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).