  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.
