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We discuss the possibility of reaching consensus in finite time using onlylinear iterations, with the additional restrictions that the update matricesmust be stochastic with positive diagonals and consistent with a given graphstructure. We show that finite-time average consensus can always be achievedfor connected undirected graphs. For directed graphs, we show some necessaryconditions for finite-time consensus, including strong connectivity and thepresence of a simple cycle of even length.

Finite-time consensus using stochastic matrices with positive diagonals