Graph alignment refers to the task of finding the vertex correspondence
between two positively correlated graphs. Extensive study has been done on
polynomial-time algorithms for the graph alignment problem under the
Erd\H{o}s--R\'enyi graph pair model, where the two graphs are
Erd\H{o}s--R\'enyi graphs with edge probability quβ, correlated
under certain vertex correspondence. To achieve exact recovery of the vertex
correspondence, all existing algorithms at least require the edge correlation
coefficient Οuβ between the two graphs to satisfy
Οuβ>Ξ±β, where Ξ±β0.338 is Otter's
tree-counting constant. Moreover, it is conjectured in [1] that no
polynomial-time algorithm can achieve exact recovery under weak edge
correlation Οuβ<Ξ±β.
In this paper, we show that with a vanishing amount of additional attribute
information, exact recovery is polynomial-time feasible under vanishing edge
correlation Οuββ₯nβΞ(1). We identify a local tree
structure, which incorporates one layer of user information and one layer of
attribute information, and apply the subgraph counting technique to such
structures. A polynomial-time algorithm is proposed that recovers the vertex
correspondence for all but a vanishing fraction of vertices. We then further
refine the algorithm output to achieve exact recovery. The motivation for
considering additional attribute information comes from the widely available
side information in real-world applications, such as the user's birthplace and
educational background on LinkedIn and Twitter social networks