Partial Recovery of Erdős-Rényi Graph Alignment via $k$-Core Alignment
This addresses a graph alignment problem in network analysis, providing theoretical conditions for partial recovery, which is incremental as it builds on prior work on exact alignment.
The paper tackles the problem of partially recovering the alignment between two correlated sparse Erdős-Rényi graphs, showing that recovery for a fraction of vertices tending to one is possible when the average degree of the intersection tends to infinity, with a matching converse bound.
We determine information theoretic conditions under which it is possible to partially recover the alignment used to generate a pair of sparse, correlated Erdős-Rényi graphs. To prove our achievability result, we introduce the $k$-core alignment estimator. This estimator searches for an alignment in which the intersection of the correlated graphs using this alignment has a minimum degree of $k$. We prove a matching converse bound. As the number of vertices grows, recovery of the alignment for a fraction of the vertices tending to one is possible when the average degree of the intersection of the graph pair tends to infinity. It was previously known that exact alignment is possible when this average degree grows faster than the logarithm of the number of vertices.