I am tryng to make sense of what you say here
Well in Loop approach and the related spinfoam approach they use SPIN NETWORKS (a spin network extends throughout all space) and SPIN FOAMS (a spin foam is co-extensive with spacetime). These things serve to represent a state of the gravitational field, or a state of spacetime geometry. They have a certain combinatorial aspect because you can in principle describe them by listing data.
and using spin networks leads to area and volume operators with discrete spectra.
but I have never heard anyone describe a spin network or a spinfoam as a PARTICLE.
I never read where Ashtekar said that a spin network was "like a graviton".
That is just one sample approach, LQG, and one person. I am trying to see what you might mean.
ohwilleke, if there is "considerable dispute" then does that include the non-perturbative QG crowd?
If you would, look down the list of representative QG people, the invited speakers, and tell me who would dispute with whom, about whether spacetime should be described as made of something "like gravitons" or some kind of point particle. You may be right! I dont know the work of everybody on the list, far from it.