Synergistic contagions are common in natural and social systems, with network topology playing a crucial role in shaping these dynamics. Despite extensive researches, a comprehensive understanding of how specific structural features influence synergistic contagions, especially their phase transitions, remains lacking. This study demonstrates that the spectral approach can serve as a high-precision analytical tool to describe the impact of network structure, particularly degree heterogeneity and degree correlations, on synergistic contagions. By comparing this approach with numerical simulations and conventional analytical approaches, such as the heterogeneous mean-field (HMF) approach, its high accuracy in predicting the critical synergistic strength is illustrated. Furthermore, the analysis based on the spectral approach reveals that the heterogeneity of the principal eigenvector of the network is the key structural factor determining the phase transition of synergistic contagions. As this heterogeneity increases, which can be promoted by increasing degree heterogeneity or degree correlations, the critical synergistic strength required for an explosive phase transition is enhanced.

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• Continuous phase transition
• Degree correlations
• Discontinuous phase transition
• Network phase transitions
• Barabasi-Albert model
• Epidemic spreading
• Preferential attachment models
• SIS model