In this talk we discuss a problem in Combinatorial Probability, that concerns some finer details of the so-called 'giant component' phase transition in random graphs. More precisely, it is well-known that the size $L_1(G_{n,p})$ of the largest component of the binomial random graph $G_{n,p}$ has a scaling limit for $p=c/n$, i.e., that $L_1(G_{n,p})/n$ converges in probability to some limiting function $\rho(c)$. It is of interest to understand finer details of this limiting function, in particular if $\rho(c)$ is well-behaved for some range of $c$, say analytic. Analyticity can be shown directly for the binomial random graph $G_{n,p}$, since explicit descriptions and formulas for $\rho(c)$ are available. In this talk I will outline a somewhat more robust approach, that also works in models where explicit formulas are not available. Our approach combines tools from random graph theory (multi-round exposure arguments), stochastic processes (differential equation approximation), generating functions, and partial differential equations (Cauchy-Kovalevskaya Theorem).