Individual-centered partial information in social networks

Most existing statistical network analysis literature assumes a global view of the network, under which community detection, testing, and other statistical procedures are developed. Yet in the real world, people frequently make decisions based on their partial understanding of network information. As individuals barely know beyond friends' friends, we assume that an individual of interest knows all paths of length up to $L=2$ that originate from them. As a result, this individual's perceived adjacency matrix $\bbB$ differs significantly from the usual adjacency matrix $\bbA$ based on the global information. The new individual-centered partial information framework sparks an array of fascinating endeavors from theory to practice. Key general properties on the eigenvalues and eigenvectors of $\bbB_E$, a major term of $\bbB$, are derived. These general results, coupled with the classic stochastic block model, lead to a new theory-backed spectral approach to detecting the community memberships based on an anchored individual's partial information. Real data analysis delivers interesting insights that result from individuals' heterogeneous knowledge, yet these insights cannot be obtained from global network analysis.