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confirmed that these zones exists and further more showed that the shape of the
zones, and therefore the subsequent properties of the network could be controlled
by altering the shape of the domain of anity exerted by a cell.
However, in recent years, as the AIS community has focussed it's attention
more and more on producing tools to solve engineering problems, it is almost
always the case that anity in a real-valued shape-space has been re-defined
in terms of
similarity
. Thus, for example, Timmis [10] introduces an idiotypic
network model in which real-valued vectors represent B-Cells (for example, at-
tributes of a data-set). In this discrete immune network, cells are connected
simply if the Euclidean distance between two cells is less than some threshold
they refer to as the
network a
nity threshold
. This approach is now endemic in
most practical applications of AIS that utilise vector representations. It seems
surprising that such little attention has been paid to whether the use of com-
plementarity of similarity has any effect on the dynamics of network formation
and performance - in fact, it is even observed by [4] that “(surprisingly) it is not
that important in most cases”.
In this paper, we show that contrary to opinion, the definition of anity im-
poses a very definite topology on an emerging network, which has subsequent
important consequences for the properties that we can expect a network to ex-
hibit. The paper is organised as follows. First, two different network models are
introduced, in 2D and in a bit-string universe. We then show how the 2D model
with complementary matching gives rise to tolerant and intolerant zones in the
shape-space. This is then contrasted to the bit-string shape-space with both com-
plementary and similar matching functions. Finally, we explain the anomalous
results we find by analysing a 2D model with a similarity-based anity function
which can be visualised in a straightforward manner.
2N tworkMod s
In this section, we describe the 2D and binary network models in which we
obtain our results.
2D Shape-space model.
The following 2D shape-space model was first proposed
byBersiniin[2]andsubsequentlyadoptedinfurtherworkby[6,5]inwhich
the effect of the shape of the cell recognition region was explored. The shape-
space is defined on a 2D integer-grid of dimension
X, Y
. A cell is specified by a
position (
x, y
) on the grid. The potential network therefore consists of a possible
X
Y
cells. Cells can be considered as connected nodes on a graph if one cell
is
stimulated
by another cell. The manner in which one cell stimulates another
depends on the anity function defined. If anity is defined as complementary,
then a cell
A
stimulates another
B
if
B
lies within a circular region of radius
r
centeredonthepoint(
X
×
y
). On the other hand, if anity is defined
between
similar
cells, then
A
stimulates
B
if
B
lies within a circular region of
radius
r
centered on
A
itself. Using these definitions, the following algorithm
can be used to simulate the growth on an idiotypic network in which there are
potential interactions between both cells and cells, and cells and antigens:
−
x, Y
−
