Generating graphs that preserve characteristic structures while promoting sample diversity can be challenging, especially when the number of graph observations is small. Here, we tackle the problem of graph generation from only one observed graph. The classical approach of graph generation from parametric models relies on the estimation of parameters, which can be inconsistent or expensive to compute due to intractable normalisation constants. Generative modelling based on machine learning techniques to generate high-quality graph samples avoids parameter estimation but usually requires abundant training samples. Our proposed generating procedure, SteinGen, which is phrased in the setting of graphs as realisations of exponential random graph models, combines ideas from Stein's method and MCMC by employing Markovian dynamics which are based on a Stein operator for the target model. SteinGen uses the Glauber dynamics associated with an estimated Stein operator to generate a sample, and re-estimates the Stein operator from the sample after every sampling step. We show that on a class of exponential random graph models this novel "estimation and re-estimation" generation strategy yields high distributional similarity (high fidelity) to the original data, combined with high sample diversity.

翻译：摘要：在保留特征结构的同时促进样本多样性的图生成具有挑战性，尤其当观测图的数量较少时。本文探讨仅从单个观测图出发的图生成问题。基于参数化模型的经典图生成方法依赖参数估计，但由于难以计算的归一化常数，这类估计可能不一致或计算成本高昂。基于机器学习技术的生成式建模虽能生成高质量图样本、避免参数估计，但通常需要充足的训练样本。我们提出的生成过程SteinGen将图视为指数随机图模型的实现，融合了斯坦方法（Stein's method）与马尔可夫链蒙特卡洛（MCMC）的思想，采用基于目标模型斯坦算子构建的马尔可夫动力学。SteinGen利用与估计斯坦算子相关的格劳伯动力学（Glauber dynamics）生成样本，并在每次采样步骤后根据样本重新估计斯坦算子。实验表明，在一类指数随机图模型上，这种新颖的“估计-再估计”生成策略既能保持与原始数据的高度分布相似性（高保真度），又能实现高样本多样性。