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network structures was not really successful. Parallel with the rapid success of
molecular immunobiology the initial enthusiasm of experimentally working im-
munologists decayed. Today there is a renewed interest in idiotypic interactions,
for example in the context of autoimmune diseases [3,4]. The progress in ex-
perimental methods seems to make a new generation of experiments feasible.
An excellent review and a thorough discussion of the historical development of
immunological paradigms has been given in [5], cf. also [6].
Idiotypic networks stayed always attractive for theoretical biologists interested
in the systems behaviour, but they attracted also the interest of theoretical
physicists. Also computer scientists are interested in the concepts that living
organisms have developed to fight against foreign invaders and develop artificial
immune systems.
The estimated size of the potential idiotypic repertoireofmenisoftruly
macroscopic order 10
12
, the expressed repertoire is of order 10
8
[7,8]. Interac-
tions between B-cells of complementary idiotype are genuinely nonlinear. Thus,
modeling idiotypic networks is an inviting playground for statistical physics,
nonlinear dynamics, and complex systems. More generally, networks, especially
random and randomly growing networks, with applications in a plethora of dif-
ferent, multidisciplinary fields [9,10,11] experience rapidly increasing interest in
the community of statistical physicists.
A minimalistic model of the idiotypic network was proposed in [12] where
idiotypes are represented by bitstrings which can interact with complementary
bitstrings allowing for a few mismatches [13]. In the model, an idiotype popula-
tion may be present or absent.
For survival it needs stimulation by suciently many complementary idio-
types, but becomes extinct if too many complementary idiotypes are present.
The dynamics is driven by the influx of new idiotypes generated by mutations
in the bone marrow.
The model has a minimal number of parameters, namely the length of the
bitchain, the allowed number of mismatches, upper and lower thresholds for
stimulation, and the influx of new idiotypes. This allows us to also derive some
analytical results. However, unrealistic features, such as the extinction of a clone
within one time step, are the price of simplicity.
A first study for one and two allowed mismatches was presented in [12]. For
typical parameter settings a random evolution towards a highly nontrivial com-
plex functional architecture of the emerging network was observed. To character-
ize this architecture the nodes can be classified into different groups with clearly
distinct properties. They include densely connected core groups and peripheral
groups of isolated nodes, resembling the notion of central and peripheral part of
the biological network [14,15].
The potential idiotypic network consisting of all idiotypes an organism is
able to generate and the links connecting complementary idiotypes allowing a
few mismatches is modeled as in [12] by an undirected base graph
G
=(
V
,
E
).
Each idiotype
v
∈V
in the network is characterised by a bitstring of length
d
:
b
d
b
d−
1
···
b
1
,
with
b
i
∈{
0
,
1
}
for all
i
∈{
1
,
2
,...,d
}
. For every pair of
