Hi!

I thought I would give what, from my perspective, could be considered the â€śstraight-forwardâ€ť answer to the question. But I might be coming at it from a different angle than the one you had in mind.

Graphs can be thought of as either graphical or topological realizations of binary relations over some set of elements. A â€śbinary relation over a setâ€ť is given by just saying which pairs of elements of that set are related, much like stating which nodes of a graph are connected by an edge. But relations are not necessarily binary: they can in principle relate many items simultaneously, such as in â€śif node X is relevant for thinking about something and node Y isnâ€™t, then node Z is relevant tooâ€ť. So thereâ€™s a sense in which graphs cannot express â€śhigher-orderâ€ť relations like that one (which would be a relation of size 3).

Something to keep in mind is that for these higher-order relations to be something really different, they should be â€śsynergisticâ€ť in some way, i.e. not decomposable into pairwise relations. For instance, one could think of three nodes X, Y, Z as related because of sharing a tag, and that this defines a relation of size 3. But this doesnâ€™t add something truly new, because in fact â€śsharing a tagâ€ť is a binary relation and one can see graphically that the three nodes share a tag by noting that they form a â€ścliqueâ€ť in the graph (that is, they are all connected to each other) (this is assuming a graph has an edge between two nodes if they share a tag).

One might want to use other, more complicated mathematical objects and their geometric realizations to depict relationships between items, which allow for more than two items to be related simultaneously (hypergraphs and simplicial complexes come to my mind). But this might be pointless if the resulting visualizations arenâ€™t intuitive or manageable enough.

On the other hand one can represent higher-order relations in a graph by adding nodes of another type to represent those relations. One could have a graph with â€śnote nodesâ€ť and â€śrelation nodesâ€ť, and let note nodes have edges to/from a common relation node if they are synergistically related with each other. If we dictate that *all* edges are of this kind, so that a â€śstandardâ€ť edge between note nodes X and Y must be replaced for a relation node to which both X and Y are connected, the result would be what is called a bipartite graph. This too might turn out to be cumbersome, and one might need even more additional stuff in the graph to make it workâ€”but it would let you represent higher-order relationships without leaving the graph formalism if you want to.