In this paper, we present a new bounded-confidence model of opinion dynamics where agents with a substantially different opinion may still be able to influence each other along a static graph. Our intention is to account for hard and fast ties present due to physical or social proximity. This additional feature allows weak but persistent interaction between disagreeing agents. Albeit simple, the model remains difficult to analyse due to its separation of time scales. We introduce an appropriate notion of stability, give some properties of the system and formulate a conjecture supported by numerical simulations.