Efficient Algorithms for Attributed Graph Alignment With Vanishing Edge Correlation
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Abstract
Graph alignment refers to the task of finding the vertex correspondence between two correlated graphs of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> vertices. Extensive study has been done on polynomial-time algorithms for the graph alignment problem under the Erdős–Rényi graph pair model, where the two graphs are Erdős–Rényi graphs with edge probability <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{\mathrm {u}}$ </tex-math></inline-formula>, correlated under certain vertex correspondence. To achieve exact recovery of the correspondence, all existing algorithms at least require the edge correlation coefficient <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho _{\mathrm {u}}$ </tex-math></inline-formula> between the two graphs to be <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">non-vanishing</i> as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\rightarrow \infty $ </tex-math></inline-formula>. Moreover, it is conjectured that no polynomial-time algorithm can achieve exact recovery under vanishing edge correlation <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\rho _{\mathrm {u}}}\lt 1/\mathrm {polylog}(n)$ </tex-math></inline-formula>. In this paper, we show that with a vanishing amount of additional <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">attribute information</i>, exact recovery is polynomial-time feasible under <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vanishing</i> edge correlation <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\rho _{\mathrm {u}}}\ge n^{-\Theta (1)}$ </tex-math></inline-formula>. We identify a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">local</i> 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 most of the vertices, and then refines the output to achieve exact recovery. The consideration of attribute information is motivated by real-world applications like LinkedIn and Twitter, where user attributes like birthplace and education background can aid alignment.
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