<s>
The	O
CheiRank	B-Application
is	O
an	O
eigenvector	O
with	O
a	O
maximal	O
real	O
eigenvalue	O
of	O
the	O
Google	B-Application
matrix	I-Application
constructed	O
for	O
a	O
directed	O
network	O
with	O
the	O
inverted	O
directions	O
of	O
links	O
.	O
</s>
<s>
It	O
is	O
similar	O
to	O
the	O
PageRank	B-Algorithm
vector	O
,	O
which	O
ranks	O
the	O
network	O
nodes	O
in	O
average	O
proportionally	O
to	O
a	O
number	O
of	O
incoming	O
links	O
being	O
the	O
maximal	O
eigenvector	O
of	O
the	O
Google	B-Application
matrix	I-Application
with	O
a	O
given	O
initial	O
direction	O
of	O
links	O
.	O
</s>
<s>
Due	O
to	O
inversion	O
of	O
link	O
directions	O
the	O
CheiRank	B-Application
ranks	O
the	O
network	O
nodes	O
in	O
average	O
proportionally	O
to	O
a	O
number	O
of	O
outgoing	O
links	O
.	O
</s>
<s>
Since	O
each	O
node	O
belongs	O
both	O
to	O
CheiRank	B-Application
and	O
PageRank	B-Algorithm
vectors	O
the	O
ranking	O
of	O
information	O
flow	O
on	O
a	O
directed	O
network	O
becomes	O
two-dimensional	O
.	O
</s>
<s>
For	O
a	O
given	O
directed	O
network	O
the	O
Google	B-Application
matrix	I-Application
is	O
constructed	O
in	O
the	O
way	O
described	O
in	O
the	O
article	O
Google	B-Application
matrix	I-Application
.	O
</s>
<s>
The	O
PageRank	B-Algorithm
vector	O
is	O
the	O
eigenvector	O
with	O
the	O
maximal	O
real	O
eigenvalue	O
.	O
</s>
<s>
It	O
was	O
introduced	O
in	O
and	O
is	O
discussed	O
in	O
the	O
article	O
PageRank	B-Algorithm
.	O
</s>
<s>
In	O
a	O
similar	O
way	O
the	O
CheiRank	B-Application
is	O
the	O
eigenvector	O
with	O
the	O
maximal	O
real	O
eigenvalue	O
of	O
the	O
matrix	O
built	O
in	O
the	O
same	O
way	O
as	O
but	O
using	O
inverted	O
direction	O
of	O
links	O
in	O
the	O
initially	O
given	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Both	O
matrices	O
and	O
belong	O
to	O
the	O
class	O
of	O
Perron	O
–	O
Frobenius	O
operators	O
and	O
according	O
to	O
the	O
Perron	O
–	O
Frobenius	O
theorem	O
the	O
CheiRank	B-Application
and	O
PageRank	B-Algorithm
eigenvectors	O
have	O
nonnegative	O
components	O
which	O
can	O
be	O
interpreted	O
as	O
probabilities	O
.	O
</s>
<s>
Thus	O
all	O
nodes	O
of	O
the	O
network	O
can	O
be	O
ordered	O
in	O
a	O
decreasing	O
probability	O
order	O
with	O
ranks	O
for	O
CheiRank	B-Application
and	O
PageRank	B-Algorithm
respectively	O
.	O
</s>
<s>
In	O
average	O
the	O
PageRank	B-Algorithm
probability	O
is	O
proportional	O
to	O
the	O
number	O
of	O
ingoing	O
links	O
with	O
.	O
</s>
<s>
The	O
CheiRank	B-Application
was	O
introduced	O
for	O
the	O
procedure	O
call	O
network	O
of	O
Linux	O
Kernel	O
software	O
in	O
,	O
the	O
term	O
itself	O
was	O
used	O
in	O
Zhirov	O
.	O
</s>
<s>
While	O
the	O
PageRank	B-Algorithm
highlights	O
very	O
well	O
known	O
and	O
popular	O
nodes	O
,	O
the	O
CheiRank	B-Application
highlights	O
very	O
communicative	O
nodes	O
.	O
</s>
<s>
Top	O
PageRank	B-Algorithm
and	O
CheiRank	B-Application
nodes	O
have	O
certain	O
analogy	O
to	O
authorities	O
and	O
hubs	O
appearing	O
in	O
the	O
HITS	B-Application
algorithm	I-Application
but	O
the	O
HITS	O
is	O
query	B-Library
dependent	O
while	O
the	O
rank	O
probabilities	O
and	O
classify	O
all	O
nodes	O
of	O
the	O
network	O
.	O
</s>
<s>
Since	O
each	O
node	O
belongs	O
both	O
to	O
CheiRank	B-Application
and	O
PageRank	B-Algorithm
we	O
obtain	O
a	O
two-dimensional	O
ranking	O
of	O
network	O
nodes	O
.	O
</s>
<s>
There	O
had	O
been	O
early	O
studies	O
of	O
PageRank	B-Algorithm
in	O
networks	O
with	O
