Skepticism toward childhood vaccines and genetically modified food has grown despite scientific evidence of their safety. Beliefs about scientific issues are difficult to change because they are entrenched within many interrelated moral concerns and beliefs about what others think. We propose a cognitive network model that estimates network ties between all interrelated beliefs to calculate the overall dissonance and interdependence. Using a probabilistic nationally representative longitudinal study, we test whether our model can be used to predict belief change and find support for our model's predictions: High network dissonance predicts subsequent belief change, and people are driven toward lower network dissonance. We show the advantages of measuring dissonance using the belief network structure compared to traditional measures. This study is the first to combine a unifying predictive model with an experimental intervention and to shed light on the dynamics of dissonance reduction leading to belief change. A cognitive network model tested in a longitudinal study shows that belief network dissonance predicts belief change.