多路签名网络中$Z_2$同步和社区发现
Synchronization over $Z_2$ and community detection in multiplex signed networks with constraints
摘要：
从成对比的噪声测量中找到群元素也被称为群同步问题，首先在平面旋转的群SO（2）的背景下引入，其有用性最近在工程和结构生物学中得到了证明。在这里，我们专注于同步在$\mathbb{Z}_2$，并考虑在一个多路复用网络中识别社区的问题时，节点之间的相互作用所描述的签署（可能加权）的相似性的措施，当多路复用网络有一个自然的分区成两个社区。当一个有额外的信息，某些子集的节点表示相同的未知组元素，我们考虑和比较几种算法的基础上频谱，半定规划（SDP）和消息传递算法。换句话说，这样的子集内的所有节点表示相同的未知群元素，并且在属于不同的非重叠子集的节点对之间具有可用的噪声成对测量。继最近对SO（2）上同步的特征向量方法的分析之后，我们利用随机矩阵理论文献的结果，分析了当成对测量的基础图是Erdens-Rényi随机图时，$\mathbb{Z}_2$上同步的特征向量方法对噪声的鲁棒性.我们还提出了一个消息传递同步算法，优于现有的特征向量同步算法，只为某些类的图形和噪声模型，并享有的灵活性，将其他频谱或基于SDP的方法可能不容易容纳的约束。我们将我们的方法应用于几个合成模型和一个真实的数据集的唱名投票模式在美国国会跨时间，以确定两个现有的社区，即，民主党和共和党。
英文摘要：
Finding group elements from noisy measurements of their pairwise ratios is also known as the group synchronization problem, first introduced in the context of the group SO(2) of planar rotations, whose usefulness has been demonstrated recently in engineering and structural biology. Here, we focus on synchronization over $\mathbb{Z}_2$, and consider the problem of identifying communities in a multiplex network when the interaction between the nodes is described by a signed (possibly weighted) measure of similarity, and when the multiplex network has a natural partition into two communities. When one has the additional information that certain subsets of nodes represent the same unknown group element, we consider and compare several algorithms based on spectral, semidefinite programming (SDP) and message passing algorithms. In other words, all nodes within such a subset represent the same unknown group element, and one has available noisy pairwise measurements between pairs of nodes that belong to different non-overlapping subsets. Following a recent analysis of the eigenvector method for synchronization over SO(2), we analyze the robustness to noise of the eigenvector method for synchronization over $\mathbb{Z}_2$, when the underlying graph of pairwise measurements is the Erdős-Rényi random graph, using results from the random matrix theory literature. We also propose a message passing synchronization algorithm that outperforms the existing eigenvector synchronization algorithm only for certain classes of graphs and noise models, and enjoys the flexibility of incorporating constraints that may not be easily accommodated by other spectral or SDP-based methods. We apply our methods both to several synthetic models and a real data set of roll call voting patterns in the U.S. Congress across time, to identify the two existing communities, i.e., the Democratic and Republican parties.
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