The Large Deviation Principle for Inhomogeneous Erd?s–Rényi Random Graphs

Consider the inhomogeneous Erd\H{o}s-R\'enyi random graph (ERRG) on $n$ vertices for which each pair $i,j\in\{1,\ldots,n\}$, $i\neq j$ is connected independently by an edge with probability $r_n(\frac{i-1}{n},\frac{j-1}{n})$, where $(r_n)_{n\in\mathbb{N}}$ a sequence of graphons converging to reference graphon $r$. As generalization celebrated LDP ERRGs Chatterjee and Varadhan (2010), Dhara Sen (2019) proved large deviation principle (LDP) such graphs under assumption that $r$ bounded away from 0 1, rate function in form lower semi-continuous envelope. We further extend results Sen. relax conditions $\log r,\log(1-r)\in L^1([0,1]^2)$. also show that, this condition, their equals different, more tractable function. then apply these largest eigenvalue weaken part analysis Chakrabarty, Hazra, Den Hollander Sfragara (2020).