In the real world as in the virtual one, trust is a fundamental concept. Without it, humans can neither act nor interact. So unsurprisingly, this concept received in the last years a growing interest from researchers in security and distributed artificial intelligence that gave rise to numerous models. The principal aim of these models was to assist users in making safe decisions at the individual level. However, studies have shown that the behavior of an individual within collective structures (e.g., a group, a community, a coalition or an organization) is affected (directly or indirectly) by the behavior of other members, creating a social influence dynamics within these structures. In this article, we study the impact of social influence phenomena when they are applied to trust management within open distributed communities of self-organized and self-governed agents.