability $\rho$. We study the expected size of the largest strongly connected component, $\textbf{E}[\#\Phi(\mathcal{D}_\rho(N))]$. Our main result gives a lower bound for this quantity in terms of the distribution of values of the divisor function $\tau(n)$. As a consequence, we show that for any fixed $\rho \in (0,1)$, the largest strongly connected component has expected size asymptotic to $N$. To obtain explicit bounds, we prove an effective version of a theorem of Hardy and Ramanujan on the normal order of $\log \tau(n)$, which may be of independent interest.