Randomly Evolving Idiotypic Networks: Analysis of Building Principles - Artificial Immune Systems - page 88

Figure 4 illustrates the concept of pattern modules in the smallest possible
two-mismatch graph G (2)
3

.

110 111

occupied

vertices potential

hubs

011

010

101

100

stable
holes

000 001

Fig. 4. Thecompletegraph G (2)

3 with a 2-cluster configuration. On G (2 3 every vertex
is connected to any other. We find two congruently occupied two-dimensional modules
( solid links ), each consisting of one occupied vertex ( black , · 10 ), two potential hubs

( gray , · 00 and · 11 ) and one stable hole ( white , · 01 ). The upper threshold has to be
adjusted to t u =1.

The 2-cluster pattern resembles in a sense the structures found in [16]. There,
chains of complementary idiotypes emerge with a fixed distance, which amounts
to the preferred occurrence of idiotype-anti-idiotype pairs with a given mis-
match. In the ideal case our 2-cluster pattern consists of an ordered array of
idiotype-anti-idiotype pairs with the maximal number of mismatches. However,
this is only the simplest of a multitude of possible patterns, which occur for
larger values of the main control parameter, the influx I . As described in the
following, all of these can be explained in a similar way.

3.2 Generalizations and Combinatorics

Many results for 2-clustered patterns on the G (2)

12 base graph can be generalized
to other choices of d and m . For instance, the 2-cluster pattern on 1-mismatch
graphs described in [12] can be explained in a similar way. For base graphs G ( m )
d
we proved: We can construct 2-cluster patterns by means of pattern modules with
exactly one occupied corner. The dimension of the pattern module d M equals the
number of allowed mismatches m , the number of qualitatively distinguishable

groups is d M +1,andthesizeofgroup S i is 2 d−d M d M
i 1 . A 2-cluster pattern

can emerge if the lower threshold is t l = 1 and the upper threshold obeys
1

−

d M .
In the static pattern regime there exists a dominating 8-cluster pattern, in
which the clusters of occupied vertices appear as cubes. Furthermore, 24- and
30-cluster patterns appear, cf. Fig. 5.

≤ t u ≤ d −