In a susceptible-infected epidemic on an evolving graph, each susceptible-infected edge transmits infection at rate lambda. Below the critical rate, outbreaks die quickly. Above it, they sweep through the network. At the critical rate, the behavior depends on something subtler than the rate itself.
Chen, Hou, Ma, and Yao show that at the critical infection rate, the probability of a major outbreak starting from a single infected individual is governed by a parameter Delta derived from the third and fourth moments of the degree distribution. When Delta is positive, the outbreak probability decays as Cn^{-1/3} — polynomially in network size, meaning large outbreaks remain possible even at criticality. When Delta is negative, the probability decays faster than any polynomial — outbreaks are exponentially suppressed. The sign of one number, computed from the shape of the degree distribution, determines whether critical epidemics are common or vanishing.
The result is reminiscent of critical Erdos-Renyi graphs, where a similar cubic-root scaling governs the emergence of giant components. The analogy is structural: the epidemic at criticality and the random graph at its percolation threshold face the same mathematical question — whether the local structure supports long-range connectivity.
The through-claim is about what governs criticality. The critical rate lambda_c marks where the transition happens, but at the transition itself, the infection rate is no longer the relevant variable. The degree distribution's shape — its higher moments, not its mean — controls the outcome. Two networks with the same critical rate can have opposite behaviors at that rate, one permitting large outbreaks and the other suppressing them, depending on the tail of their degree distributions. The threshold is the same; the behavior at the threshold is different.