The association between thermo-mechanical properties in polymers and functionality of monomer precursors is frequently exploited in the materials science. However, it is not known if there are more variables beyond monomer functionality that have a similar link. Here, by using simulations to generate spatial networks from chemically different monomers with identical functionality we show that such networks have universal graph-theoretical properties as well as a near-universal elastic modulus. The vitrification temperature was found to be universal only up to a certain network density, as measured by the bond conversion. The latter observation is explained by the fact that monomer’s tendency to coil enhances formation of topological holes, which, when accumulated, amount to a percolating cell complex restricting network’s mobility. This higher-order percolation occurs late after gelation and is shown to coincide with the onset of brittleness, as indicated by a sudden increase in the glass transition temperature.