| Tóm tắt: | 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 balanced 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 balanced 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.
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