State estimation of linear time-invariant (LTI) systems by using a network of distributed observers is studied in this paper. We assume that each observer has access to a local measurement which may be insufficient to provide the observability of the system, but the ensemble of all measurements in the network guarantees the observability. In this condition, the objective is to design a distributed state estimation approach such that, while the observers can exchange their estimated state vectors under a communication network, the estimated state vector of each observer converges to the state vector of the system. We consider a scenario when the communication links may fail and rebuild over time and the communication network does not stay connected constantly. Accordingly, the main contribution of the paper is to propose a distributed approach (with guarantees on the feasibility of the design) such that the state vector of the system is estimated by each observer if the union/joint of communication links in bounded intervals of time makes the network communication graph connected. Moreover, we also consider a scenario when the LTI system is subject to external disturbances and measurement noise. In this case, we derive sufficient conditions on the proposed approach such that if the communication topology stays connected during links failure, a desired
performance to attenuate the effect of external disturbances and measurement noise on estimation errors is guaranteed. Simulation results show the effectiveness of the proposed estimation approach.