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 binomial 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大小是一个著名的困难组合问题，已针对各类图结构进行了深入研究。在随机图领域，仅有经典二项随机图$G_{n,p}$被分析过，并证明其最大独立集大小以高概率为$\Theta(\log{n})$。该经典模型无法捕捉现实网络中边之间可能存在的依赖结构。我们通过定义随机图$G^{r}_{n,p}$（其边的存在性由同时受衰减参数$r\in(0,1]$控制的马尔可夫过程决定）开启此方向的研究。我们证明：以高概率，$G^{r}_{n,p}$对任意$\epsilon > 0$存在大小为$(\frac{1-r}{2+\epsilon}) \frac{n}{\log{n}}$的独立集，这给出了一个渐近下界$\Omega(\pi(n))$（其中$\pi(n)$为素数计数函数）。该结果基于调和级数项的界、稳定数的Turán界以及某特定依赖伯努利变量序列的浓度分析（该分析本身可能具有独立意义）。由于无衰减时$G^{r}_{n,p}$退化为$G_{n,p}$，因此随机图构造中即使存在最轻微的依赖（任意$r < 1$）也会导致大独立集的出现，从而我们的随机模型在其边界值$r=1$处发生相变。对于贪心算法输出的最大独立集，我们证明：当每次迭代选取最低度顶点时，其性能比以高概率不超过$1 + \frac{\log{n}}{(1-r)}$；并进一步表明，在任意其他顶点排序下，算法输出大小为$\Omega(n^{1/1+\tau})$的集合（其中$\tau=1/(1-r)$），从而其性能比为$O(n^{\frac{1}{2-r}})$。