We have combined ab initio molecular dynamics with the intrinsic reaction coordinate in order to investigate the mechanisms of stability and pyrolysis of N$_{4}$-- N$_{120}$ fullerene-like nitrogen cages. The stability of the cages was evaluated in terms of the activation barriers and the activation Gibbs energies of their thermal-induced breaking. We found that binding energies, bond lengths, and quantum-mechanical descriptors failed to predict the stability of the cages. However, we derived a simple topological rule that adjacent hexagons on the cage surface resulted in its instability. For this reason, the number of stable nitrogen cages is significantly restricted in comparison with their carbon counterparts. As a rule, smaller clusters are more stable, whereas earlier proposed rather large cages collapse at room temperature. The most stable all-nitrogen cages are N$_{4}$ and N$_{6}$ clusters, which can form the van-der-Waals crystals with the densities of 1.23 and 1.36 g/cm$^{3}$, respectively. Examination of their band structures and densities of electronic states shows that they are both insulators. Their power and sensitivity are not inferior to the modern advanced high-energy nanosystems.
