Percolation in random hypergraphs with anchor nodes

In this paper, we introduce the concept of anchor nodes in hypergraphs, providing a natural framework to unify and transition between the two primary classes of hypergraph percolation processes explored in the literature. The removal of anchor nodes deletes the hyperedges they belong to, whereas removing nonanchor nodes only reduces the size of the hyperedges without affecting their existence. Using a bipartite factor graph representation, we establish a versatile analytical framework based on message-passing theory to determine percolation thresholds, the critical probability of anchor nodes, the percolation critical exponent, and the size of giant clusters and finite clusters. Our analysis reveals that hypergraphs with heterogeneous cardinality distributions exhibit heightened sensitivity to the presence of anchor nodes, compared to those with homogeneous distributions. Intriguing crossover phenomena emerge, governed by the fraction of anchor nodes, underscoring the complex interplay between reinforcing and fragmenting effects of large hyperedges. Extending this framework to multiplex random hypergraphs, we uncover additional crossover behaviors and demonstrate how anchor nodes mediate the robustness of hypergraphs across varying structural properties. In both hypergraph models, we find strikingly that the absence of anchor nodes leads to a second-order phase transition, while any nonzero fraction results in a higher-order phase transition. These findings deepen our understanding of hypergraph percolation, providing a basis for future investigation into their application in real-world scenarios.