Abstract: Based on the network paradigm of complexity, a systematic analysis of the dynamics of the largest stock markets in the world has been carried out in the work. According to the algorithm of the visibility graph, the daily values of stock indices are converted into a network, the spectral and topological properties of which are sensitive to the critical and crisis phenomena of the studied complex systems. It is shown that some of the spectral and topological characteristics can serve as measures of the complexity of the stock market, and their specific behaviour in the pre-crisis period is used as indicators-precursors of crisis phenomena. The influence of globalization processes on the world stock market is taken into account by calculating the interconnection (multiplex) measures of complexity, which modifies in some way, but does not change the fundamentally predictive possibilities of the proposed indicators-precursors.