This paper investigates the event-driven sampled-data consensus in switching networks of multiple integrators and studies both the bidirectional interaction and leader-following passive reaction topologies in a unified framework. In these topologies, each information link is modelled by an edge of the information graph and assigned a sequence of edge events, which activate the mutual data sampling and controller updates of the two linked agents. Two kinds of edge-event-detecting rules are proposed for the general asynchronous data-sampling case and the synchronous periodic event-detecting case. They are implemented in a distributed fashion, and their effectiveness in reducing communication costs and solving consensus problems under a jointly connected topology condition is shown by both theoretical analysis and simulation examples.