Jim writes:
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For each positive integer n > 0, what is the maximum value of N for which
there exists an n-by-n stochastic matrix A and a probability vector v such
that A^N v is the first forward image of v under repeated multiplication
by A that lies in the open positive orthant (i.e., has all its components
positive)?
>>
For n > 1 couldn't we have A = nxn identity, and v = (1,0,...,0)
so that for no N is A^N v in the strictly positive orthant? Or is this
like the Busy Beaver number -- we're only considering A's & v's
such that A^N v is in the strictly positive orthant for *some* N ?
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This can be recast as a purely combinatorial question. Suppose a finite
directed graph with n vertices has the property that for every pair of
vertices v,w, there is a way to get from v to w in N moves. Define the
"inconvenience" of the digraph as the smallest N that works for all v,w.
How large can the inconvenience of an n-vertex directed graph be? (One
must assume that the digraph has finite inconvenience to begin with.)
. . .
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I am a wee bit puzzled. Suppose we have a digraph with nodes {A,B,C,D}
and arrows AB, BC, AD. Is its inconvenience = 2 ? (But there is no way
to get from A to D in 2 moves, unless a move is allowed to be not moving.)
--Dan
