Performing distributed consensus in a network has been an important research problem for several years, and is directly applicable to sensor networks, autonomous vehicle formation, etc. While there exists a wide variety of algorithms that can be proven to asymptotically reach consensus, in applications involving time-varying parameters and tracking, it is often crucial to reach consensus "as quickly as possible". In [?] it has been shown that, with global knowledge of the network topology, it is possible to optimize the convergence time in distributed averaging algorithms via solving a semi-definite program (SDP) to obtain the optimal averaging weights. Unfortunately, in most applications, nodes do not have knowledge of the full network topology and cannot implement the required SDP in a distributed fashion. In this paper, we present a symmetric adaptive weight algorithm for distributed consensus averaging on bi-directional noiseless networks. The algorithm uses an LMS (Least Mean Squares) approach to adaptively update the edge weights used to calculate each node's values. The derivation shows that global error can be minimized in a distributed fashion and that the resulting adaptive weights are symmetric-symmetry being critical for convergence to the true average. Simulations show that convergence time is nearly equal to that of a non-symmetric adaptive algorithm developed in [?], and significantly better than that of the non-adaptive Metropolis-Hastings algorithm. Most importantly, our symmetric adaptive algorithm converges to the sample mean, whereas the method of [?] converges to an arbitrary value and results in significant error.
