  How can we tell when accounts are fake or real in a social network? And how
can we tell which accounts belong to liberal, conservative or centrist users?
Often, we can answer such questions and label nodes in a network based on the
labels of their neighbors and appropriate assumptions of homophily ("birds of a
feather flock together") or heterophily ("opposites attract"). One of the most
widely used methods for this kind of inference is Belief Propagation (BP) which
iteratively propagates the information from a few nodes with explicit labels
throughout a network until convergence. One main problem with BP, however, is
that there are no known exact guarantees of convergence in graphs with loops.
  This paper introduces Linearized Belief Propagation (LinBP), a linearization
of BP that allows a closed-form solution via intuitive matrix equations and,
thus, comes with convergence guarantees. It handles homophily, heterophily, and
more general cases that arise in multi-class settings. Plus, it allows a
compact implementation in SQL. The paper also introduces Single-pass Belief
Propagation (SBP), a "localized" version of LinBP that propagates information
across every edge at most once and for which the final class assignments depend
only on the nearest labeled neighbors. In addition, SBP allows fast incremental
updates in dynamic networks. Our runtime experiments show that LinBP and SBP
are orders of magnitude faster than standard
