An arXiv preprint reports that the modeled share of anchor links is associated with whether a hypergraph core emerges gradually or in a sudden jump. In its (2, 2) example, an anchor probability of 0.2 showed continuous emergence, while 0.9 showed a discontinuous jump at a larger threshold. Here, continuous means a smooth rise, while discontinuous means a sudden change as the modeled anchor level rises.
The paper asks whether that transition pattern changes when the pruning rule looks beyond first neighbors. It examines first-neighbor pruning and second-neighbor pruning applied to nodes or hyperedges, linking the comparison to broader questions about hypergraph-core robustness and phase diagrams.
How the model defines an anchor
The framework represents the system as a bipartite factor graph, separating nodes from the hyperedges that connect them. Anchor status is assigned independently to each hyperedge-node link with probability theta. If an anchor node fails, its associated hyperedge is removed automatically. Non-anchor nodes instead have to meet a minimum connectivity requirement.
The study analyzes three pruning processes: first-neighbor pruning, second-neighbor pruning of nodes and second-neighbor pruning of hyperedges. The distinction is important because node and hyperedge pruning do not feed the core equations in exactly the same way.
To locate transition points, the analysis uses self-consistency equations, reduces them to a fixed-point equation and applies tangency conditions. It then compares the analytical predictions with Monte Carlo simulations.
A random test bed for the equations
The numerical work uses random hypergraphs built with the configuration model. The simulated degree distribution is Poisson, while hyperedge cardinality follows a shifted Poisson distribution. This is the model setting used for the reported phase diagrams and numerical checks.
The simulation caption reports N = 105 as printed, along with a mean degree of 7.6 and a mean cardinality of 3.8. Across the reported parameter regimes, the analytical predictions were in excellent agreement with the Monte Carlo simulations.
That agreement checks the equations against the simulations, but it does not turn the exercise into an observation of a real hypergraph. Both the predictions and the numerical comparison are made within the random-hypergraph setup described by the study.
The tipping point depends on the thresholds
For (k, n) = (2, 3) and (3, 2), the transitions were discontinuous regardless of theta. The abrupt behavior was therefore not confined to the high-anchor example used for (2, 2); it also appeared in the two other threshold combinations examined.
The phase diagrams showed a larger non-percolating region, where a giant core does not emerge, as n and theta increased. Higher anchor densities also corresponded to a higher connectivity threshold for the emergence of the (k, n)-core. Stronger connectivity requirements and more anchor-heavy conditions were paired with a denser hypergraph requirement.
The framework also treats the limiting anchor cases explicitly. At theta = 0, there are no anchor nodes; at theta = 1, all nodes are anchors. These limits give the model clear endpoints for comparison.
Looking one step farther changes the bookkeeping
Second-neighbor node pruning leaves the local self-consistency equations unchanged from first-neighbor pruning. Hyperedge pruning is different: it requires a modified hyperedge order parameter because inactive hyperedges can still contribute to the core. The model therefore separates the effect of pruning range from the bookkeeping needed for the object being pruned.
The transition pattern survives the change in pruning range. Under second-neighbor pruning, the (2, 2) case shifts from continuous to discontinuous as theta increases, while (2, 3) and (3, 2) remain discontinuous for all theta.
For that comparison, the analytical and numerical tests used random Poisson hypergraphs with mean degree 7.6 and mean cardinality 3.8.
What remains open
The central result is conditional on the model's assumptions. Anchor links are assigned independently with a common probability theta, and the simulated hypergraphs use the stated random degree and cardinality distributions. Those choices define the reported behavior within the framework, but do not establish that empirical hypergraphs will show the same transitions.
The evidence is therefore limited to equations, phase-diagram analysis and simulations of configuration-model random hypergraphs. The study does not compare its predictions with observed data or establish the same phase changes in a specific real system.
The document is an arXiv version 1 preprint dated 26 Aug 2026.
The research received support from two National Research Foundation of Korea programs, including an MSIT-funded grant No. RS-2025-25433094 and a LAMP Program grant No. RS-2024-00445180.
Paper data and sources
Original title: $(k,n)$-core percolation on hypergraphs with anchor nodes
Authors: Hoseung Jang, Byungjoon Min, Ginestra Bianconi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text