I'm not sure I agree with your assessment of the criteria of a fixed number of edges. Say we keep the total number of fixed, but allow the number of vertices to change. These rules for vertices and edges would correspond to the global average of number of edges per vertice to decrease, which means that the energy density of the universe was actually decreasing. A decreasing energy density, seems to me, to potentially indicate an inflationary behavior.

Additionally, past the criteria for replacement rules to globally increase vertices while conserving edges (which may hint toward families of initial conditions), the simple model of energy also provides an avenue for some kind of quantized spin. Since we can imagine a particle as a discrete structure, we will then have a quantized 'motion' in that changing edges, through replacement rules, must occur in a discrete manner. So, if we are to take the case for rest matter seriously and purpose that edges undergo some kind of 'rotation' of edges, then this rotation much in tern be discrete. This would be a very intuitive model for spin since the edges that construct a particle are not, due to their stable nature, a component of space. Therefore, the spin of the particle is not only discrete, resemble an actual angular system, but also is intrinsic due to the inaccessibility of the interior edges of from exterior edges or 'space'.

Another point I'd like to bring up is that we have modeled a particle as a kind of structure which undergoes some periodic dynamics. If particles can be modeled as stable periodic localities of the hypergraph, then we may, potentially, be able to model force carriers (the bosons) as a kind of 'extension' of this periodic behavior into the exterior edges that surround the particle(to be considered fermion obviously). In this way, fermions modeled as a kind of discrete waveform would be 'driving' periodic patterns in the surrounding 'space' which, themselves, would be discrete waveforms and could be considered bosons. The behavior of fermions and bosons would then be defined by wave-like phenomenon in a discrete structure.

As far as I knew the conservation of energy, even at cosmological scales, is still held. I know of theories that use violations to explain the origins of dark energy (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.021102), but am unaware of any experimental evidence of violations. So I was coming from the point of view that energy conservation holds for the entire universe. If you have any evidence to convince me that energy conservation is not held at cosmological levels, I'd gladly see it.

What I had written above was under the assumption that energy is conserved at all scales of the universe. Holding that assumption, I proposed a very simple model for edge flux as simply the changes in the number of edges on vertices and explains that if one vertice lost an edge, then another vertice gained one and this was the interpretation of energy conservation as an edge conservation through updating events. If energy conservation was indeed synonymous with a conservation of edges, then if energy was to be conserved globally by the updating events, the number of edges did as well. (If, at this point we how the universe could expand without changes the quantity of edges, we simply give it a very large number of vertices with a comparably small number of edges as the initial conditions for the universe.)

If however, we are assuming that energy is not to be conversed globally, then it still must be nearly conserved globally because the measured value of the cosmological constant is still very small. Models, like the one I already gave the link to above, show how early violation and a leveling off violations could occur and explain dark energy. Therefore, whether we decide to keep energy conservation globally or not, we must still construct some replacement rule and initial condition that either maintains a constant quantity of edges or, after some amount of time, produces some logistic feature for the population of edges, leaving one with a nearly constant number of edges. Either will give global criteria for replacement rules as far as their long-term structures global structures are concerned.

Furthermore, even if we have assumed that energy conservation does not hold globally, then it is held extremely well at small scales because we have, as far as I'm aware, never found experimental evidence of such a violation. Therefore, the arguments I gave for rest mass, the internal rotations of edges, interpretations of quantized spin, and the difference of fermions and bosons are still up for grabs.