Supporting information.

Regime switching in a time series is an important and challenging issue in complex financial system analysis. Existing regime models have focused on the features of fluctuations at a single point in financial time series, often neglecting time series nonlinearity and uncertainties from a dynamic perspective. This study proposes a heteroskedastic network combined with a Hidden Markov Model, the ARMA-GARCH model, and a machine learning algorithm to characterize the dynamic process of a fluctuation in a time series which can uncover the hidden structure of a nonlinear time series with uncertainty. The network community structure can be used to detect regime switching and its early warning signals. We select the S&P 500 time series as our sample data. Our findings indicate that the critical switches between regimes can be detected across various typical periods, and we analyze them from the perspective of the fundamentals and trader expectations in financial markets. The evolution features of regime switching and its early warning signals are also analyzed over the entire sample period. In particular, the critical features of early warning signals can be extracted. This study not only expands regime switching research in time series analysis but also provides a strong theoretical basis for early warning of risk in financial markets for policy-makers and market investors.