Comments I made on a blog post about computer science and social networks have a connection with the work in progress on this thread, so it may gain a march to append them here.
I have been interested for a long time now in using graphs to do logic. For that you need more than positive links — negative relations are more generative than positive relations. This is analogous to social networks where people can “unlike” or “¬like” other people, or website networks where the information at one node can contradict the information at another node. In my pursuits it turns out that particular species of graph-theoretic “cacti” are much more useful than the usual trees and unsigned graphs.
For what it’s worth, here is my somewhat dry exposition of “painted cacti” and their application to propositional calculus.
Painted cacti are rooted cacti with any number of symbols from a finite alphabet attached to each node. In the intended interpretations these symbols (“paints”) are boolean variables.
Triangles are interesting in this context because they arise in logical case expressions. In one of the customary interpretations of the cactus graphs, a rooted triangular lobe says that the values of the two non-root nodes are logically inequivalent.