Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well-studied for various classes of graphs. When it comes to random graphs, only the classical Erd\H{o}s-R\'enyi-Gilbert random graph $G_{n,p}$ has been analysed and shown to have largest independent sets of size $\Theta(\log{n})$ w.h.p. This classical model does not capture any dependency structure between edges that can appear in real-world networks. We initiate study in this direction by defining random graphs $G^{r}_{n,p}$ whose existence of edges is determined by a Markov process that is also governed by a decay parameter $r\in(0,1]$. We prove that w.h.p. $G^{r}_{n,p}$ has independent sets of size $(\frac{1-r}{2+\epsilon}) \frac{n}{\log{n}}$ for arbitrary $\epsilon > 0$, which implies an asymptotic lower bound of $\Omega(\pi(n))$ where $\pi(n)$ is the prime-counting function. This is derived using bounds on the terms of a harmonic series, Tur\'an bound on stability number, and a concentration analysis for a certain sequence of dependent Bernoulli variables that may also be of independent interest. Since $G^{r}_{n,p}$ collapses to $G_{n,p}$ when there is no decay, it follows that having even the slightest bit of dependency (any $r < 1$) in the random graph construction leads to the presence of large independent sets and thus our random model has a phase transition at its boundary value of $r=1$. For the maximal independent set output by a greedy algorithm, we deduce that it has a performance ratio of at most $1 + \frac{\log{n}}{(1-r)}$ w.h.p. when the lowest degree vertex is picked at each iteration, and also show that under any other permutation of vertices the algorithm outputs a set of size $\Omega(n^{1/1+\tau})$, where $\tau=1/(1-r)$, and hence has a performance ratio of $O(n^{\frac{1}{2-r}})$.

翻译：计算图中独立集的最大规模是一个著名的困难组合问题，已在各类图中得到深入研究。对于随机图，目前仅经典 Erdős–Rényi–Gilbert 随机图 $G_{n,p}$ 被分析并证明其最大独立集规模高概率为 $\Theta(\log{n})$。该经典模型未能刻画现实网络可能存在的边间依赖结构。我们通过定义随机图 $G^{r}_{n,p}$ 在此方向展开研究：其边的存在性由马尔可夫过程决定，该过程同时受衰减参数 $r\in(0,1]$ 调控。我们证明高概率下 $G^{r}_{n,p}$ 具有规模为 $(\frac{1-r}{2+\epsilon}) \frac{n}{\log{n}}$（$\epsilon > 0$ 任意）的独立集，这推出渐近下界 $\Omega(\pi(n))$，其中 $\pi(n)$ 为素数计数函数。该结论通过调和数列项界、独立数的 Turán 界，以及对一列可能具有独立研究价值的相依伯努利变量的集中性分析得到。由于无衰减时 $G^{r}_{n,p}$ 退化为 $G_{n,p}$，这表明随机图构造中即使存在最微弱的依赖性（任意 $r < 1$）也会导致大独立集的出现，因此我们的随机模型在边界值 $r=1$ 处存在相变。对于贪心算法输出的最大独立集，我们推导出当每次迭代选取最低度顶点时，其性能比高概率不超过 $1 + \frac{\log{n}}{(1-r)}$；同时证明在任何其他顶点排列下，算法输出集的规模为 $\Omega(n^{1/1+\tau})$（其中 $\tau=1/(1-r)$），因而具有 $O(n^{\frac{1}{2-r}})$ 的性能比。