It is shown here that given any vector space retrieval model with dot product as its object similarity function, there exists a semantic network retrieval model such that the two models are equivalent under ranked retrieval.
Proof:Let 
, 
, and 
. For each 
,
define 
where 
and 
are defined to be matrices
of all zeros, 
| 
, and 
. If 
, 
where 
completes 
, 
if 
contains 
,
and f is the dot product of two sequences. If there is no 
such that
completes , then 
.
It is shown that 
by proving that for all 
and for 
, 
.
( 
) Let 
such that 
. Since 
, 
.
Since 
is the dot product, 
. By definition of 
, 
.
The ordering of the labels of nodes in the semantic network model may only decrease the
similarity between an object and the retrieval object. The dot product of two vectors
representing objects containing the same set of tokens will have non-zero entries in all
of those dimensions. In the spreading activation model two objects with the same set of
tokens may have labels that list the tokens in different orders. Only the label with the
tokens listed in the same order will allow completion. Hence 
, and 
.
But by construction there is at least one label for 
and
one label 
for 
such that 
, 
.
Thus 
and 
.
Hence 
so that 
,
and 
.
( 
) Let 
such that . By definition of 
, 
where , , 
and completes both
and 
.
Since completes and , it must be the case that
and = .
By the definition of 
this means that 
.
This means that 
so that 
< 
.
Thus 
= 
as claimed. 
