  We present an extension of the Kuramoto-Sakaguchi model for networks,
deriving the second-order phase approximation for a paradigmatic model of
oscillatory networks - an ensemble of non-identical Stuart-Landau oscillators
coupled pairwisely via an arbitrary coupling matrix. We explicitly demonstrate
how this matrix translates into the coupling structure in the phase equations.
To illustrate the power of our approach and the crucial importance of
high-order phase reduction, we tackle a trendy setup of non-locally coupled
oscillators exhibiting a chimera state. We reveal that our second-order phase
model reproduces the dependence of the chimera shape on the coupling strength
that is not captured by the typically used first-order Kuramoto-like model. Our
derivation contributes to a better understanding of complex networks' dynamics,
establishing a relation between the coupling matrix and multi-body interaction
terms in the high-order phase model.
