Connectivity threshold conjecture for heterogeneous random key graphs

Let $r$ be the number of classes, let $\pmb{\mu}=(\mu_1,\mu_2,\ldots,\mu_r)$ be a probability distribution with $\mu_i>0$ for $i=1,\ldots,r$, and let $K_1,\ldots,K_r,P$ and $\pmb{\alpha}=(\alpha_{ij})$ be the scaling functions specified in the conjecture. Assume the scaling condition $\Lambda_m(n)=c_n\frac{\log n}{n}$ with $\lim_{n\to\infty}c_n=c>0$, and write $\alpha_{md}(n)$ and $\alpha_{mm}(n)$ for the corresponding minimum cross-class and within-class link probabilities. Suppose either

$$\lim_{n\to\infty}\alpha_{md}(n)\log n=0,$$

or

$$\lim_{n\to\infty}\alpha_{md}(n)\log n=\alpha^*\in(0,\infty],\qquad \lim_{n\to\infty}\alpha_{mm}(n)\log n=\alpha^{**}\in(0,\infty].$$

Connectivity threshold conjecture. Possibly under some additional conditions, the heterogeneous random graph $\mathbb{H}(n;\pmb{\mu},\pmb{\Theta}_n)$ should satisfy

$$\lim_{n\to\infty}\mathbb{P}\left[\mathbb{H}(n;\pmb{\mu},\pmb{\Theta}_n)\text{ is connected}\right] = \begin{cases} 0,&c<1,\\ 1,&c>1. \end{cases}$$

The conjecture proposes that connectivity has the same zero-one threshold as the absence of isolated nodes. The zero-law for connectivity already follows from the isolated-node result, while the one-law remains to be established, potentially subject to additional conditions.