Information Technology Reference
In-Depth Information
We find that the bitstrings of a vertex
v
i
∈
S
i
always deviates in exactly
i
−
1
determinant bits from those of a vertex
v
1
∈
S
1
. When calculating the number
of links of vertex
v
i
to vertices
v
j
∈
S
j
we have to take into account that the
bitstring of
v
j
also deviate in
j
S
1
.
Among the vertices in
S
j
there exists a vertex
v
j
with a minimum number of
mismatches with respect to
v
i
. The non-determinant bits of
v
i
and
v
j
can be
chosen to be inverse, but for the determinant bits there are constraints. The
lowest number of mismatches between
v
i
and
v
j
can be achieved if we arrange
without loss of generality all deviating bits of
v
i
to the left and all deviating bits
of
v
j
to the right:
−
1 determinant bit positions from that of a
v
1
∈
b
k
i
... ...
... ...
b
k
d
M
−j
+1
... ...
b
k
d
M
b
k
d
M
−j
+2
...
b
k
d
M
v
i
:
v
j
:
b
k
1
b
k
2
...
b
k
i−
1
b
k
1
b
k
2
...
...
1
bits
j
+2
bits
j−
1
bits
i
−
m
min
ij
:=
d
M
−
i
−
If
m
min
ij
<
0, the deviating bits will overlap in the arrangement, and if
m
min
ij
>
0,
there will be a gap. Considering all allowed arrangements of bits we can thus
calculate the number of links of a given vertex
v
i
∈
S
i
to vertices in
S
j
by
elementary combinatorics:
d
,
k
l
1
k
+max(0
,m
min
i
−
i
+1
k
+max(0
,
d
M
−
−
d
M
(5)
m
min
ij
)
−
ij
)
l
k
=0
l
=0
where
k
=
(
m−|m
min
ij
|
)
/
2
,and
l
=
m−|m
min
ij
|−
2
k
. Details of the calculation
are given in [19] and in a forthcoming publication.
3.3 The Six-Group Pattern
A remarkable pattern found empirically in [12] on
G
(2)
12
for
I
= 90 is the dynamic
pattern consisting of a six-groups, cf. Table 3.
Table 3.
Characterization of the six empirical groups. Data from [12].
S
1
S
2
S
3
S
4
S
5
S
6
group size
|
S
i
|
1124
924
924
134
330
660
life time
τ
(
v
)
0.0
3.8
5.4
10.0
18.1
35.6
S
i
We now denote the empirically found groups by
S
i
to distinguish them from
the groups
S
i
defined analyzing the pattern modules.
S
1
is the group of stable
holes,
S
2
and
S
3
are central groups, which have connections among each other,
as well as to the peripheral group
S
5
.
S
2
additionally has got links to the other
peripheral group
S
6
. The group
S
4
is somewhat special, because it is entirely
surrounded by stable holes, cf. Fig. 6. Occupied vertices of this group are sus-
tained solely by the random influx. Figure 7 shows a snapshot of the occupied
