A polynomial-time iterative algorithm for random graph matching with non-vanishing correlation

We propose an efficient algorithm for matching two correlated ErdHos--R'enyi graphs with n vertices whose edges are correlated through a latent vertex correspondence. When the edge density q= n- α+o(1) for a constant α ∈ [0,1), we show that our algorithm has polynomial running time and succeeds to recover the latent matching as long as the edge correlation is non-vanishing. This is closely related to our previous work on a polynomial-time algorithm that matches two Gaussian Wigner matrices with non-vanishing correlation, and provides the first polynomial-time random graph matching algorithm (regardless of the regime of q) when the edge correlation is below the square root of the Otter's constant (which is ≈ 0.338).