# Tight Bounds for Consensus Systems Convergence

Research paper by **Pierre-Yves Chevalier, Julien M. Hendrickx, Raphaël M. Jungers**

Indexed on: **19 Jan '16**Published on: **19 Jan '16**Published in: **Computer Science - Discrete Mathematics**

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#### Abstract

We analyze the asymptotic convergence of all infinite products of matrices
taken in a given finite set, by looking only at finite or periodic products. It
is known that when the matrices of the set have a common nonincreasing
polyhedral norm, all infinite products converge to zero if and only if all
infinite periodic products with period smaller than a certain value converge to
zero, and bounds exist on that value.
We provide a stronger bound holding for both polyhedral norms and polyhedral
seminorms. In the latter case, the matrix products do not necessarily converge
to 0, but all trajectories of the associated system converge to a common
invariant space. We prove our bound to be tight, in the sense that for any
polyhedral seminorm, there is a set of matrices such that not all infinite
products converge, but every periodic product with period smaller than our
bound does converge.
Our technique is based on an analysis of the combinatorial structure of the
face lattice of the unit ball of the nonincreasing seminorm. The bound we
obtain is equal to half the size of the largest antichain in this lattice.
Explicitly evaluating this quantity may be challenging in some cases. We
therefore link our problem with the Sperner property: the property that, for
some graded posets, -- in this case the face lattice of the unit ball -- the
size of the largest antichain is equal to the size of the largest rank level.
We show that some sets of matrices with invariant polyhedral seminorms lead
to posets that do not have that Sperner property. However, this property holds
for the polyhedron obtained when treating sets of stochastic matrices, and our
bound can then be easily evaluated in that case. In particular, we show that
for the dimension of the space $n \geq 8$, our bound is smaller than the
previously known bound by a multiplicative factor of $\frac{3}{2 \sqrt{\pi
n}}$.