We develop an operator-description for the linear stability in finite networks of Kuramoto oscillators on circulant graphs. This mathematical approach offers analytical predictions for the linear stability of $q$-states, which include phase synchronization $(q=0)$ and phase-locked states with different spatial frequencies $\left(\right|q|>0)$. This approach seamlessly incorporates the presence of time delays (represented by phase lags in the coupling). With this, we are able to determine the specific combination of connectivity and time delays (phase lags) that leads to any given $q$-state to be linearly stable. We apply our framework to a variety of networks, including k-ring graphs, distance-dependent graphs, and random circulant graphs. This approach offers a geometric perspective of linear stability in finite networks in terms of the connectivity and delays (phase lag), and it opens a path to designing and controlling the spatiotemporal dynamics of individual and finite oscillator networks.

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• Collective dynamics
• Coupled oscillators
• Dynamics of networks
• Synchronization