How universal is the mean-field universality class for percolation in complex networks?

Clustering and degree correlations are ubiquitous in real-world complex networks. Yet, understanding their role in critical phenomena remains a challenge for theoretical studies. Here, we provide the exact solution of site percolation in a model for strongly clustered random graphs, with many overlapping loops and heterogeneous degree distribution. We systematically compare the exact solution with heterogeneous mean-field predictions obtained from a treelike random rewiring of the network, which preserves only the degree sequence. Our results demonstrate a non-trivial interplay between degree heterogeneity, correlations and network topology, which can significantly alter both the percolation threshold and the critical exponents predicted by the heterogeneous mean field. These findings reveal limitations of heterogeneous mean-field theory, demonstrating that the degree distribution alone is insufficient to determine universality classes in complex networks with realistic structural features.