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. 
