Group Sparse β-Model for Network

Sparsity, homogeneity and heterogeneity are three important characteristics of many real-life networks. The recently proposed Sparse β-Model divides nodes into core ones and peripheral ones to accommodate sparsity, but the parameters of core nodes are assumed to be of similar magnitude, which may not be in line with applications. In this paper, we propose the Group Sparse β-Model that splits the core nodes into groups and assumes different orders of magnitude of parameters in different groups, accounting for the heterogeneity among core nodes. When the groups are known, we provide consistent and asymptotically normal moment estimators of the parameters that control the global and local density. Based on that, consistency and asymptotic normality of the maximum likelihood estimators of the remaining parameters are derived. We also establish finite-sample error bounds results. When the groups are unknown, a ratio method is proposed to detect groups, which is computationally efficient. Simulations show competitive results and the analysis of a corporate inter-relationships network illustrates the usefulness of the proposed model.

稀疏性、同质性与异质性是诸多现实网络的三大重要特征。近期提出的Sparse β-Model将网络节点划分为核心节点与外围节点以适配稀疏性，但该模型假设核心节点的参数具有相近的量级，这一点可能与实际应用场景不符。本文提出Group Sparse β-Model，将核心节点进一步划分为多个群组，并假设不同群组内的参数具有不同的量级，以此刻画核心节点之间的异质性。当群组结构已知时，本文给出了控制全局与局部密度的参数的相合渐近正态矩估计量。在此基础上，推导得到剩余参数的极大似然估计量的相合性与渐近正态性。本文还建立了有限样本误差界相关结论。当群组结构未知时，本文提出一种比值法进行群组检测，该方法具有较高的计算效率。仿真实验展现出具有竞争力的实验结果，而对企业间关联网络的案例分析则验证了所提模型的实际应用价值。