  Two competing types of interactions often play an important part in shaping
system behavior, such as activatory or inhibitory functions in biological
systems. Hence, signed networks, where each connection can be either positive
or negative, have become popular models over recent years. However, the primary
focus of the literature is on the unweighted and structurally unbalanced ones,
where all cycles have an even number of negative edges. Hence here, we first
introduce a classification of signed networks into balanced, antibalanced or
strictly unbalanced ones, and then characterize each type of signed networks in
terms of the spectral properties of the signed weighted adjacency matrix. In
particular, we show that the spectral radius of the matrix with signs is
smaller than that without if and only if the signed network is strictly
unbalanced. These properties are important to understand the dynamics on signed
networks, both linear and nonlinear ones. Specifically, we find consistent
patterns in a linear and a nonlinear dynamics theoretically, depending on their
type of balance. We also propose two measures to further characterize strictly
unbalanced networks, motivated by perturbation theory. Finally, we numerically
verify these properties through experiments on both synthetic and real
networks.
