This paper develops a simplified dependency model for sources on social networks that is shown to improve the quality of fact-finding - assessing veracity of observations shared on social media. Recent literature developed a mathematical approach for exploiting social networks, such as Twitter, as noisy sensor networks that report observations on the state of the physical world. It was shown that the quality of state estimation from such noisy data, known as fact-finding, was a function of assumptions made regarding the independence of sources or lack thereof. When sources propagate information they hear from others (without verification), correlated errors may arise that degrade fact-finding performance. This work advances the state of the art by developing a simplified model of dependencies between sources and designing an improved dependency-aware estimator to assess veracity of observations, taking into account the observed dependency structure. A fundamental error bound is derived for this estimator to understand the gap in its performance from optimal. It is shown that the new estimator outperforms state of the art fact-finders and, in some cases, yields an accuracy close to the fundamental error bound.