<s>
SimRank	B-Algorithm
is	O
a	O
general	O
similarity	O
measure	O
,	O
based	O
on	O
a	O
simple	O
and	O
intuitive	O
graph-theoretic	O
model	O
.	O
</s>
<s>
SimRank	B-Algorithm
is	O
applicable	O
in	O
any	O
domain	B-Application
with	O
object-to-object	O
relationships	B-Algorithm
,	O
that	O
measures	O
similarity	O
of	O
the	O
structural	O
context	O
in	O
which	O
objects	O
occur	O
,	O
based	O
on	O
their	O
relationships	B-Algorithm
with	O
other	O
objects	O
.	O
</s>
<s>
Effectively	O
,	O
SimRank	B-Algorithm
is	O
a	O
measure	O
that	O
says	O
"	O
two	O
objects	O
are	O
considered	O
to	O
be	O
similar	O
if	O
they	O
are	O
referenced	O
by	O
similar	O
objects.	O
"	O
</s>
<s>
Although	O
SimRank	B-Algorithm
is	O
widely	O
adopted	O
,	O
it	O
may	O
output	O
unreasonable	O
similarity	O
scores	O
which	O
are	O
influenced	O
by	O
different	O
factors	O
,	O
and	O
can	O
be	O
solved	O
in	O
several	O
ways	O
,	O
such	O
as	O
introducing	O
an	O
evidence	O
weight	O
factor	O
,	O
inserting	O
additional	O
terms	O
that	O
are	O
neglected	O
by	O
SimRank	B-Algorithm
or	O
using	O
PageRank-based	O
alternatives	O
.	O
</s>
<s>
Many	O
applications	B-Application
require	O
a	O
measure	O
of	O
"	O
similarity	O
"	O
between	O
objects	O
.	O
</s>
<s>
More	O
generally	O
,	O
a	O
similarity	O
measure	O
can	O
be	O
used	O
to	O
cluster	B-Algorithm
objects	I-Algorithm
,	O
such	O
as	O
for	O
collaborative	B-Algorithm
filtering	I-Algorithm
in	O
a	O
recommender	B-Application
system	I-Application
,	O
in	O
which	O
“	O
similar	O
”	O
users	O
and	O
items	O
are	O
grouped	O
based	O
on	O
the	O
users’	O
preferences	O
.	O
</s>
<s>
Various	O
aspects	O
of	O
objects	O
can	O
be	O
used	O
to	O
determine	O
similarity	O
,	O
usually	O
depending	O
on	O
the	O
domain	B-Application
and	O
the	O
appropriate	O
definition	O
of	O
similarity	O
for	O
that	O
domain	B-Application
.	O
</s>
<s>
In	O
a	O
document	O
corpus	O
,	O
matching	O
text	O
may	O
be	O
used	O
,	O
and	O
for	O
collaborative	B-Algorithm
filtering	I-Algorithm
,	O
similar	O
users	O
may	O
be	O
identified	O
by	O
common	O
preferences	O
.	O
</s>
<s>
SimRank	B-Algorithm
is	O
a	O
general	O
approach	O
that	O
exploits	O
the	O
object-to-object	O
relationships	B-Algorithm
found	O
in	O
many	O
domains	O
of	O
interest	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
recommender	B-Application
systems	I-Application
,	O
a	O
user	O
’s	O
preference	O
for	O
an	O
item	O
constitutes	O
a	O
relationship	O
between	O
the	O
user	O
and	O
the	O
item	O
.	O
</s>
<s>
Such	O
domains	O
are	O
naturally	O
modeled	O
as	O
graphs	O
,	O
with	O
nodes	O
representing	O
objects	O
and	O
edges	O
representing	O
relationships	B-Algorithm
.	O
</s>
<s>
The	O
intuition	O
behind	O
the	O
SimRank	B-Algorithm
algorithm	O
is	O
that	O
,	O
in	O
many	O
domains	O
,	O
similar	O
objects	O
are	O
referenced	O
by	O
similar	O
objects	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
note	O
that	O
SimRank	B-Algorithm
is	O
a	O
general	O
algorithm	O
that	O
determines	O
only	O
the	O
similarity	O
of	O
structural	O
context	O
.	O
</s>
<s>
SimRank	B-Algorithm
applies	O
to	O
any	O
domain	B-Application
where	O
there	O
are	O
enough	O
relevant	O
relationships	B-Algorithm
between	O
objects	O
to	O
base	O
at	O
least	O
some	O
notion	O
of	O
similarity	O
on	O
relationships	B-Algorithm
.	O
</s>
<s>
Obviously	O
,	O
similarity	O
of	O
other	O
domain-specific	O
aspects	O
are	O
important	O
as	O
well	O
;	O
these	O
can	O
—	O
and	O
should	O
be	O
combined	O
with	O
relational	O
structural-context	O
similarity	O
for	O
an	O
overall	O
similarity	O
measure	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
Web	O
pages	O
SimRank	B-Algorithm
can	O
be	O
combined	O
with	O
traditional	O
textual	O
similarity	O
;	O
the	O
same	O
idea	O
applies	O
to	O
scientific	O
