Spreading of electronic viruses, among computers and mobile phones, typically depends on address/phone number lists. The network formed by these lists is not symmetric: the fact that A has B’s address does not ensure that B has A’s address. Thus the underlying network on which such spreading takes place is directed: the links are in general one-way. We present an extension of our analysis for spreading on undirected graphs, to the case of directed graphs. We find that some ideas from Web link analysis lead us to a concrete prediction: that the epidemic coverage changes qualitatively when the rate of infections from ”outside” the network exceeds a threshold rate. Specifically, for low rate of infections from outside, with high probability, only the giant component and its out-components are infected; while for above-threshold infection rate from outside, the whole graph is likely infected.