We study the critical behavior of inhomogeneous random graphs where edges are
present independently but with unequal edge occupation probabilities. The edge
probabilities are moderated by vertex weights, and are such that the degree of
vertex i is close in distribution to a Poisson random variable with parameter
w_i, where w_i denotes the weight of vertex i. We choose the weights such that
the weight of a uniformly chosen vertex converges in distribution to a limiting
random variable W, in which case the proportion of vertices with degree k is
close to the probability that a Poisson random variable with random parameter W
takes the value k. We pay special attention to the power-law case, in which
P(W\geq k) is proportional to k^{-(\tau-1)} for some power-law exponent \tau>3,
a property which is then inherited by the asymptotic degree distribution.
We show that the critical behavior depends sensitively on the properties of
the asymptotic degree distribution moderated by the asymptotic weight
distribution W. Indeed, when P(W\geq k) \leq ck^{-(\tau-1)} for all k\geq 1 and
some \tau>4 and c>0, the largest critical connected component in a graph of
size n is of order n^{2/3}, as on the Erd\H{o}s-R\'enyi random graph. When,
instead, P(W\geq k)=ck^{-(\tau-1)}(1+o(1)) for k large and some \tau\in (3,4)
and c>0, the largest critical connected component is of the much smaller order
n^{(\tau-2)/(\tau-1)}.Comment: 26 page