AI 中文总结
研究稀疏非齐次随机图模型中图对齐何时可行，开发通用框架，利用平衡负载函数给出下界，用自重叠衡量阻碍，证明该标准在多种模型中尖锐，恢复相关现象并给出阈值。
AI 中文摘要
我们通过研究可恢复潜在匹配的顶点集，开发了一个通用框架，用于理解在稀疏非齐次随机图模型中图对齐何时在信息理论上可行。我们的主要定理通过利用Hajek（1990）引入的平衡负载函数给出了该集合的一般下界。相应的阻碍由一个新的图参数——自重叠来捕获，它衡量了图在非平凡重新标记下模仿自身的程度。然后我们表明，在广泛的稀疏非齐次模型中，该标准是尖锐的，恢复了已知的Erdős-Rényi现象，并为Chung-Lu图和随机块模型产生了尖锐阈值。
英文摘要
We develop a general framework for understanding when graph alignment is information-theoretically feasible in sparse inhomogeneous random graph models, by studying the set of vertices on which the underlying matching can be recovered. Our main theorem gives a general lower bound on this set by leveraging the balanced load function introduced by Hajek (1990). The corresponding obstruction is captured by a new graph parameter, the self-overlap, which measures the extent to which a graph can imitate itself under a non-trivial relabelling. We then show that this criterion is sharp in a broad class of sparse inhomogeneous models, recovering known Erdős--Rényi phenomena and yielding sharp thresholds for Chung--Lu graphs and stochastic block models.
Comments31 pages, 1 figure
