Network data has attracted growing interest across scientific domains, prompting the development of various network models. Existing network analysis methods mainly focus on unsigned networks, whereas signed networks, consisting of both positive and negative edges, have been frequently encountered in practice but much less investigated. In this paper, we formally define strong and weak balance in signed networks, and propose a signed block $β$-model, which is capable of modeling strong- and weak-balanced signed networks simultaneously. We establish the identifiability of the proposed model by leveraging properties of bipartite graphs, and develop an efficient alternating updating algorithm to optimize the resulting log-likelihood function. More importantly, we establish the asymptotic consistencies of the proposed model in terms of both probability estimation and community detection. Its advantages are also demonstrated through extensive numerical experiments and the application to a real-world international relationship network.

翻译：网络数据在科学领域引起了日益增长的关注，推动了各种网络模型的发展。现有的网络分析方法主要关注无符号网络，而由正边和负边组成的符号网络在实践中频繁出现，但研究却少得多。在本文中，我们正式定义了符号网络中的强平衡与弱平衡，并提出了一种符号区块$β$-模型，该模型能够同时建模强平衡与弱平衡的符号网络。我们利用二分图的特性建立了所提模型的可识别性，并开发了一种高效的交替更新算法来优化所得的对数似然函数。更重要的是，我们建立了所提模型在概率估计和社区检测两方面的渐近一致性。其优势也通过广泛的数值实验和一个现实世界国际关系网络的应用得到了验证。