Many fundamental concepts in network-based epidemic models depend on the branching factor, which captures a sense of dispersion in the network con- nectivity and quantifies the rate of spreading across the network. Moreover, contact network information generally is available only up to some level of error. We study the propagation of such errors to the estimation of the branching factor. Specifically, we characterize the impact of network noise on the bias and variance of the observed branching factor for arbitrary true networks, with examples in sparse, dense, homo- geneous and inhomogeneous networks. In addition, we propose two estimators, by method-of-moments and bootstrap sampling. We illustrate the practical performance of our estimators through simulation studies and social contact networks in British secondary schools.