  We introduce a new directed graph structure into the set of alternating sign
matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as
a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave
characterizations of Bruhat order in terms of total nonnegativity (TNN) and
subtraction-free Laurent (SFL) expressions for permutation monomials. With our
directed graph, we extend their idea in two ways: first, from permutations to
alternating sign matrices; second, $q$-analogs (which we name $q$TNN and $q$SFL
properties). %In our discussion, essential sets, introduced by Fulton in a
rather different context, play a key role. As a by-product, we obtain a new
kind of permutation statistic, the signed bigrassmannian statistics, using
Dodgson's condensation on determinants.
