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high
connectivity. Their model suggests that above the percolation transition,
the network consists of on a large connected cluster (the central part of the id-
iotypic network) with a number of weakly connected constituents, and co-exists
with a number of small isolated clusters. Figure 6 shows some similarities to this
view; in the homogeneous case (lower graph), the network is disconnected and
it consists of two isolated clusters. The upper graph showing a heterogeneous
network consists of one large cluster which clearly has a highly connected central
part weakly connected to a number of smaller clusters. In addition, from a purely
visual inspection, the network appears to shows sign of being disassortative, i.e.
that nodes with high degree are connected preferentially with nodes with low
degree (note several nodes in the lower diagram connected to a large number
of nodes which have degree 1). This is a trait which is frequently observed in
topological analysis of biological networks (e.g for protein-protein interactions
in yeast [11]).
6
Conclusions
An in-depth analysis of a growth model for an idiotypic network and the resultant
architecture has been presented, and provides an addition to existing literature
in building a picture of how an idiotypic network might emerge and function.
Despite the simplicity of the model, we find networks which are in accordance
with biology; both homogeneous and heterogenous network models stabilise to
a relatively constant size following an initial growth period, and do not either
collapse or expand indefinitely. Although the model is simplistic compared to
those proposed a decade ago, the topologies of the resultant networks at least
contain glimpses of those features we observe from immunological studies —
in heterogeneous networks we observe the formation of a large cluster with a
number of weakly connected constituents, and the networks show signs of being
disassortative. However, more work is needed before definite conclusions can
be made in relation to this, particularly in the light of the important role the
network topology may play in influencing immunological memory.
Surprisingly, we find some results which contradict the observations made
by [4] using an exogenous production model. In particular, our model suggests
that hubs can emerge, although they are clearly transient, and that a power-
law degree-distribution emerges at least over some range of degrees, even if it is
somewhat truncated. The hubs do not arise through a preferential attachment
mechanism related to the degree of a node as in the growth model proposed by [1].
However, neither can they be explained through a positive feedback mechanism
which rewards nodes with high concentration as in the endogenous production
model of [4,13]. It seems likely that the identity of the hubs is in part a lucky acci-
dent of the placing of the first few random nodes in the simulation, which sets up
the environmental conditions for nodes to exist at certain points in the shape-
space where they are able to maintain a balance between becoming over and
under stimulated. However, the role that concentration plays needs further inves-
tigation, to explain the relationships observed between degree and concentration,
