  We introduce a class of weighted graphs whose properties are meant to mimic
the topological features of idiotypic networks, namely the interaction networks
involving the B-core of the immune system. Each node is endowed with a
bit-string representing the idiotypic specificity of the corresponding B cell
and a proper distance between any couple of bit-strings provides the coupling
strength between the two nodes. We show that a biased distribution of the
entries in bit-strings can yield fringes in the (weighted) degree distribution,
small-worlds features, and scaling laws, in agreement with experimental
findings. We also investigate the role of ageing, thought of as a progressive
increase in the degree of bias in bit-strings, and we show that it can possibly
induce mild percolation phenomena, which are investigated too.
