In artificial multi-agent systems, the ability to learn collaborative policies is predicated upon the agents' communication skills: they must be able to encode the information received from the environment and learn how to share it with other agents as required by the task at hand. We present a deep reinforcement learning approach, Connectivity Driven Communication (CDC), that facilitates the emergence of multi-agent collaborative behaviour only through experience. The agents are modelled as nodes of a weighted graph whose state-dependent edges encode pair-wise messages that can be exchanged. We introduce a graph-dependent attention mechanisms that controls how the agents' incoming messages are weighted. This mechanism takes into full account the current state of the system as represented by the graph, and builds upon a diffusion process that captures how the information flows on the graph. The graph topology is not assumed to be known a priori, but depends dynamically on the agents' observations, and is learnt concurrently with the attention mechanism and policy in an end-to-end fashion. Our empirical results show that CDC is able to learn effective collaborative policies and can over-perform competing learning algorithms on cooperative navigation tasks.