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We present the theory of the k-core pruning process (progressive removal ofnodes with degree less than k) in uncorrelated random networks. We derive exactequations describing this process and the evolution of the network structure,and solve them numerically and, in the critical regime of the process,analytically. We show that the pruning process exhibits three differentbehaviors depending on whether the mean degree <q> of the initial network isabove, equal to, or below the threshold <q>_c corresponding to the emergence ofthe giant k-core. We find that above the threshold the network relaxesexponentially to the k-core. The system manifests the phenomenon known as"critical slowing down", as the relaxation time diverges when <q> tends to<q>_c. At the threshold, the dynamics become critical characterized by apower-law relaxation (1/t^2). Below the threshold, a long-lasting transientprocess (a "plateau" stage) occurs. This transient process ends with a collapsein which the entire network disappears completely. The duration of the processdiverges when <q> tends to <q>_c. We show that the critical dynamics of thepruning are determined by branching processes of spreading damage. Clusters ofnodes of degree exactly k are the evolving substrate for these branchingprocesses. Our theory completely describes this branching cascade of damage inuncorrelated networks by providing the time dependent distribution function ofbranching. These theoretical results are supported by our simulations of the$k$-core pruning in Erdos-Renyi graphs.

Critical dynamics of the k-core pruning process