We propose two methods for the unsupervised detection of communities in undirected multiplex networks. These networks consist of multiple layers that record different relationships between the same entities or incorporate data from different sources. Both methods are formulated as gradient flows of suitable energy functionals: the first (MPBTV) builds on the minimization of a balanced total variation functional, which we show to be equivalent to multiplex modularity maximization, while the second (DGFM3) directly maximizes multiplex modularity. The resulting non-linear matrix-valued ordinary differential equations (ODEs) are solved efficiently by a graph Merriman--Bence--Osher (MBO) scheme. Key to the efficiency is the approximate integration of the discrete linear differential operators by truncated eigendecompositions in the matrix exponential function. Numerical experiments on several real-world multiplex networks show that our methods are competitive with the state of the art with respect to various metrics. Their major benefit is a significant reduction of computational complexity leading to runtimes that are orders of magnitude faster for large multiplex networks.

翻译：本文提出了两种无监督检测无向多层网络社区的方法。多层网络由多个层组成，这些层记录了相同实体间的不同关系或整合了来自不同来源的数据。两种方法均被表述为适当能量泛函的梯度流：第一种方法（MPBTV）基于平衡总变差泛函的最小化，我们证明其等价于多层模块度最大化；而第二种方法（DGFM3）直接最大化多层模块度。通过图Merriman--Bence--Osher（MBO）格式可高效求解由此产生的非线性矩阵值常微分方程（ODEs）。其效率的关键在于通过矩阵指数函数中的截断特征分解来近似积分离散线性微分算子。在多个真实世界多层网络上的数值实验表明，我们的方法在各项指标上均与当前最优技术具有竞争力。其主要优势在于显著降低了计算复杂度，使得处理大型多层网络的运行时间缩短了数个数量级。