Analytical Results of k-core Pruning Process on Multi-layer Networks

TL;DR

This paper extends the analytical solution of the k-core pruning process to multi-layer networks using a novel Multicolor Nonbacktracking Expand Branch method, applicable to various correlation structures.

Contribution

It introduces the MNEB method for exact analytical results of k-core pruning in multi-layer networks, including correlated and uncorrelated cases.

Findings

Exact solutions for k-core pruning at each step in multi-layer networks.

Method applies to networks with interlayer and in-layer correlations.

Extends previous NBEB approach to multi-layer scenarios.

Abstract

Multi-layer networks or multiplex networks are generally considered as the networks that have the same set of vertices but different types of edges. Multi-layer networks are especially useful when describing the systems with several kinds of interactions. In this paper we study the analytical solution of $\textbf{k}$-core pruning process on multi-layer networks. $k$-core decomposition is a widely used method to find the dense core of the network. Previously the Nonbacktracking Expand Branch (NBEB) is found to be able to easily derive the exact analytical results in the $k$-core pruning process. Here we further extend this method to solve the $\textbf{k}$-core pruning process on multi-layer networks by designing a variation of the method called Multicolor Nonbacktracking Expand Branch (MNEB). Our results show that, given any initial multi-layer network, Multicolor Nonbacktracking Expand Branch can offer the exact solution for each intermediate state of the pruning process, these results do not only apply to uncorrelated network, but also apply to networks with either interlayer correlations or in-layer correlations.