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technological networks such as the Internet, it has serious flaws from a biolog-
ical perspective as it implies an mechanism by which a cell or protein decides
to attach to another cell based on a knowledge of the other's connectivity. In
order to address this, a number of more biologically focussed models have been
proposed. For example, [18] put forward the gene-duplication model, in which
preferential attachment arises as a result of similarity between genes producing
proteins and the initial topology of the network [4]. This model has been shown
to explain biological structure in the case of gene-duplication, yet it has yet to
be generalised to other biological areas, for example the idiotypic network pro-
posed by Jerne in [12]. In response to this, [13,4] propose a more generalised
growth model which can be extended to a number of different biological net-
works, yet retains the important property that it makes no implicit assumption
of preferential attachment based on current node connectivity. Using this model,
they show that under certain conditions, networks can be produced that have
scale-free properties; however, these conditions are reminiscent of those used in
a gene-duplication network in which there is an endogenous production of new
nodes. The results do not extend to networks such as the idiotypic immune net-
work in which there is an exogenous production of new cells (in the immune case
from the bone-marrow).
Yet, idiotypic networks may play a crucial role in advancing our understanding
of the natural immune system. For example, they have been postulated to play
a crucial role creating immunological memory [12], in preventing auto-immunity
[17], and knowledge of their architecture is critical for describing population dy-
namics of B-lymphocytes and antibodies [5]. Thus they have received a great deal
of attention from the immunological community, e.g [17]. At the other extreme,
the properties that are integral to the idiotypic network have also captured the
attention of engineers and computer scientists; thus we see them deployed in
applications ranging from robotics [19] to data-classification [14]. Attempts to
unify understanding and thus progress both disciplines have been made by [3,10],
whose work has gradually begun to build a picture of the properties of idiotypic
networks. In this paper, we extend a previous analysis concerning the dynamics
of emergent idiotypic networks and their resulting properties with an in-depth
analysis of the physical properties of the underlying network itself. We attempt
to map our observations to those that have been made in theoretical immunology
and other studies of biological networks in the hope that the work can impact on
both immunological and engineering studies of the immune system. In the next
section we review some related work on growth models for idiotypic networks,
and then present our model and the experimental results derived from it.
2
Related Work
Interest in modelling idiotypic networks is not new — over a decade ago models
were proposed independently for example by [7,16] and the emergent properties
of these models analysed. These models raised interesting questions regarding
the properties of idiotypic networks, but tended to focus on explaining observed
