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

This paper introduces a theoretical framework for $(k,n)$ -core percolation on hypergraphs with anchor nodes to model how essential and non-essential node roles affect network robustness, deriving self-consistency equations and phase diagrams that reveal how functional heterogeneity and interaction ranges influence the emergence of giant cores through both continuous and discontinuous transitions.

Original authors: Hoseung Jang, Byungjoon Min, Ginestra Bianconi

Published 2026-08-27

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CC BY 4.0

Original authors: Hoseung Jang, Byungjoon Min, Ginestra Bianconi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Complex systems, from the neurons firing in a brain to the suppliers in a global manufacturing chain, are rarely just simple chains of two. They are webs where groups of elements interact all at once. In the language of network science, these are called hypergraphs. Unlike a standard map of connections where a line joins only two points, a hypergraph allows a single link to bind a whole cluster of nodes together. This structure is essential for understanding how collective systems function, but it also hides a critical vulnerability: not every member of a group is equally important. In many real-world scenarios, a group's survival depends on specific, irreplaceable members. If a key player leaves, the entire group collapses. If a regular member leaves, the group simply shrinks but continues to function. Understanding how these different roles affect the stability of the whole system has long been a challenge for scientists.

A team of researchers has now built a comprehensive framework to explore this exact problem. They developed a model for "percolation," which is the study of how a network holds together or falls apart when parts are removed. Specifically, they looked at a scenario where every member of a group has a chance of being an "anchor." An anchor is a node that is essential to the group's existence; if it fails, the entire group vanishes. Non-anchor nodes are less critical; their failure merely reduces the group's size. The researchers asked a fundamental question: how does the presence of these essential anchors change the point at which a network suddenly collapses? They tested this by simulating the removal of nodes and groups under different rules, tracking how the network's ability to stay connected changed as the number of anchors increased.

The team discovered that the presence of these essential anchors dramatically alters the nature of a network's failure. In a system without anchors, where every member is equally replaceable, the network tends to degrade gradually. As more parts are removed, the giant connected structure shrinks slowly and predictably. However, as the researchers increased the proportion of anchors, the behavior shifted. The network became more fragile, and the transition from a connected state to a collapsed state became abrupt. Instead of a slow decline, the system would reach a tipping point and then suddenly shatter, losing its giant connected component in a single, catastrophic jump. This shift from a smooth, continuous decline to a sudden, discontinuous collapse was particularly evident when the network had specific requirements for how many members a group needed to stay active.

The researchers also explored how the definition of "connected" changes the outcome. In their first scenario, a group is considered active only if enough of its own members are still present and connected to the wider network. In a second, more complex scenario, they allowed a group to remain part of the network even if it had lost some members, provided those members were still connected to other active groups through different pathways. They found that this extended range of connectivity offered some protection, but the fundamental lesson remained: the more the system relied on specific, irreplaceable anchors, the more likely it was to suffer a sudden, total failure rather than a slow erosion. The study confirmed these findings through extensive computer simulations on random networks, showing that the mathematical predictions matched the simulated reality perfectly.

The implications of this work reach far beyond abstract mathematics. In biological systems, a protein complex might stop working entirely if a single catalytic subunit fails, regardless of how many other parts remain. In social groups, a team might lose its ability to function if a key leader departs, even if the rest of the team is intact. In supply chains, the loss of a critical supplier can halt production immediately, while the loss of a minor supplier might only cause a delay. The research suggests that systems with high concentrations of these essential, non-replaceable components are inherently more prone to sudden, catastrophic breakdowns. The more a network depends on these anchors, the less it can tolerate damage, and the more likely it is to fail all at once rather than fading away. This insight provides a new way to understand why some complex systems are surprisingly resilient while others are dangerously fragile, depending entirely on the functional roles of their individual parts.

Technical Summary: (k, n)-core Percolation on Hypergraphs with Anchor Nodes

Problem Statement
Higher-order networks, represented by hypergraphs, model interactions involving more than two nodes. A critical gap in existing literature is the assumption that all nodes within a hyperedge play identical functional roles. In reality, nodes often exhibit functional heterogeneity: some are "essential" (anchors), while others are non-essential. The failure of an essential node completely dismantles the hyperedge, whereas the failure of a non-essential node merely reduces the hyperedge's size. Previous models either treated all nodes as non-essential (Lee et al.) or all as essential (Bianconi and Dorogovtsev). This paper addresses the need for a framework that captures this heterogeneity and explores how it affects the robustness of hypergraphs under different connectivity constraints.

