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：本文针对有符号有向网络提出了一种新颖的带符号$β$模型，这类网络在实际应用领域中频繁出现，但在现有文献中很大程度上被忽视。所提出的带符号$β$模型将带符号有向网络分解为两个无符号网络的差值，并为每个节点嵌入两个潜在因子，分别表示入度状态和出度状态。负边（negative edges）的存在导致对数似然函数呈非凹性，为此开发了一种一步估计算法以促进参数估计，该算法在理论上和计算上均具备高效性。我们还在带符号$β$模型下开发了针对成对和多重节点比较的推断程序，填补了节点排名缺乏不确定性量化的空白。理论结果涵盖了置信区间的覆盖概率以及多重节点比较中错误发现率（False Discovery Rate, FDR）的控制。通过针对合成网络和真实网络的广泛数值实验，我们进一步检验了带符号$β$模型在有限样本下的性能。