When modeling network data using a latent position model, it is typical to assume that the nodes' positions are independently and identically distributed. However, this assumption implies the average node degree grows linearly with the number of nodes, which is inappropriate when the graph is thought to be sparse. We propose an alternative assumption -- that the latent positions are generated according to a Poisson point process -- and show that it is compatible with various levels of sparsity. Unlike other notions of sparse latent position models in the literature, our framework also defines a projective sequence of probability models, thus ensuring consistency of statistical inference across networks of different sizes. We establish conditions for consistent estimation of the latent positions, and compare our results to existing frameworks for modeling sparse networks.

翻译：在利用潜位置模型对网络数据进行建模时，通常假设节点的位置是独立同分布的。然而，这一假设意味着节点的平均度数随节点数量线性增长，这在图被认为稀疏的情况下并不适用。我们提出另一种假设——潜位置是根据泊松点过程生成的——并证明该假设与不同稀疏程度兼容。与文献中其他稀疏潜位置模型的概念不同，我们的框架还定义了一个投影概率模型序列，从而确保不同规模网络之间统计推断的一致性。我们建立了潜位置一致估计的条件，并将我们的结果与现有稀疏网络建模框架进行了比较。