Interactions among networked agents influence the beliefs of agents about the state of nature. For example, in deciding whether the state of nature, denoted by θ, is either θ = 1 or θ = 0, an agent observes some data whose probability distribution is dependent on the unknown θ and, additionally, consults with its neighboring agents about their opinion on the most plausible value for θ. By combining their local measurements with the information from their neighbors, agents update their belief about θ continuously. In this work, we examine social learning and the diffusion of information over weakly-connected graphs where the flow of information is asymmetric. This scenario is common over social networks. For example, it is not unusual for some influential agents (such as celebrities) to have a large number of followers, while the influential agent may not be following most of them. A similar effect arises when social networks operate in the presence of stubborn agents, which insist on their opinion regardless of the evidence provided by observations or by neighboring agents. It turns out that weak graphs influence the evolution of the agents’ learning in an interesting manner and facilitate the spread of false information. Under some circumstances, a scenario arises where the influential agents in the network drive the learning behavior and determine the limiting state for all agents regardless of their local observations. For example, some agents may be driven to believe erroneously that “it is raining” although they are observing “sunny conditions.” At the same time, the graph topology endows non-influential agents with a resistive mechanism where they cannot be driven to any arbitrary belief. This presentation examines these patterns of behavior over multi-agent inference networks analytically and illustrates the results with examples and simulations.