This paper builds on the work of Acemoglu et al. (2012) and considers a production network with unobserved common technological factor and establishes general conditions under which the network structure contributes to aggregate fluctuations. It introduces the notions of strongly and weakly dominant units, and shows that at most a finite number of units in the network can be strongly dominant, while the number of weakly dominant units can rise with N (the cross section dimension). This paper further establishes the equivalence between the highest degree of dominance in a network and the inverse of the shape parameter of the power law. A new extremum estimator for the degree of pervasiveness of individual units in the network is proposed, and is shown to be robust to the choice of the underlying distribution. Using Monte Carlo techniques, the proposed estimator is shown to have satisfactory small sample properties. Empirical applications to US input-output tables suggest the presence of production sectors with a high degree of pervasiveness, but their effects are not sufficiently pervasive to be considered as strongly dominant.