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  1. We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of verticesn. Moreover, we show the number of seeds needed for perfect recovery in polynomial‐time can be as low asin the sparse graph regime (with the average degree smaller than) andin the dense graph regime, for a small positive constant. Unlike previous work on graph matching, which used small neighborhoods or small subgraphs with a logarithmic number of vertices in order to match vertices, our algorithms match vertices if their large neighborhoods have a significant overlap in the number of seeds.

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    This paper studies seeded graph matching for power-law graphs. Assume that two edge-correlated graphs are independently edge-sampled from a common parent graph with a power-law degree distribution. A set of correctly matched vertex-pairs is chosen at random and revealed as initial seeds. Our goal is to use the seeds to recover the remaining latent vertex correspondence between the two graphs. Departing from the existing approaches that focus on the use of high-degree seeds in $1$-hop neighborhoods, we develop an efficient algorithm that exploits the low-degree seeds in suitably-defined D-hop neighborhoods. Specifically, we first match a set of vertex-pairs with appropriate degrees (which we refer to as the first slice) based on the number of low-degree seeds in their D-hop neighborhoods. This approach significantly reduces the number of initial seeds needed to trigger a cascading process to match the rest of graphs. Under the Chung-Lu random graph model with n vertices, max degree Θ(√n), and the power-law exponent 2<β<3, we show that as soon as D> 4-β/3-β, by optimally choosing the first slice, with high probability our algorithm can correctly match a constant fraction of the true pairs without any error, provided with only Ω((log n)4-β) initial seeds. Our result achieves an exponential reduction in the seed size requirement, as the best previously known result requires n1/2+ε seeds (for any small constant ε>0). Performance evaluation with synthetic and real data further corroborates the improved performance of our algorithm. 
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