We investigate signed networks with community structure with respect to their spectra and their evolution under a dynamical model of structural balance, a prominent theory of signed social networks. The spectrum of the adjacency matrix generated by a stochastic block model with two equal-size communities shows detectability transitions in which the community structure becomes manifest when its signal eigenvalue appears outside the main spectral band. The spectrum also exhibits “sociality” transitions involving the homogeneous structure representing the average tie value. We derive expressions for the eigenvalues associated with the community and homogeneous structure as well as the transition boundaries, all in good agreement with numerical results. Using the stochastically generated networks as initial conditions for a simple model of structural balance dynamics yields three outcome regimes: two hostile factions that correspond with the initial communities, two hostile factions uncorrelated with those communities, and a single harmonious faction of all nodes. The detectability transition predicts the boundary between the assortative and mixed two-faction states and the sociality transition predicts that between the mixed and harmonious states. Our results may yield insight into the dynamics of cooperation and conflict among actors with distinct social identities.

中文翻译：

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签名网络中的社区可检测性和结构平衡动态。

我们在结构平衡的动态模型（一种著名的社交网络理论）下，研究具有社区结构的签名网络的频谱和演化。由具有两个相等大小的社区的随机块模型生成的邻接矩阵的频谱显示出可检测性转换，其中，当其信号特征值出现在主频谱带之外时，社区结构变得明显。该频谱还表现出“社会性”过渡，涉及代表平均联系值的均质结构。我们推导了与社区和同质结构以及过渡边界相关的特征值的表达式，所有这些表达式都与数值结果非常吻合。使用随机生成的网络作为结构平衡动力学简单模型的初始条件会产生三个结果机制：两个与初始社区相对应的敌对派别，两个与这些社区不相关的敌对派别以及所有节点的单个和谐派别。可检测性过渡预测了混合派和混合派两个国家之间的界限，而社会性过渡则预测了混合派和和谐派之间的界限。我们的研究结果可能会深入了解具有独特社会身份的行为者之间的合作与冲突动态。可检测性过渡预测了混合派和混合派两个国家之间的界限，而社会性过渡则预测了混合派和和谐派之间的界限。我们的研究结果可能会深入了解具有独特社会身份的行为者之间的合作与冲突动态。可检测性过渡预测了混合派和混合派两个国家之间的界限，而社会性过渡则预测了混合派和和谐派之间的界限。我们的研究结果可能会深入了解具有独特社会身份的行为者之间的合作与冲突动态。