Knowledge-sharing communities are a fundamental element of any knowledge-based society. Understanding how they emerge, function, and disappear is thus of crucial importance. Many social and economic factors influence sustainable knowledge-sharing communities. Here we explore the role of the structure of social interactions and social trust in the emergence of these communities. Using tools from complex network theory, we analyze the early evolution of social structure in four pairs of StackExchange communities, each corresponding to one active and one closed community on the same topic. We adapt the dynamical reputation model to quantify the evolution of social trust in these communities. Our analysis shows that active communities have higher local cohesiveness and develop stable and more strongly connected cores. The average reputation is higher in sustainable communities. In these communities, the trust between core members develops early and remains high over time. Our results imply that efforts to create a stable and trustworthy core may be crucial for building a sustainable knowledge-sharing community.