Random node reinforcement and $K$-core structure of complex networks

TL;DR

This paper investigates how random node reinforcement influences the mesoscopic $K$-core structure of complex networks, providing an analytical framework to optimize reinforcement strategies for enhanced robustness.

Contribution

It introduces a systematic analytical approach to evaluate the impact of random node reinforcement on $K$-core structures and derives an optimal reinforcement fraction based on a gain function.

Findings

Reinforcement smooths the $K$-core emergence from abrupt to continuous.

A phase diagram for $K$-core behavior under reinforcement is derived.

An optimal reinforcement fraction maximizes the gain function in cost-benefit analysis.

Abstract

To enhance robustness of complex networked systems, a simple method is introducing reinforced nodes which always function during failure propagation. A random scheme of node reinforcement can be considered as a benchmark for finding an optimal reinforcement solution. Yet there still lacks a systematic evaluation on how node reinforcement affects network structure at a mesoscopic level upon failures. Here we study this problem through the lens of $K$-cores of networks. Based on an analytical percolation framework, we first show that, on uncorrelated random graphs, with a critical size of reinforced nodes, an abrupt emergence of $K$-cores is smoothed out to a continuous one, and a detailed phase diagram is derived. We then show that, with a cost-benefit analysis on random reinforcement, for proper weight factors in cost functions with constant and increasing marginal costs, a gain function shows a unimodality, thus we can analytically find an optimal reinforcement fraction by locating the maximal gain. In all, our framework offers a gain-oriented analytical perspective to designing robust interconnected systems.