Caching contents in edge networks can reduce latency and lighten the burden on backhaul links. Since the capacity of cache nodes is limited, accurate content popularity distribution is crucial to the effectual usage of cache capacity. However, existing popularity prediction models stem from big data and, hence, may suffer poor accuracy due to the small population in edge caching. In this letter, we propose a social-driven propagation dynamics-based prediction model, which requires neither training phases nor prior knowledge. Specifically, we first explore social relationships to bridge the gap between small population and prediction accuracy under susceptible-infected-recovery model. Then, a discrete-time markov chain approach is proposed to predict the viewing probability of certain contents from the perspective of individuals. Simulations validate that our proposed model outperforms other solutions significantly, by improving up to 94% in accuracy and 99% less runtime overhead.