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K-core percolation is a fundamental dynamical process in complex networkswith applications that span numerous real-world systems. Earlier studies focusprimarily on random networks without spatial constraints and reveal intriguingmixed-order transitions. However, real-world systems, ranging fromtransportation and communication networks to complex brain networks, are notrandom but are spatially embedded. Here, we study k-core percolation ontwo-dimensional spatially embedded networks and show that, in contrast toregular percolation, the length of connections can control the transition type,leading to four different types of phase transitions associated with novelphenomena and a rich phase diagram. A key finding is the existence of ametastable phase in which microscopic localized damage, independent of systemsize, can cause a macroscopic phase transition, a result which cannot beachieved in traditional percolation. In this case, local failures canspontaneously propagate the damage radially until the system entirelycollapses, a phenomenon analogous to the nucleation process. These findingssuggest novel features and extreme vulnerabilities of spatially embedded k-corenetwork systems, and highlight the necessity to take into account thecharacteristic length of links when designing robust spatial networks.Furthermore, our insight about the microscopic processes and their originduring the mixed order and first order abrupt transitions in k-core networkscould shed light on the mechanisms of many systems where such transitionsoccur.

Nucleation phenomena and extreme vulnerability of spatial k-core systems