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The central theme of complex systems research is understanding the emergentmacroscopic properties of a system from the interplay of its microscopicconstituents. Here, we ask what conditions a complex network of microscopicdynamical units has to meet to permit stationary macroscopic dynamics, such asstable equilibria or phase-locked states. We present an analytical approachwhich is based on a graphical notation that allows rewriting Jacobi's signaturecriterion in an interpretable form. The derived conditions pertain totopological structures on all scales, ranging from individual nodes to theinteraction network as a whole. Our approach can be applied to many systems ofsymmetrically coupled units. For the purpose of illustration, we consider theexample of synchronization, specifically the (heterogeneous) Kuramoto model andan adaptive variant. Moreover, we discuss how the graphical notation can beemployed to study isospectrality in Hermitian matrices. The results completeand extend the previous analysis of Do et al. [Phys. Rev. Lett. 108, 194102(2012)].

Signature stability analysis for networks of coupled dynamical systems with Hermitian Jacobian