Causal graphs provide one kind of summary of the evolution of a system, based on capturing the causal relationships between events. What we call branchial graphs provide another kind of summary, based on capturing relationships between states on different branches of a multiway system. And whereas causal graphs capture relationships between events at different steps in the evolution of a system, branchial graphs capture relationships between states on different branches at a given step. And in a sense they define a map for exploring branchial space in a multiway system. page 
One might perhaps have imagined that states on different branches of a multiway system would be completely independent. But when causal invariance is present they are definitely not—because for example whenever they split (to form a branch pair), they will always merge again.