Phase transition in inhomogenous Erd\H{o}s-R\'enyi random graphs via tree counting

Consider the complete graph $K_n$ on $n$ vertices where each edge $e$ is independently open with probability $p_n(e)$ or closed otherwise. Here $\frac{C-\alpha_n}{n} \leq p_n(e) \leq \frac{C+\alpha_n}{n}$ where $C > 0$ is a constant not depending on~$n$ or~$e$ and $0 \leq \alpha_n \longrightarrow 0$ as $n \rightarrow \infty.$ The resulting random graph~$G$ is inhomogenous and we use a tree counting argument to establish phase transition in $G.$ We also obtain that the critical value for phase transition is one in the following sense. For $C < 1,$ all components of $G$ are small (i.e. contain at most $M\log{n}$ vertices) with high probability, i.e., with probability converging to one as $n \rightarrow \infty.$ For $C > 1,$ with high probability, there is at least one giant component (containing at least $\epsilon n$ vertices for some $\epsilon > 0$) and every component is either small or giant. For $C > 8,$ with positive probability, the giant component is unique and every other component is small. As a consequence of our method, we directly obtain the fraction of vertices present in the giant component in the form of an infinite series.