inverted	O
direction	O
of	O
links	O
but	O
the	O
properties	O
of	O
two-dimensional	O
ranking	O
had	O
not	O
been	O
analyzed	O
in	O
detail	O
.	O
</s>
<s>
An	O
example	O
of	O
nodes	O
distribution	O
in	O
the	O
plane	O
of	O
PageRank	B-Algorithm
and	O
CheiRank	B-Application
is	O
shown	O
in	O
Fig.1	O
for	O
the	O
procedure	O
call	O
network	O
of	O
Linux	O
Kernel	O
software	O
.	O
</s>
<s>
The	O
distribution	O
of	O
these	O
articles	O
in	O
the	O
plane	O
of	O
PageRank	B-Algorithm
and	O
CheiRank	B-Application
is	O
shown	O
in	O
Fig.3	O
from	O
Zhirov	O
.	O
</s>
<s>
The	O
difference	O
between	O
PageRank	B-Algorithm
and	O
CheiRank	B-Application
is	O
clearly	O
seen	O
from	O
the	O
names	O
of	O
Wikipedia	O
articles	O
(	O
2009	O
)	O
with	O
highest	O
rank	O
.	O
</s>
<s>
At	O
the	O
top	O
of	O
PageRank	B-Algorithm
we	O
have	O
1.United	O
States	O
,	O
2.United	O
Kingdom	O
,	O
3.France	O
while	O
for	O
CheiRank	B-Application
we	O
find	O
1.Portal:Contents/Outline	O
of	O
knowledge/Geography	O
and	O
places	O
,	O
2.List	O
of	O
state	O
leaders	O
by	O
year	O
,	O
3.Portal:Contents/Index/Geography	O
and	O
places	O
.	O
</s>
<s>
Clearly	O
PageRank	B-Algorithm
selects	O
first	O
articles	O
on	O
a	O
broadly	O
known	O
subject	O
with	O
a	O
large	O
number	O
of	O
ingoing	O
links	O
while	O
CheiRank	B-Application
selects	O
first	O
highly	O
communicative	O
articles	O
with	O
many	O
outgoing	O
links	O
.	O
</s>
<s>
The	O
horizontal	O
and	O
vertical	O
lines	O
correspond	O
to	O
PageRank	B-Algorithm
and	O
CheiRank	B-Application
,	O
2DRank	O
combines	O
properties	O
of	O
CheiRank	B-Application
and	O
PageRank	B-Algorithm
as	O
it	O
is	O
discussed	O
in	O
Zhirov	O
.	O
</s>
<s>
According	O
to	O
the	O
PageRank	B-Algorithm
the	O
top	O
100	O
personalities	O
described	O
in	O
Wikipedia	O
articles	O
have	O
in	O
5	O
main	O
category	O
activities	O
:	O
58	O
(	O
politics	O
)	O
,	O
10	O
(	O
religion	O
)	O
,	O
17	O
(	O
arts	O
)	O
,	O
15	O
(	O
science	O
)	O
,	O
0	O
(	O
sport	O
)	O
and	O
thus	O
the	O
importance	O
of	O
politicians	O
is	O
strongly	O
overestimated	O
.	O
</s>
<s>
The	O
CheiRank	B-Application
gives	O
respectively	O
15	O
,	O
1	O
,	O
52	O
,	O
16	O
,	O
16	O
while	O
for	O
2DRank	O
one	O
finds	O
24	O
,	O
5	O
,	O
62	O
,	O
7	O
,	O
2	O
.	O
</s>
<s>
CheiRank	B-Application
and	O
PageRank	B-Algorithm
naturally	O
appear	O
for	O
the	O
world	O
trade	O
network	O
,	O
or	O
international	O
trade	O
,	O
where	O
they	O
and	O
linked	O
with	O
export	O
and	O
import	O
flows	O
for	O
a	O
given	O
country	O
respectively	O
.	O
</s>
<s>
Possibilities	O
of	O
development	O
of	O
two-dimensional	O
search	B-Application
engines	I-Application
based	O
on	O
PageRank	B-Algorithm
and	O
CheiRank	B-Application
are	O
considered	O
.	O
</s>
<s>
Directed	O
networks	O
can	O
be	O
characterized	O
by	O
the	O
correlator	O
between	O
PageRank	B-Algorithm
and	O
CheiRank	B-Application
vectors	O
:	O
in	O
certain	O
networks	O
this	O
correlator	O
is	O
close	O
to	O
zero	O
(	O
e.g.	O
</s>
<s>
A	O
simple	O
example	O
of	O
the	O
construction	O
of	O
the	O
Google	O
matrices	O
and	O
,	O
used	O
for	O
determination	O
of	O
the	O
related	O
PageRank	B-Algorithm
and	O
CheiRank	B-Application
vectors	O
,	O
is	O
given	O
below	O
.	O
</s>
<s>
in	O
the	O
article	O
Google	B-Application
matrix	I-Application
,	O
is	O
shown	O
in	O
Fig.5	O
;	O
</s>
<s>
In	O
a	O
similar	O
way	O
,	O
to	O
determine	O
the	O
CheiRank	B-Application
eigenvector	O
all	O
directions	O
of	O
links	O
in	O
Fig.4	O
are	O
inverted	O
,	O
</s>
