We consider the problem of community detection in overlapping weighted networks, where nodes can belong to multiple communities and edge weights can be finite real numbers. To model such complex networks, we propose a general framework - the mixed membership distribution-free (MMDF) model. MMDF has no distribution constraints of edge weights and can be viewed as generalizations of some previous models, including the well-known mixed membership stochastic blockmodels. Especially, overlapping signed networks with latent community structures can also be generated from our model. We use an efficient spectral algorithm with a theoretical guarantee of convergence rate to estimate community memberships under the model. We also propose fuzzy weighted modularity to evaluate the quality of community detection for overlapping weighted networks with positive and negative edge weights. We then provide a method to determine the number of communities for weighted networks by taking advantage of our fuzzy weighted modularity. Numerical simulations and real data applications are carried out to demonstrate the usefulness of our mixed membership distribution-free model and our fuzzy weighted modularity.

翻译：我们研究了重叠加权网络中的社区检测问题，其中节点可同时属于多个社区，边权重可为有限实数。为建模此类复杂网络，我们提出一个通用框架——混合隶属度无分布（MMDF）模型。MMDF对边权重无分布约束，可视为包括著名混合隶属度随机块模型在内的若干前期模型的推广。特别地，具有潜在社区结构的重叠符号网络亦可由此模型生成。我们采用具有理论收敛速率保证的高效谱算法来估计模型下的社区隶属度。同时提出模糊加权模块度，用于评估包含正负边权重的重叠加权网络的社区检测质量。进而利用该模糊加权模块度，提供一种确定加权网络社区数量的方法。通过数值模拟和真实数据应用，验证了混合隶属度无分布模型与模糊加权模块度的实用性。