In recent times we hear increasingly often about cyber attacks on various commercial and strategic sites that manage to escape any defense. In this article, we model such attacks on networks via stochastic processes and predict the time of a total or partial failure of a network including the magnitude of losses (such as the number of compromised nodes, lost weights, and a loss of other associated components relative to some fixed thresholds). To make such modeling more realistic we also assume that the information about the attacks is delayed as per random observations. We arrive at analytically and numerically tractable results demonstrated by examples and comparative simulation.