An Analytical Solution to the $k$-core Pruning Process

TL;DR

This paper derives exact analytical solutions for the $k$-core pruning process in large uncorrelated networks, providing insights into its dynamics and resulting structural properties.

Contribution

It simplifies the theoretical framework of $k$-core pruning to a recursive form and solves it analytically for uncorrelated networks, revealing key statistical properties.

Findings

Exact solutions for degree distribution at each pruning step

Analytical expressions for the size of the remaining subgraph

Resolution of the $k$-core pruning dynamics puzzle

Abstract

$k$-core decomposition is widely used to identify the center of a large network, it is a pruning process in which the nodes with degrees less than $k$ are recursively removed. Although the simplicity and effectiveness of this method facilitate its implementation on broad applications across many scientific fields, it produces few analytical results. We here simplify the existing theoretical framework to a simple iterative relationship and obtain the exact analytical solutions of the $k$-core pruning process on large uncorrelated networks. From these solutions we obtain such statistical properties as the degree distribution and the size of the remaining subgraph in each of the pruning steps. Our theoretical results resolve the long-lasting puzzle of the $k$-core pruning dynamics and provide an intuitive description of the dynamic process.