It is now shown that given any semantic network retrieval model, there exists a vector space retrieval model such that the two models are equivalent under ranked retrieval.
Proof:The idea is to have retrieve objects from the vector space
whose dimensions are the elements of the
-closure. Let
and
. Let the
-closure be
. Let
. Define
>, where
is
1 if
, and is 0 otherwise. Let
and
. Then
. Finally, put
.
and
are equivalent under ranked retrieval
because their similarity functions are the same. This is true since
iff both
and
are defined along the
dimension. So by definition
where
completes
. So for every
,
.
It follows that
as claimed.