  Many real-world systems of coupled agents exhibit directed interactions,
meaning that the influence of an agent on another is not reciprocal.
Furthermore, interactions usually do not have identical amplitude and/or sign.
To describe synchronization phenomena in such systems, we use a generalized
Kuramoto model with oriented, weighted and signed interactions. Taking a
bottom-up approach, we investigate the simplest possible oriented networks,
namely acyclic oriented networks and oriented cycles. These two types of
networks are fundamental building blocks from which many general oriented
networks can be constructed. For acyclic, weighted and signed networks, we are
able to completely characterize synchronization properties through necessary
and sufficient conditions, which we show are optimal. Additionally, we prove
that if it exists, a stable synchronous state is unique. In oriented, weighted
and signed cycles with identical natural frequencies, we show that the system
globally synchronizes and that the number of stable synchronous states is
finite.