papers	O
or	O
other	O
document	O
corpora	O
.	O
</s>
<s>
For	O
recommendation	B-Application
systems	I-Application
,	O
there	O
may	O
be	O
built-in	O
known	O
similarities	O
between	O
items	O
(	O
e.g.	O
,	O
both	O
computers	O
,	O
both	O
clothing	O
,	O
etc	O
.	O
</s>
<s>
Let	O
be	O
the	O
similarity	O
matrix	O
whose	O
entry	O
denotes	O
the	O
similarity	O
score	O
,	O
and	O
be	O
the	O
column	O
normalized	O
adjacency	B-Algorithm
matrix	I-Algorithm
whose	O
entry	O
if	O
there	O
is	O
an	O
edge	O
from	O
to	O
,	O
and	O
0	O
otherwise	O
.	O
</s>
<s>
A	O
solution	O
to	O
the	O
SimRank	B-Algorithm
equations	O
for	O
a	O
graph	O
can	O
be	O
reached	O
by	O
iteration	B-Algorithm
to	O
a	O
fixed-point	O
.	O
</s>
<s>
For	O
each	O
iteration	B-Algorithm
,	O
we	O
can	O
keep	O
entries	O
,	O
where	O
gives	O
the	O
score	O
between	O
and	O
on	O
iteration	B-Algorithm
.	O
</s>
<s>
We	O
start	O
with	O
where	O
each	O
is	O
a	O
lower	O
bound	O
on	O
the	O
actual	O
SimRank	B-Algorithm
score	O
:	O
</s>
<s>
To	O
compute	O
from	O
,	O
we	O
use	O
the	O
basic	O
SimRank	B-Algorithm
equation	O
to	O
get	O
:	O
</s>
<s>
That	O
is	O
,	O
on	O
each	O
iteration	B-Algorithm
,	O
we	O
update	O
the	O
similarity	O
of	O
using	O
the	O
similarity	O
scores	O
of	O
the	O
neighbours	O
of	O
from	O
the	O
previous	O
iteration	B-Algorithm
according	O
to	O
the	O
basic	O
SimRank	B-Algorithm
equation	O
.	O
</s>
<s>
It	O
was	O
shown	O
in	O
that	O
the	O
values	O
converge	B-Algorithm
to	O
limits	B-Algorithm
satisfying	O
the	O
basic	O
SimRank	B-Algorithm
equation	O
,	O
the	O
SimRank	B-Algorithm
scores	O
,	O
i.e.	O
,	O
for	O
all	O
,	O
.	O
</s>
<s>
The	O
original	O
SimRank	B-Algorithm
proposal	O
suggested	O
choosing	O
the	O
decay	O
factor	O
and	O
a	O
fixed	O
number	O
of	O
iterations	O
to	O
perform	O
.	O
</s>
<s>
However	O
,	O
the	O
recent	O
research	O
showed	O
that	O
the	O
given	O
values	O
for	O
and	O
generally	O
imply	O
relatively	O
low	O
accuracy	O
of	O
iteratively	O
computed	O
SimRank	B-Algorithm
scores	O
.	O
</s>
<s>
CoSimRank	O
is	O
a	O
variant	O
of	O
SimRank	B-Algorithm
with	O
the	O
advantage	O
of	O
also	O
having	O
a	O
local	O
formulation	O
,	O
i.e.	O
</s>
<s>
Let	O
be	O
the	O
similarity	O
matrix	O
whose	O
entry	O
denotes	O
the	O
similarity	O
score	O
,	O
and	O
be	O
the	O
column	O
normalized	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Step	O
one	O
can	O
be	O
seen	O
a	O
simplified	O
version	O
of	O
Personalized	O
PageRank	B-Algorithm
.	O
</s>
<s>
Step	O
two	O
sums	O
up	O
the	O
vector	O
similarity	O
of	O
each	O
iteration	B-Algorithm
.	O
</s>
<s>
Fogaras	O
and	O
Racz	O
suggested	O
speeding	O
up	O
SimRank	B-Algorithm
computation	O
through	O
probabilistic	O
computation	O
using	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
extended	O
SimRank	B-Algorithm
equations	O
to	O
take	O
into	O
consideration	O
(	O
i	O
)	O
evidence	O
factor	O
for	O
incident	O
nodes	O
and	O
(	O
ii	O
)	O
link	O
weights	O
.	O
</s>
<s>
further	O
improved	O
SimRank	B-Algorithm
computation	O
via	O
a	O
fine-grained	O
memoization	O
method	O
to	O
share	O
small	O
common	O
parts	O
among	O
different	O
partial	O
sums	O
.	O
</s>
<s>
Chen	O
and	O
Giles	O
discussed	O
the	O
limitations	O
and	O
proper	O
use	O
cases	O
of	O
SimRank	B-Algorithm
.	O
</s>
<s>
proposed	O
three	O
optimization	O
techniques	O
for	O
speeding	O
up	O
the	O
computation	O
of	O
SimRank	B-Algorithm
:	O
</s>
<s>
In	O
particular	O
,	O
the	O
second	O
observation	O
of	O
partial	O
sums	O
memoization	O
plays	O
a	O
paramount	O
role	O
in	O
greatly	O
speeding	O
up	O
the	O
computation	O
of	O
SimRank	B-Algorithm
from	O
to	O
,	O
where	O
is	O
the	O
number	O
of	O
iterations	O
,	O
is	O
average	O
degree	O
of	O
a	O
graph	O
,	O
and	O
is	O
the	O
number	O
of	O
nodes	O
in	O
a	O
graph	O
.	O
</s>