Methodology
The authors formulate a comprehensive theoretical framework for (k, n)-core percolation on hypergraphs with anchor nodes.

Model Definition: In a hypergraph $H=(V, E)$ , each node within a hyperedge is independently designated as an anchor node with probability $\theta$ . If an anchor node fails (is damaged or removed), the hyperedge fails immediately. Non-anchor nodes contribute to the hyperedge's survival only if a minimum number $n$ of nodes remain active. A node remains active if it is undamaged and belongs to at least $k$ active hyperedges.
Pruning Processes: The study analyzes three distinct recursive pruning scenarios:
1. First-Neighbor Pruning: The standard approach where the state of a node or hyperedge depends on the immediate state of its neighbors in the factor graph (bipartite representation of nodes and hyperedges).
2. Second-Neighbor Pruning (Node-based): Only nodes are recursively pruned. A hyperedge remains part of the giant component if its anchor nodes are active and at least one node is active, even if the hyperedge itself has fewer than $n$ active nodes, provided the nodes satisfy the $k$ -core condition via other hyperedges.
3. Second-Neighbor Pruning (Hyperedge-based): Only hyperedges are recursively pruned. Nodes remain part of the core as long as they are attached to the surviving structure, but they become inactive if they belong to fewer than $k$ surviving hyperedges.
Analytical Approach: The authors derive self-consistency equations using the cavity method on locally tree-like random hypergraphs. They define probabilities ( $W$ and $V$ ) for reaching the giant component via links from hyperedges and nodes, respectively. These equations are solved iteratively to find fixed points corresponding to the size of the giant $(k, n)$ -core.
Validation: Analytical predictions are validated against Monte-Carlo simulations on random hypergraphs generated via the configuration model with Poisson degree and shifted Poisson cardinality distributions.

Key Contributions

Unified Framework: The paper introduces a general model parameterized by $\theta$ that interpolates between the $(k, n)$ -hypergraph percolation (where $\theta=0$ , all nodes non-essential) and the factor graph $(k, n)$ -core (where $\theta=1$ , all nodes essential).
Extended Connectivity Analysis: The work extends percolation theory to second-neighbor problems, distinguishing between node-based and hyperedge-based pruning to capture long-range connectivity effects in higher-order networks.
Phase Diagram Characterization: The authors derive the full phase diagrams in the $\langle q \rangle$ - $\langle m \rangle$ plane (average degree vs. average cardinality), identifying regions of continuous and discontinuous transitions.

Results

Transition Nature: The nature of the percolation transition is highly sensitive to both the core thresholds $(k, n)$ $(k, n)$ and the anchor density $\theta$ $θ$ .
- For $(k, n) = (2, 2)$ , the system exhibits a continuous transition at low anchor densities ( $\theta$ ). As $\theta$ increases, the transition shifts to discontinuous (hybrid), separated by a tricritical point.
- For higher core constraints (e.g., $(2, 3)$ , $(3, 2)$ , or $k > 2$ ), the transition is discontinuous regardless of $\theta$ .
Robustness and Fragility: Increasing the anchor density $\theta$ makes the hypergraph more fragile. Higher $\theta$ raises the connectivity threshold required for the emergence of a giant core, expanding the non-percolating region in the phase diagram.
Pruning Differences: The second-neighbor pruning processes reveal that robustness depends not only on structure but also on the range of interactions ensuring connectivity. The phase diagrams for second-neighbor hyperedge pruning show that increasing $\theta$ or connectivity requirements consistently delays the percolation threshold, requiring denser structures for the giant core to emerge.
Agreement: The analytical self-consistency equations show excellent agreement with numerical simulations across all parameter regimes.

Significance and Claims
The paper claims that the functional heterogeneity of nodes (captured by anchor nodes) fundamentally alters the robustness of higher-order networks. Specifically:

Anchor nodes do not merely shift the percolation threshold; they can transform the nature of the degradation from a progressive (continuous) process to an abrupt, catastrophic (discontinuous) collapse.
The interplay between higher-order structure and functional roles is critical for understanding real-world systems, such as biological complexes (where a core subunit failure stops the complex) or supply chains (where essential supplier failure halts production).
The framework provides a necessary tool for assessing the vulnerability of systems where connectivity is ensured by extended ranges of interactions (second-neighbor effects), which standard first-neighbor models cannot fully describe.

The authors conclude that this framework opens avenues for studying heterogeneous anchor probabilities, correlations between anchor roles, and targeted damage scenarios in empirical higher-order networks.

Technical Summary: (k, n)-core Percolation on Hypergraphs with Anchor Nodes

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