Kevin Topley

Kevin Topley, Vikram Krishnamurthy

We consider the average-consensus problem in a multi-node network of finite size. Communication between nodes is modeled by a sequence of directed signals with arbitrary communication delays. Four distributed algorithms that achieve average-consensus are proposed. Necessary and sufficient communication conditions are given for each algorithm to achieve average-consensus. Resource costs for each algorithm are derived based on the number of scalar values that are required for communication and storage at each node. Numerical examples are provided to illustrate the empirical convergence rate of the four algorithms in comparison with a well-known "gossip" algorithm as well as a randomized information spreading algorithm when assuming a fully connected random graph with instantaneous communication.

Kevin Topley

Given a network of fixed size $n$ and an initial distribution of data, we derive sufficient connectivity conditions on a sequence of time-varying digraphs for (a) data collection and (b) data dissemination, within at most $(n-1)$ iterations. The former is shown to enable distributed computation of the network size $n$, while the latter does not. Knowledge of $n$ subsequently enables each node to acknowledge the earliest time point at which they can cease communication, specifically we find the number of redundant signals can be truncated at the finite time $n$. Using a probabilistic approach, we obtain tight upper and lower bounds for the expected time until the $\textit{last}$ node obtains the entire collection of data, in other words complete data dissemination. Similarly tight upper and lower bounds are also found for the expected time until the $\textit{first}$ node obtains the entire collection of data. Interestingly, these bounds are both $Θ(\text{log}_2(n))$ and in fact differ by only two iterations. Numerical results are explored and verify each result.