This paper proposes a novel signed $\beta$-model for directed signed network, which is frequently encountered in application domains but largely neglected in literature. The proposed signed $\beta$-model decomposes a directed signed network as the difference of two unsigned networks and embeds each node with two latent factors for in-status and out-status. The presence of negative edges leads to a non-concave log-likelihood, and a one-step estimation algorithm is developed to facilitate parameter estimation, which is efficient both theoretically and computationally. We also develop an inferential procedure for pairwise and multiple node comparisons under the signed $\beta$-model, which fills the void of lacking uncertainty quantification for node ranking. Theoretical results are established for the coverage probability of confidence interval, as well as the false discovery rate (FDR) control for multiple node comparison. The finite sample performance of the signed $\beta$-model is also examined through extensive numerical experiments on both synthetic and real-life networks.

翻译：本文针对有向符号网络提出了一种新颖的有符号$β$-模型。此类网络在应用领域中频繁出现，但在文献中却长期被忽视。所提出的有符号$β$-模型将有向符号网络分解为两个无符号网络的差异，并为每个节点赋予两个潜在因子，分别代表内向地位和外向地位。负边的存在导致对数似然函数非凹，为此开发了一种一步估计算法以促进参数估计，该算法在理论和计算上均具高效性。我们还基于该模型建立了配对节点比较与多重节点比较的推断方法，填补了节点排序不确定性量化的空白。本文建立了置信区间覆盖概率的理论结果，以及多重节点比较中错误发现率（FDR）控制的保证。通过对合成网络和真实网络的大量数值实验，验证了有符号$β$-模型在有限样本下的性能。