Information Technology Reference
In-Depth Information
Depending on the influx
I
we observe with varying probability either a decay
of the giant cluster into numerous small identical clusters or the formation of
one large cluster accompanied by many isolated occupied nodes.
In any case, the system reaches a stationary state in which the influx of new
idiotypes and the loss of old ones stay well-balanced, cf. Fig. 1. The stationary
states may have a complex architecture, in which we can classify the nodes into
groups with clearly distinct statistical characteristics.
2500
2000
Occ. vertices
Largest cluster
Av. cluster size
Stable holes
1500
1000
500
0
0
50 100 150 200 250 300 350 400 450 500
Time
Fig. 1.
The time series of the number of occupied vertices
n
T
(
G
), the size of the
currently largest cluster
|C
max
T
|
, the average cluster size
|C|
C
T
, and the number of
stable holes
h
T
(
G
) on a base graph
G
(2)
12
with
t
l
=1,
t
u
=10 and
I
= 110
The empty base graph is a highly symmetric object. Due to the random influx
the symmetry is broken and the system falls into a network configuration of lower
symmetry depending on the individual history. Increasing the influx may lead
to transitions between different patterns where the formation of intermediate
unstable giant clusters play a role. For a more detailed account of the transient
behavior and the transitions see [12].
In the simulations we measured the behavior of the whole system, as well as
the time averages of local quantities characterizing every single node. In this way
groups of nodes can be distinguished with clearly distinct properties. Figure 2
shows the time average of the number of occupied neighbours of every node as
a function of the influx. We find distinct regions in dependence of the influx
I
. For small and moderate influx a clear group structure is visible. Considering
also other characteristics, e.g. the mean life time, we can describe them as static
(
I<
90) and dynamic (90
I<
260). For higher influx the clear distinction
of groups becomes impossible, we call these patters transient (260
≤
≤
I
), and
random (350
I
). Static patterns have groups of occupied nodes which have
a high mean life time. Many of the other groups are stable holes or sparsely
occupied vertices. In dynamic patterns there still are some stable hole groups,
