Critical Dynamics of the k-Core Pruning Process

Abstract

We present the theory of the k-core pruning process (progressive removal of nodes with degree less than k) in uncorrelated random networks. We derive exact equations 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 different behaviors depending on whether the mean degree ⟨q⟩ of the initial network is above, equal to, or below the threshold ⟨q⟩_{c} corresponding to the emergence of the giant k-core. We find that above the threshold the network relaxes exponentially 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 a power-law relaxation (∝1/t^{2}). Below the threshold, a long-lasting transient process (a “plateau” stage) occurs. This transient process ends with a collapse in which the entire network disappears completely. The duration of the process diverges when ⟨q⟩→⟨q⟩_{c}. We show that the critical dynamics of the pruning are determined by branching processes of spreading damage. Clusters of nodes of degree exactly k are the evolving substrate for these branching processes. Our theory completely describes this branching cascade of damage in uncorrelated networks by providing the time-dependent distribution function of branching. These theoretical results are supported by our simulations of the k-core pruning in Erdős-Rényi graphs.