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

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. • We define a K -core percolation model with reinforced nodes on a network. • We develop a theory for the model on random graphs with random node reinforcement. • We analyze hybrid-to-continuous transition behaviors in the model. • We locate optimal fractions for random reinforcement with a cost–benefit analysis. [ABSTRACT FROM AUTHOR]