In statistical network analysis, we often assume either the full network is available or multiple subgraphs can be sampled to estimate various global properties of the network. However, in a real social network, people frequently make decisions based on their local view of the network alone. Here, we consider a partial information framework that characterizes the local network centered at a given individual by path length $L$ and gives rise to a partial adjacency matrix. Under $L=2$, we focus on the problem of (global) community detection using the popular stochastic block model (SBM) and its degree-corrected variant (DCSBM). We derive theoretical properties of the eigenvalues and eigenvectors from the signal term of the partial adjacency matrix and propose new spectral-based community detection algorithms that achieve consistency under appropriate conditions. Our analysis also allows us to propose a new centrality measure that assesses the importance of an individual's partial information in determining global community structure. Using simulated and real networks, we demonstrate the performance of our algorithms and compare our centrality measure with other popular alternatives to show it captures unique nodal information. Our results illustrate that the partial information framework enables us to compare the viewpoints of different individuals regarding the global structure.

翻译：在统计网络分析中，我们通常假设整个网络可用或多个子图可被采样以估计网络的各种全局性质。然而，在真实社交网络中，人们往往仅基于其局部网络视图做出决策。本文提出一种部分信息框架，该框架通过路径长度$L$刻画以给定个体为中心的网络局部视图，并生成部分邻接矩阵。在$L=2$的条件下，我们重点研究使用流行的随机块模型（SBM）及其度校正变体（DCSBM）进行（全局）社区检测的问题。我们推导了部分邻接矩阵信号项特征值与特征向量的理论性质，并提出新的基于谱的社区检测算法，该算法在适当条件下能达到一致性。我们的分析还使我们能够提出一种新的中心性度量，用于评估个体部分信息在决定全局社区结构中的重要性。通过模拟和真实网络，我们展示了算法的性能，并将我们的中心性度量与其他常用方法进行比较，以证明其能捕捉独特的节点信息。我们的结果表明，部分信息框架使我们能够比较不同个体对全局结构的观点。