  This paper studies the effect of boundary value conditions on consensus
networks. Consider a network where some nodes keep their estimates constant
while other nodes average their estimates with that of their neighbors. We
analyze such networks and show that in contrast to standard consensus networks,
the network estimate converges to a general harmonic function on the graph.
Furthermore, the final value depends only on the value at the boundary nodes.
This has important implications in consensus networks -- for example, we show
that consensus networks are extremely sensitive to the existence of a single
malicious node or consistent errors in a single node. We also discuss
applications of this result in social and sensor networks. We investigate the
existence of boundary nodes in human social networks via an experimental study
involving human subjects. Finally, the paper is concluded with the numerical
studies of the boundary value problems in consensus networks.
