Liren Yu

Liren Yu, Jiaming Xu, Xiaojun Lin

There is a growing interest in designing Graph Neural Networks (GNNs) for seeded graph matching, which aims to match two unlabeled graphs using only topological information and a small set of seed nodes. However, most previous GNNs for this task use a semi-supervised approach, which requires a large number of seeds and cannot learn knowledge that is transferable to unseen graphs. In contrast, this paper proposes a new supervised approach that can learn from a training set how to match unseen graphs with only a few seeds. Our SeedGNN architecture incorporates several novel designs, inspired by theoretical studies of seeded graph matching: 1) it can learn to compute and use witness-like information from different hops, in a way that can be generalized to graphs of different sizes; 2) it can use easily-matched node-pairs as new seeds to improve the matching in subsequent layers. We evaluate SeedGNN on synthetic and real-world graphs and demonstrate significant performance improvements over both non-learning and learning algorithms in the existing literature. Furthermore, our experiments confirm that the knowledge learned by SeedGNN from training graphs can be generalized to test graphs of different sizes and categories.

Liren Yu, Jiaming Xu, Xiaojun Lin

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 significantly reduces the number of initial seeds needed to trigger a cascading process to match the rest of the graphs. Under the Chung-Lu random graph model with $n$ vertices, max degree $Θ(\sqrt{n})$, and the power-law exponent $2<β<3$, we show that as soon as $D> \frac{4-β}{3-β}$, 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 $n^{1/2+ε}$ seeds (for any small constant $ε>0$). Performance evaluation with synthetic and real data further corroborates the improved performance of our algorithm.