We consider the problem of learning the interaction strength between the nodes of a network based on dependent binary observations residing on these nodes, generated from a Markov Random Field (MRF). Since these observations can possibly be corrupted/noisy in larger networks in practice, it is important to robustly estimate the parameters of the underlying true MRF to account for such inherent contamination in observed data. However, it is well-known that classical likelihood and pseudolikelihood based approaches are highly sensitive to even a small amount of data contamination. So, in this paper, we propose a density power divergence (DPD) based robust generalization of the computationally efficient maximum pseudolikelihood (MPL) estimator of the interaction strength parameter, and derive its rate of consistency under the pure model. Along the way, we establish consistency and asymptotics for a class of general $Z$-estimators, covering our proposed DPD based estimators, under flexible assumptions that hold for a substantial class of standard models. To the best of our knowledge, these are the first central limit theorems for the class of general $Z$-estimators in such settings. Moreover, we show that the gross error sensitivities of the proposed DPD based estimators are significantly smaller than that of the MPL estimator, thereby theoretically justifying the greater (local) robustness of the former under contaminated settings. Finally, we demonstrate the superior (finite sample) performance of the DPD based variants over the traditional MPL estimator in a number of synthetically generated contaminated network datasets, and apply them to learn the network interaction strength in several real datasets from diverse domains of social science, neurobiology and genomics.