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This paper considers the average-consensus problem within the same framework as the companion paper [K. Topley and V. Krishnamurthy, “Average-Consensus Algorithms in a Deterministic Framework-Part I: Strong Connectivity, IEEE Trans. Signal Process., vol. 60, no. 12, Dec. 2012]. Two distributed algorithms are proposed and shown to be analogous to the algorithms presented in the Part I of the paper with respect to the communication costs and conditions sufficient for average-consensus. We provide convergence proofs, as well as numerical examples that (i) illustrate the empirical convergence rate of all four algorithms, and (ii) show that consensus algorithms in the past literature can fail to achieve average-consensus within our framework. Three applications from the literature that motivate the proposed algorithms are discussed, and also we show how all four algorithms allow each node to compute an upper bound on the error of their current local consensus estimate.