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immunological phenomena. Over a decade later, a resurgent interest in networks
has come about in which new understanding in the area of statistical physics
has led to a greater focus on undertanding the properties of the network itself.
Thus, Brede and Behn in [5,6] focus on analysing the dynamics and architec-
ture of an idiotypic network. Their model incorporates two important principles
specific to immune networks; the first is that the dynamics and network evolution
should be driven by a continuous influx of new idiotypes from the bone marrow,
and that secondly, that idiotypes should die out if they become
under
or
over
stimulated
. Their model adopts a bit-string approach: for a bit-string of length
d
,thereare2
d
possible antibodies (representing vertices of a hyper-cube). By
defining recognition to occur between vertices which are either perfectly com-
plementary or have only
n
matching bits ('
n
-mismatch'), the network can be
represented as a graph in which some vertices are connected (e.g a “1-mismatch”
rule on a hypercube of dimension 3 has all space and side-diagonals connected).
Growth dynamics are simulated by simply selecting at random a set of vertices of
the hypercube and occupying them. The neighbourhood of each occupied vertex
is then checked, and any vertex having a degree less than
t
l
or greater than
t
u
is deleted. The upper bound
t
u
prevents unlimited growth of the network in the
first instance and can lead to instantaneous removal of nodes, whilst the lower
bound
t
l
is responsible for maintaining a memory of perturbations which can
last over many iterations. They obtain results which show that their model pro-
duces a non-trivial seemingly realistic network topology. Although the model is
appealing in it's simplicity it has some drawbacks from a biological perspective
in that it makes no reference to the concentration of cells, and instead appeals
to cell degree as the deciding factor in determining whether cells survive or not.
On the other hand, Bersini
et al
[4] propose a general model that fits well
with the biological perspective and has the added advantage of being generalis-
able to either exogenous production of nodes (such as the immune network) or
endogenous production (as in protein networks) and to both homogeneous and
heterogeneous networks. The model again utilises a binary shape-space. Bit-
strings are able to bind if the Hamming distance between two strings is greater
than some threshold
t
. The key features of the model are that: each node has a
different identity based on it's physical properties which define it's
type
and an
associated
concentration
that changes over time; the model is
type-based
rather
than
instance-based
as in technological or social networks; nodes connect based
on mutual attractiveness (anity); the nodes that are added to the network de-
pend on the dynamics of the existing network. At each iteration of the model,
new instances of types are introduced to the network, and they are added only
if they can bind to other instances in the network — links only appear if the
types of the two instances had not previously been bound. This essentially forms
a biological interpretation of the preferential attachment rules proposed in [1].
Using this model, they obtain results which suggest that scale-free distributions
are only obtained using an endogenous production scheme and that exogenous
production models as observed in the immune system can lead to in the worst
case, an exponential distribution. However, their model is incomplete in the sense
