  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.
