Graph representation learning (GRL) has become a prominent tool for furthering the understanding of complex networks providing tools for network embedding, link prediction, and node classification. In this paper, we propose the Hybrid Membership-Latent Distance Model (HM-LDM) by exploring how a Latent Distance Model (LDM) can be constrained to a latent simplex. By controlling the edge lengths of the corners of the simplex, the volume of the latent space can be systematically controlled. Thereby communities are revealed as the space becomes more constrained, with hard memberships being recovered as the simplex volume goes to zero. We further explore a recent likelihood formulation for signed networks utilizing the Skellam distribution to account for signed weighted networks and extend the HM-LDM to the signed Hybrid Membership-Latent Distance Model (sHM-LDM). Importantly, the induced likelihood function explicitly attracts nodes with positive links and deters nodes from having negative interactions. We demonstrate the utility of HM-LDM and sHM-LDM on several real networks. We find that the procedures successfully identify prominent distinct structures, as well as how nodes relate to the extracted aspects providing favorable performances in terms of link prediction when compared to prominent baselines. Furthermore, the learned soft memberships enable easily interpretable network visualizations highlighting distinct patterns.

翻译：图表示学习（GRL）已成为深化复杂网络理解的重要工具，为网络嵌入、链接预测及节点分类提供了方法支撑。本文通过探究如何将潜距离模型（LDM）约束至潜在单纯形，提出混合成员-潜距离模型（HM-LDM）。通过控制单纯形顶点边的长度，可系统性地调控潜在空间的体积。随着空间约束增强，社区结构得以显现；当单纯形体积趋近于零时，硬成员关系得以恢复。我们进一步探索了基于Skellam分布的有符号网络似然函数构建方法，以表征有符号加权网络，并将HM-LDM扩展至有符号混合成员-潜距离模型（sHM-LDM）。关键在于，所构建的似然函数能够显式吸引具有正向链接的节点，同时抑制节点间的负向交互。我们在多个真实网络上验证了HM-LDM与sHM-LDM的有效性。实验表明，该方法不仅能成功识别显著的不同结构，还揭示了节点与提取特征间的关联，在链接预测任务中相较于主流基线模型展现出更优性能。此外，学习得到的软成员关系可生成易于解释的网络可视化结果，突显了网络中的独特模式。