social network graph problem
This is known as majority dynamics. Usually the assumption is that all nodes adopt the majority opinion simultaneously, and this is known as the synchronous model. For an arbitrary tie-breaking rule, this converges either to a fixed point or to a cycle of length 2; see for example pages 5-6 of Ginosar and Holzman's The majority action on in nite graphs: strings and puppets. If you break ties in a biased way, then the dynamic probably always converges.

What you describe is the asynchronous model, in which the majority rule is applied in sequence rather than in parallel. In that case the process always converges. See for example Tamuz and Tesler, though their methods are probably overkill for you, since in your case you get to choose the sequence, whereas in their case the sequence is chosen at random.