We extend the latent position random graph model to the line graph of a random graph, which is formed by creating a vertex for each edge in the original random graph, and connecting each pair of edges incident to a common vertex in the original graph. We prove concentration inequalities for the spectrum of a line graph, as well as limiting distribution results for the largest eigenvalue and the empirical spectral distribution in certain settings. For the stochastic blockmodel, we establish that although naive spectral decompositions can fail to extract necessary signal for edge clustering, there exist signal-preserving singular subspaces of the line graph that can be recovered through a carefully-chosen projection. Moreover, we can consistently estimate edge latent positions in a random line graph, even though such graphs are of a random size, typically have high rank, and possess no spectral gap. Our results demonstrate that the line graph of a stochastic block model exhibits underlying block structure, and in simulations, we synthesize and test our methods against several commonly-used techniques, including tensor decompositions, for cluster recovery and edge covariate inference. By naturally incorporating information encoded in both vertices and edges, the random line graph improves network inference.

翻译：我们将潜在位置随机图模型推广至随机图的线图，该线图通过为原始随机图中每条边创建一个顶点，并连接原始图中共享同一顶点的每对边构建而成。我们证明了线图谱的浓度不等式，以及在特定设定下最大特征值和经验谱分布的极限分布结果。对于随机块模型，我们证实：尽管朴素谱分解无法提取边聚类所需的必要信号，但线图中存在可通过精心选择的投影恢复的保信号奇异子空间。此外，即便随机线图具有随机大小、通常高秩且无谱隙，我们仍能一致估计其中的边潜在位置。结果表明随机块模型的线图展现出内在块结构，并通过仿真实验，我们将所提方法与包括张量分解在内的多种常用技术进行对比测试，用于聚类恢复和边协变量推断。通过自然融合顶点与边中编码的信息，随机线图有效提升了网络推理性能。