$(k,n)$-core percolation on hypergraphs with anchor nodes

What the math gives to ML

The paper gives a cavity-method characterization of robustness in higher-order networks where hyperedge members have unequal functional importance: anchor failure destroys an entire hyperedge, while non-anchor failure only reduces its effective size. Its transferable asset is a coupled fixed-point recursion for node-to-hyperedge and hyperedge-to-node survival probabilities, with explicit dependence on degree distributions, hyperedge cardinalities, node and hyperedge retention, and anchor probability. This can become a differentiable structural regularizer or inference-time pruning rule for hypergraph neural networks, higher-order attention, or grouped MoE routing, encouraging the active computational graph to retain a giant anchor-aware core rather than merely preserving average degree. The first implementation should use the recursion as a cheap ensemble-level proxy, then compare it against actual message passing and task accuracy after structured sparsification.