Architecture of idiotypic networks: Percolation and scaling Behavior - art. no. 011908

Wt: investigate a model where idiotypes (characterizing B lymphocytes and a ntibodies of an immune system) and anti-idiotypes are represented by comple mentary bit strings of a given length d allowing for a number of mismatches (matching rules). In this model, the vertices of the hypercube in dimensio n d represent the potential repertoire of idiotypes. A random set of (with probability p,) occupied vertices corresponds to the expressed repertoire o f idiotypes at a given moment. Vertices of this set linked by the above mat ching rules build random clusters. We give a structural and statistical cha racterization of these clusters, or in other words of the architecture of t he idiotypic network. Increasing the probability p one finds at a critical p a percolation transition where for the first time a large connected graph occurs with probability 1. Increasing p further, there is a second transit ion above Which the repertoire is complete in the sense that any newly intr oduced idiotype rinds a complementary anti-idiotype. We introduce structura l characteristics such as the mass distribution and the fragmentation rate for random clusters, and determine the scaling behavior of the cluster size distribution near the percolation transition. including finite size correc tions. We find that slightly above the percolation transition the large con nected cluster (the central part of the idiotypic network) consists typical ly of one highly connected part and a number of weakly connected constituen ts and coexists with a number of small, isolated clusters. This is in accor dance with the picture of a central and a peripheral part of the idiotypic network and gives some support to idealized architectures of the central pa rt used in recent dynamical mean field models.