<s>
A	O
Google	B-Application
matrix	I-Application
is	O
a	O
particular	O
stochastic	B-Algorithm
matrix	I-Algorithm
that	O
is	O
used	O
by	O
Google	B-Application
's	I-Application
PageRank	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
PageRank	B-Algorithm
of	O
each	O
page	O
can	O
then	O
be	O
generated	O
iteratively	O
from	O
the	B-Application
Google	I-Application
matrix	O
using	O
the	O
power	B-Language
method	I-Language
.	O
</s>
<s>
However	O
,	O
in	O
order	O
for	O
the	O
power	B-Language
method	I-Language
to	O
converge	O
,	O
the	O
matrix	O
must	O
be	O
stochastic	O
,	O
irreducible	O
and	O
aperiodic	O
.	O
</s>
<s>
In	O
order	O
to	O
generate	O
the	B-Application
Google	I-Application
matrix	O
G	O
,	O
we	O
must	O
first	O
generate	O
an	O
adjacency	B-Algorithm
matrix	I-Algorithm
A	O
which	O
represents	O
the	O
relations	O
between	O
pages	O
or	O
nodes	O
.	O
</s>
<s>
A	O
matrix	O
element	O
is	O
filled	O
with	O
1	O
if	O
node	O
has	O
a	O
link	O
to	O
node	O
,	O
and	O
0	O
otherwise	O
;	O
this	O
is	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
links	O
.	O
</s>
<s>
That	O
makes	O
S	O
suitable	O
for	O
the	O
PageRank	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Then	O
the	O
final	O
Google	B-Application
matrix	I-Application
G	O
can	O
be	O
expressed	O
via	O
S	O
as	O
:	O
</s>
<s>
(	O
1	O
)	O
within	O
a	O
simple	O
network	O
is	O
given	O
in	O
the	O
article	O
CheiRank	B-Application
.	O
</s>
<s>
For	O
the	O
actual	O
matrix	O
,	O
Google	B-Application
uses	O
a	O
damping	O
factor	O
around	O
0.85	O
.	O
</s>
<s>
The	O
examples	O
of	O
Google	B-Application
matrix	I-Application
structure	O
are	O
shown	O
in	O
Fig.1	O
for	O
Wikipedia	O
articles	O
hyperlink	O
network	O
in	O
2009	O
at	O
small	O
scale	O
and	O
in	O
Fig.2	O
for	O
University	O
of	O
Cambridge	O
network	O
in	O
2006	O
at	O
large	O
scale	O
.	O
</s>
<s>
These	O
probabilities	O
ordered	O
by	O
their	O
decreasing	O
values	O
give	O
the	O
PageRank	B-Algorithm
vector	O
with	O
the	O
PageRank	B-Algorithm
used	O
by	O
Google	B-Application
search	O
to	O
rank	O
webpages	O
.	O
</s>
<s>
The	O
number	O
of	O
nodes	O
with	O
a	O
given	O
PageRank	B-Algorithm
value	O
scales	O
as	O
with	O
the	O
exponent	O
.	O
</s>
<s>
The	O
PageRank	B-Algorithm
vector	O
varies	O
with	O
but	O
other	O
eigenvectors	O
with	O
remain	O
unchanged	O
due	O
to	O
their	O
orthogonality	O
to	O
the	O
constant	O
left	O
vector	O
at	O
.	O
</s>
<s>
The	O
gap	O
between	O
and	O
other	O
eigenvalue	O
being	O
gives	O
a	O
rapid	O
convergence	O
of	O
a	O
random	O
initial	O
vector	O
to	O
the	O
PageRank	B-Algorithm
approximately	O
after	O
50	O
multiplications	O
on	O
matrix	O
.	O
</s>
<s>
Examples	O
of	O
the	O
eigenvalue	O
spectrum	O
of	O
the	B-Application
Google	I-Application
matrix	O
of	O
various	O
directed	O
networks	O
is	O
shown	O
in	O
Fig.3	O
from	O
and	O
Fig.4	O
from	O
.	O
</s>
<s>
The	B-Application
Google	I-Application
matrix	O
can	O
be	O
also	O
constructed	O
for	O
the	O
Ulam	O
networks	O
generated	O
by	O
the	O
Ulam	O
method	O
 [ 8 ] 	O
for	O
dynamical	O
maps	O
.	O
</s>
<s>
The	B-Application
Google	I-Application
matrix	O
can	O
be	O
constructed	O
also	O
for	O
other	O
directed	O
networks	O
,	O
e.g.	O
</s>
<s>
The	B-Application
Google	I-Application
matrix	O
with	O
damping	O
factor	O
was	O
described	O
by	O
Sergey	O
Brin	O
and	O
Larry	O
Page	O
in	O
1998	O
[22],	O
see	O
also	O
articles	O
on	O
PageRank	B-Algorithm
history	O
 [ 23 ]  ,  [ 24 ] 	O
.	O
</s>
