Aligning Multiple Inhomogeneous Random Graphs: Fundamental Limits of Exact Recovery

Exact Matching from Multiple Graphs

Networks often describe the same underlying objects from different viewpoints: users on social platforms, proteins across species, or maps built by collaborating robots. But when labels on the nodes of the network are hidden or scrambled, how much correlation among the networks is enough to recover the true correspondence? In “Aligning Multiple Inhomogeneous Random Graphs: Fundamental Limits of Exact Recovery,” Taha Ameen and Bruce Hajek study this question for multiple heterogeneous random graphs, where different pairs of nodes may have different connection probabilities. The paper introduces a simple principle: first obtain reliable partial pairwise matchings, then combine them through transitive closure. The authors identify conditions under which this procedure exactly matches all nodes, and show that for homogeneous Erdős–Rényi graphs these conditions are information-theoretically sharp. A key insight is that multiple graphs can succeed even when no pair of graphs alone contains enough information for exact matching. The work also develops related results on k-cores of inhomogeneous random graphs.