Exotic phase transitions of k-cores in clustered networks

The giant k-core-maximal connected subgraph of a network where each node has at least k neighbors-is important in the study of phase transitions and in applications of network theory. Unlike Erdos-Renyi graphs and other random networks where k-cores emerge discontinuously for k >= 3, we show that transitive linking (or triadic closure) leads to 3-cores emerging through single or double phase transitions of both discontinuous and continuous nature. We also develop a k-core calculation that includes clustering and provides insights into how high-level connectivity emerges.