Local structure graph models (LSGMs) describe random graphs and networks as a Markov random field (MRF)—each graph edge has a specified conditional distribution dependent on explicit neighbourhoods of other graph edges. Centred parameterizations of LSGMs allow for direct control and interpretation of parameters for large- and small-scale structures (e.g., marginal means vs. dependence). We extend this parameterization to account for triples of dependent edges and illustrate the importance of centred parameterizations for incorporating covariates and interpreting parameters. Using a MRF framework, common exponential random graph models are also shown to induce conditional distributions without centred parameterizations and thereby have undesirable features. This work attempts to advance graph models through conditional model specifications with modern parameterizations, covariates and higher-order dependencies.

中文翻译：

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具有高阶依赖的局部结构图模型

局部结构图模型 (LSGM) 将随机图和网络描述为马尔可夫随机场 (MRF)——每个图边都有一个特定的条件分布，具体取决于其他图边的显式邻域。LSGM 的中心参数化允许直接控制和解释大型和小型结构的参数（例如，边际均值与相关性）。我们扩展此参数化以解释相关边的三元组，并说明中心参数化对于合并协变量和解释参数的重要性。使用 MRF 框架，还显示常见的指数随机图模型会在没有中心参数化的情况下诱导条件分布，从而具有不良特征。