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> tends to <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 Erdos-Renyi graphs.
Highlights
Pruning algorithms for networks provide an effective way to extract subgraphs distinguished by their structural properties, connectivity, robustness against failures and damage, and other features [1,2,3,4,5,6]
We show that the pruning process exhibits three different behaviors depending on whether the mean degree hqi of the initial network is above, equal to, or below the threshold hqic corresponding to the emergence of the giant k-core
We have developed the theory of the k-core pruning process in uncorrelated, sparse random networks with an arbitrary degree distribution
Summary
Pruning algorithms for networks provide an effective way to extract subgraphs distinguished by their structural properties, connectivity, robustness against failures and damage, and other features [1,2,3,4,5,6]. The standard algorithm for finding the k-core of a network employs the following pruning process: at each step remove all nodes of degree less than k. We show that near the threshold value of the mean degree hqic, corresponding to the emergence of the giant k-core, this cascade of removals of nodes is a branching process with the mean branching coefficient close to 1. Our theory describes this process completely providing the full time-dependent distribution of branching from the beginning until the end of the pruning. A relationship with dynamical systems close to a saddle point bifurcation and details of calculations are given in the appendixes
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