Fig. 4: A novel metastable phase that yields a phase transition via a microscopic damage.

a Phase diagram of spatial 5-core percolation as a function of ζ and 〈k〉. Purple is the unstable phase, and brown is the stable phase where the system is resilient to localized attacks. The colors within the dashed line represent the metastable phase where a circular damage above a radius \({r}_{h}^{c}\) will cause the system to collapse. The value of \({r}_{h}^{c}\) separates the unstable phase (\({r}_{h}^{c}=0\), purple), the stable phase (\({r}_{h}^{c} \sim L/2\), brown), and the new metastable phase (\(0 < {r}_{h}^{c} < L/2\), blue to orange). The size of the network is N = L × L = 1000 × 1000. b Illustration of the propagation of damage initiated from localized attacks with a hole of radius r_h in the spatial network. The hole on the left is smaller than the \({r}_{h}^{c}\), and the damage does not propagate, thus maintaining the same size as the initial state. However, damage caused by holes larger than \({r}_{h}^{c}\) on the right evolves and spreads radially throughout the system over time t. c Dependence of \({r}_{h}^{c}\) on the size of the system L for ζ = 18 and varying 〈k〉 values. The fact that \({r}_{h}^{c}\) does not increase with L demonstrates that the localized damage is of microscopic size. d The dependence of \({r}_{h}^{c}\) on the system size for 〈k〉 = 7.5 and varying ζ values. In c and d, each point represents the average of 10 realizations for L < 1000, and the average of 5 realizations for L ≥ 1000.