Generation of graphs constrained by a specified graph edit distance from a source graph is important in applications such as cheminformatics, network anomaly synthesis, and structured data augmentation. Despite the growing demand for such constrained generative models in areas including molecule design and network perturbation analysis, the neural architectures required to provably generate graphs within a bounded graph edit distance remain largely unexplored. In addition, existing graph generative models are predominantly data-driven and depend heavily on the availability and quality of training data, which may result in generated graphs that do not satisfy the desired edit distance constraints. In this paper, we address these challenges by theoretically characterizing ReLU neural networks capable of generating graphs within a prescribed graph edit distance from a given graph. In particular, we show the existence of constant depth and O(n^2 d) size ReLU networks that deterministically generate graphs within edit distance d from a given input graph with n vertices, eliminating reliance on training data while guaranteeing validity of the generated graphs. Experimental evaluations demonstrate that the proposed network successfully generates valid graphs for instances with up to 1400 vertices and edit distance bounds up to 140, whereas baseline generative models fail to generate graphs with the desired edit distance. These results provide a theoretical foundation for constructing compact generative models with guaranteed validity.

翻译：从源图出发，生成受指定图编辑距离约束的图在化学信息学、网络异常合成和结构化数据增强等应用中具有重要意义。尽管在分子设计和网络扰动分析等领域对此类约束生成模型的需求日益增长，但用于在有限图编辑距离内可证明地生成图的神经架构仍 largely 未得到探索。此外，现有图生成模型主要基于数据驱动，高度依赖训练数据的可用性与质量，可能导致生成的图无法满足期望的编辑距离约束。本文通过理论刻画能够在给定图的指定图编辑距离内生成图的ReLU神经网络，从而应对这些挑战。具体而言，我们证明了存在常数深度和O(n² d)规模的ReLU网络，能够从具有n个顶点的给定输入图确定性地生成编辑距离在d以内的图，在消除对训练数据依赖的同时保证生成图的有效性。实验评估表明，所提网络成功为顶点数多达1400、编辑距离上界达140的实例生成了有效图，而基线生成模型无法生成具有期望编辑距离的图。这些结果为构建具有保证有效性的紧凑生成模型提供了理论基础。