Graph alignment - identifying node correspondences between two graphs - is a fundamental problem with applications in network analysis, biology, and privacy research. While substantial progress has been made in aligning correlated Erdős-Rényi graphs under symmetric settings, real-world networks often exhibit asymmetry in both node numbers and edge densities. In this work, we introduce a novel framework for asymmetric correlated Erdős-Rényi graphs, generalizing existing models to account for these asymmetries. We conduct a rigorous theoretical analysis of graph alignment in the sparse regime, where local neighborhoods exhibit tree-like structures. Our approach leverages tree correlation testing as the central tool in our polynomial-time algorithm, MPAlign, which achieves one-sided partial alignment under certain conditions. A key contribution of our work is characterizing these conditions under which asymmetric tree correlation testing is feasible: If two correlated graphs $G$ and $G'$ have average degrees $λs$ and $λs'$ respectively, where $λ$ is their common density and $s,s'$ are marginal correlation parameters, their tree neighborhoods can be aligned if $ss' > α$, where $α$ denotes Otter's constant and $λ$ is supposed large enough. The feasibility of this tree comparison problem undergoes a sharp phase transition since $ss' \leq α$ implies its impossibility. These new results on tree correlation testing allow us to solve a class of random subgraph isomorphism problems, resolving an open problem in the field.

翻译：图对齐——识别两个图之间的节点对应关系——是网络分析、生物学和隐私研究等领域应用中的一个基本问题。尽管在对称设置下对齐相关的Erdős-Rényi图已取得实质性进展，但现实世界网络通常在节点数量和边密度上表现出非对称性。本文中，我们针对非对称相关Erdős-Rényi图提出了一种新颖的框架，将现有模型推广至能够处理这些非对称性的情况。我们对稀疏区域（其中局部邻域呈现树状结构）下的图对齐问题进行了严格的理论分析。我们的方法以树相关性检验为核心工具，并应用于多项式时间算法MPAlign中，该算法在一定条件下实现了单边部分对齐。本研究的一个关键贡献在于刻画了非对称树相关性检验可行的条件：若两个相关图$G$和$G'$的平均度分别为$λs$和$λs'$，其中$λ$为它们的共同密度，$s,s'$为边缘相关参数，则当$ss' > α$且$λ$足够大时（其中$α$表示Otter常数），它们的树邻域可被对齐。由于$ss' \leq α$意味着该问题不可行，这一树比较问题的可行性经历了一个尖锐的相变。这些关于树相关性检验的新结果使我们能够解决一类随机子图同构问题，从而解决了该领域的一个开放性问题。