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：生成既保留特征结构又促进样本多样性的图可能富有挑战，尤其是在图观测数量较少的情况下。本文探讨了仅从单个观测图生成图的问题。基于参数模型的经典图生成方法依赖于参数估计，但由于归一化常数难以处理，该估计可能不一致或计算代价高昂。基于机器学习技术生成高质量图样本的生成式建模虽避免了参数估计，但通常需要大量训练样本。我们提出的生成过程SteinGen，是在指数随机图模型框架下将图视为其实例，结合Stein方法与MCMC的思想，采用基于目标模型Stein算子的马尔可夫动力学。SteinGen使用与估计的Stein算子相关联的Glauber动力学生成样本，并在每次采样步骤后从样本中重新估计Stein算子。我们表明，在一类指数随机图模型上，这种新颖的"估计与重估计"生成策略能产生与原始数据高度分布相似性（高保真度）的同时，兼顾高样本多样性。