在稀疏和不平衡情形下确定社区数量
Determining the Number of Communities in Sparse and Imbalanced Settings
AI总结：
本文针对稀疏和不平衡网络中社区数量K难以估计的问题，提出一种基于中心化非回溯算子的谱方法，利用最大特征值进行拟合优度检验，有效克服弱信号和度数不平衡挑战。
摘要：
社区结构是网络分析的一个关键方面，已有多种方法被开发用于识别这些社区。然而，一个常见的困难在于确定社区数量K，这一参数在实践中通常需要估计。现有估计K的方法面临两个显著挑战：稀疏网络中存在的弱社区信号，以及社区规模或边密度的不平衡所导致的每个社区期望度数不相等。我们提出一种基于新型网络算子的谱方法，该算子的谱性质能够有效克服上述两个挑战。该算子是非回溯算子的改进版本，由“中心化”邻接矩阵改造而来。在稀疏网络中，其前几个特征值比邻接矩阵的特征值更加集中；同时在不平衡情形下，它们也表现出增强的信号，这一优势归因于中心化步骤。在零模型K=1下，无论是稠密还是超稀疏情形，这一点都得到了理论或数值上的论证。基于最大特征值的拟合优度检验可用于确定社区数量K。
英文摘要：
Community structures represent a crucial aspect of network analysis, and various methods have been developed to identify these communities. However, a common hurdle lies in determining the number of communities K, a parameter that often requires estimation in practice. Existing approaches for estimating K face two notable challenges: the weak community signal present in sparse networks and the imbalance in community sizes or edge densities that result in unequal per-community expected degree. We propose a spectral method based on a novel network operator whose spectral properties effectively overcome both challenges. This operator is a refined version of the non-backtracking operator, adapted from a "centered" adjacency matrix. Its leading eigenvalues are more concentrated than those of the adjacency matrix for sparse networks, while they also demonstrate enhanced signal under imbalance scenarios, a benefit attributed to the centering step. This is justified, either theoretically or numerically, under the null model K = 1, in both dense and ultra-sparse settings. A goodness-of-fit test based on the leading eigenvalue can be applied to determine the number of communities K.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
