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We describe a protocol for the average consensus problem on any fixedundirected graph whose convergence time scales linearly in the total numbernodes $n$. The protocol is completely distributed, with the exception ofrequiring all nodes to know the same upper bound $U$ on the total number ofnodes which is correct within a constant multiplicative factor.  We next discuss applications of this protocol to problems in multi-agentcontrol connected to the consensus problem. In particular, we describeprotocols for formation maintenance and leader-following with convergence timeswhich also scale linearly with the number of nodes.  Finally, we develop a distributed protocol for minimizing an average of(possibly nondifferentiable) convex functions $ (1/n) \sum_{i=1}^nf_i(\theta)$, in the setting where only node $i$ in an undirected, connectedgraph knows the function $f_i(\theta)$. Under the same assumption about allnodes knowing $U$, and additionally assuming that the subgradients of each$f_i(\theta)$ have absolute values upper bounded by some constant $L$ known tothe nodes, we show that after $T$ iterations our protocol has error which is$O(L \sqrt{n/T})$.

Linear Time Average Consensus on Fixed Graphs and Implications for Decentralized Optimization and Multi-Agent Control