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vertices the degree of complementarity is evaluated: If the Hamming distance d H
between the bitstrings of two vertices v, w
∈V
equals the length of the bitstring
d , there is a link l =
1,
we call it a one-mismatch link, etc. Allowing m mismatches, the base graph
consisting of all bitstrings of length d and the allowed links is denoted by G ( m d .
The expressed idiotypic network is only a fraction of the potential network,
the nodes of the expressed idiotypes and their links are a subgraph of G ( m d .
Driven by the random influx of new idiotypes the network evolves towards
a stationary state of nontrivial architecture. Crucial for that is that beside the
occupation of previously empty nodes, occupied nodes can become empty if
linked with too many or too few nodes of complementary idiotype. To be specific,
the rules for (parallel) update are
{
v, w
}
representing a perfect match, if d H ( v, w )= d
(i) Choose I unoccupied sites (holes) randomly and set them occupied. They
represent the influx of new idiotypes from the bone marrow.
(ii) Count the number of occupied vertices n ( ∂v ) in the neighborhood of every
vertex v
G .If n ( ∂v ) is outside the window of lower and upper threshold
( t l ,t u ) , the vertex v will be set empty.
(iii) Iterate.
A similar model was proposed by Stewart and Varela [16], who also apply
a window update rule to simulate the internal dynamics and a 0-1 clone pop-
ulation. However, their shape space differs from our model: While we consider
a discrete d -dimensional hypercubic shape space, in [16,17] the complementary
idiotypes live on different sheets of a 2D continuous shape space.
In the following section we describe the typical course of the random evolution
of the network as found in extensive numerical simulations. The evolution tends
toward a steady state of highly organized architecture. We describe how this
architecture can be characterized classifying nodes into different groups with
clearly distinct statistical properties and how these groups are linked together.
In Sect. 3 we show that the empirical findings can be explained analytically once
the building principles are understood. In the final section concluding remarks
and an outlook are given.
2
Random Evolution of the Network
We performed simulations on the basegraph G (2)
12 for ( t l ,t u )=(1 , 10) for different
values of I starting with an empty base graph. The base graph contains 4096
nodes each of which has 79 links to other nodes. In the first step only those
nodes survive which have at least one occupied neighbor (having more than 10
occupied neighbors is unlikely in the beginning). The surviving nodes represent
seeds to which other occupied nodes easily can attach. That leads to a rapid
growing towards a giant cluster. Parallel to that many stable holes are created,
i.e. nodes with the number of occupied neighbors above the upper threshold.
Going through a state with one giant cluster determines -in a sense- the
pattern towards which the system will evolve.
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