Bond percolation on a class of correlated and clustered random graphs

We introduce a formalism for computing bond percolation properties of a class of correlated and clustered random graphs. This class of graphs is a generalization of the Configuration Model where nodes of different types are connected via different types of hyperedges, edges that can link more than 2 nodes. We argue that the multitype approach coupled with the use of clustered hyperedges can reproduce a wide spectrum of complex patterns, and thus enhances our capability to model real complex networks. As an illustration of this claim, we use our formalism to highlight unusual behaviors of the size and composition of the components (small and giant) in a synthetic, albeit realistic, social network.

💡 Research Summary

The paper introduces a novel random‑graph framework called the Multitype Clustered Hypergraph (MCHG) that extends the classic Configuration Model (CM) by allowing nodes to belong to several predefined types and to be linked through hyperedges that can connect more than two nodes. Each type‑hyperedge pair is characterized by a residual degree distribution, and the internal topology of hyperedges (e.g., complete triangles, stars, chains) can be freely chosen, thereby embedding arbitrary levels of clustering and type‑type correlations directly into the network generation process.

To study bond percolation on these graphs, the authors generalize the generating‑function and message‑passing formalism. For a given bond‑occupation probability T, the probability that a hyperedge transmits connectivity is multiplied by T and then propagated independently to all other nodes in that hyperedge. Under the locally tree‑like approximation (valid for large sparse networks), this yields a set of self‑consistent equations that can be written compactly as a matrix‑valued transition operator. The spectral radius of this operator determines the percolation threshold: when the leading eigenvalue exceeds one, a giant component emerges.

The theory provides explicit expressions for (i) the relative size S_i of the giant component contributed by each node type, obtained from the dominant eigenvector, and (ii) the size distribution of finite clusters, which follows a compound Poisson law whose parameters depend on hyperedge size and type‑specific connection probabilities. Importantly, the framework captures asymmetric effects: a small hyperedge that predominantly contains a particular type (e.g., a youth club) can disproportionately influence the emergence and composition of the giant component.

The authors validate the formalism on a synthetic social network that mixes three node attributes (age, occupation, hobby) and hyperedges of size two to four, mimicking real‑world groups such as families, clubs, and work teams. Simulations of bond percolation match the analytical predictions for the critical threshold, giant‑component size, and type composition. Compared with a standard CM, the MCHG predicts a substantially higher percolation threshold, reflecting the dampening effect of strong clustering on global connectivity.

Beyond the specific example, the paper argues that MCHG offers a versatile tool for modeling a wide spectrum of complex systems where clustering and attribute correlations are essential—ranging from epidemiological spreading to infrastructure robustness. By selecting appropriate hyperedge topologies and type‑specific attachment rules, researchers can design networks with desired resilience or controllability properties. In summary, the work provides a rigorous, analytically tractable extension of random‑graph theory that bridges the gap between idealized tree‑like models and the richly clustered, correlated structures observed in real networks.