  When communicating agents form opinions about a set of possible options,
agreement and disagreement are both possible outcomes. Depending on the
context, either can be desirable or undesirable. We show that for nonlinear
opinion dynamics on networks, and a variety of network structures, the spectral
properties of the underlying adjacency matrix fully characterize the occurrence
of either agreement or disagreement. We further show how the corresponding
eigenvector centrality, as well as any symmetry in the network, informs the
resulting patterns of opinion formation and agent sensitivity to input that
triggers opinion cascades.
