The paper introduces a Signed Generalized Random Dot Product Graph (SGRDPG) model, which is a variant of the Generalized Random Dot Product Graph (GRDPG), where, in addition, edges can be positive or negative. The setting is extended to a multiplex version, where all layers have the same collection of nodes and follow the SGRDPG. The only common feature of the layers of the network is that they can be partitioned into groups with common subspace structures, while otherwise matrices of connection probabilities can be all different. The setting above is extremely flexible and includes a variety of existing multiplex network models as its particular cases. The paper fulfills two objectives. First, it shows that keeping signs of the edges in the process of network construction leads to a better precision of estimation and clustering and, hence, is beneficial for tackling real world problems such as, for example, analysis of brain networks. Second, by employing novel algorithms, our paper ensures strongly consistent clustering of layers and high accuracy of subspace estimation. In addition to theoretical guarantees, both of those features are demonstrated using numerical simulations and a real data example.

翻译：本文提出了一种符号化广义随机点积图（SGRDPG）模型，该模型是广义随机点积图（GRDPG）的变体，其中边可带有正负符号。该设置被扩展至多重网络版本，所有层共享相同的节点集合并遵循SGRDPG特性。网络各层唯一的共同特征在于可被划分为具有公共子空间结构的组群，而除此以外的连接概率矩阵可以各不相同。上述设置具有极高的灵活性，可将现有多种多重网络模型作为其特例。本文实现两大目标：首先，证明在网络构建过程中保留边的符号信息能够提升估计与聚类的精度，因而有利于解决实际领域问题（如脑网络分析）；其次，通过采用新型算法，确保了网络层的强相合聚类与子空间估计的高精度。除理论保证外，这两项特性均通过数值仿真及真实数据实例得到验证。