Resistance Distance and Control Performance for Google bittide Synchronization

We discuss control of bittide distributed systems, which are designed to provide logical synchronization between networked machines by observing data flow rates between adjacent systems at the physical network layer and controlling local reference clock frequencies. We analyze the performance of approximate proportional-integral control of the synchronization mechanism and develop a simple continuous-time model to show the resulting dynamics are stable for any positive choice of gains. We then construct explicit formulae to show that closed-loop performance measured using the L2 norm is a product of two terms, one depending only on resistance distances in the graph, and the other depending only on controller gains.