  Water distribution networks are hydraulic infrastructures that aim to meet
water demands at their various nodes. Water flows through pipes in the network
create nonlinear dynamics on networks. A desirable feature of water
distribution networks is high resistance to failures and other shocks given to
the system. Such threats would at least transiently change the flow rate in
various pipes, potentially undermining the functionality of the whole water
distribution system. Here we carry out a linear stability analysis for a
nonlinear dynamical system representing the flow rate through pipes that are
interconnected through an arbitrary pipe network with reservoirs and consumer
nodes. We show that the steady state is always locally stable and develop a
method to calculate the eigenvalue that corresponds to the mode that decays the
most slowly towards the equilibrium, which we use as an index for resilience of
the system. We show that the proposed index is positively correlated with the
recovery rate of the pipe network, which was derived from a realistic and
industrially popular simulator. The present analytical framework is expected to
be useful for deploying tools from nonlinear dynamics and network analysis to
designing, resilience managements and scenario testings of water distribution
networks.
