Over the last two decades, the Latent Position Model (LPM) has become a prominent tool to obtain model-based visualizations of networks. However, the geometric structure of the LPM is inherently symmetric, in the sense that outgoing and incoming edges are assumed to follow the same statistical distribution. As a consequence, the canonical LPM framework is not ideal for the analysis of directed networks. In addition, edges may be weighted to describe the duration or intensity of a connection. This can lead to disassortative patterns and other motifs that cannot be easily captured by the underlying geometry. To address these limitations, we develop a novel extension of the LPM, called the Mixed Latent Position Cluster Model (MLPCM), which can deal with asymmetry and non-Euclidean patterns, while providing new interpretations of the latent space. We dissect the directed edges of the network by formally disentangling how a node behaves from how it is perceived by others. This leads to a dual representation of a node's profile, identifying its ``overt'' and ``covert'' social positions. In order to efficiently estimate the parameters of our model, we develop a variational Bayes approach to approximate the posterior distribution. Unlike many existing variational frameworks, our algorithm does not require any additional numerical approximations. Model selection is performed by introducing a novel partially integrated complete likelihood criteria, which builds upon the literature on penalized likelihood methods. We demonstrate the accuracy of our proposed methodology using synthetic datasets, and we illustrate its practical utility with an application to a dataset of international arms transfers.

翻译：在过去的二十年中，潜在位置模型已成为获取网络基于模型的可视化的重要工具。然而，潜在位置模型的几何结构本质上是对称的，即出边和入边被假定遵循相同的统计分布。因此，经典的潜在位置模型框架并不适合分析有向网络。此外，边可能被赋予权重以描述连接的持续时间或强度，这可能导致非同类匹配模式及其他无法通过底层几何结构轻易捕捉的图元。为应对这些局限性，我们开发了一种潜在位置模型的新扩展，称为混合潜在位置聚类模型，该模型能够处理非对称性和非欧几里得模式，同时为潜在空间提供新的解释。我们通过正式解耦节点自身行为与其被他人感知的方式，剖析网络的有向边，从而形成节点特征的双重表示，识别其“显性”与“隐性”社会位置。为有效估计模型参数，我们开发了一种变分贝叶斯方法来近似后验分布。与许多现有变分框架不同，我们的算法无需任何额外的数值近似。模型选择通过引入一种新颖的部分积分完全似然准则实现，该准则建立在惩罚似然方法文献的基础上。我们使用合成数据集验证了所提方法的准确性，并通过国际武器转让数据集的实例应用展示了其实用价值。