Patterns in randomly evolving networks: Idiotypic networks

Abstract

We present a model for the evolution of networks of occupied sites on undirected regular graphs. At every iteration step in a parallel update I randomly chosen empty sites are occupied and occupied sites having degree outside of a given interval (t l , tu) are set empty. Depending on the influx I and the values of both lower threshold and upper threshold of the degree different kinds of behaviour can be observed. In certain regimes stable long-living patterns appear. We distinguish two types of pattern: static patterns arising on graphs with low connectivity and dynamic patterns found on high connectivity graphs. Increasing I patterns become unstable and transitions between almost stable patterns, interrupted by disordered phases, occur. For still larger I the lifetime of occupied sites becomes very small and network structures are dominated by randomness. We develop methods to analyze nature and dynamics of these network patterns, give a statistical description of defects and fluctuations around them, and elucidate transitions between different patterns. Results and methods presented can be applied to a variety of problems in different fields and a broad class of graphs. Aiming chiefly at the modeling of functional networks of interacting antibodies and B-cells of the immune system (idiotypic networks) we focus on a class of graphs constructed by bit-chains. The biological relevance of the patterns and possible operational modes of idiotypic networks are discussed.

Recent Advances in Artificial Immune Systems: Models and Applications

“…The major contribution of this work is to explain these very complex network structures emerging during the random evolution through a detailed analytical understanding of the building principles. The building principles allow us to calculate instance size and connectivity of the idiotype groups in perfect agreement with the empirical findings reported in [56].…”

“…The idiotypic network is modeled by an undirected base graph G = ( , ε). Each idiotype v ∈ in the network is characterized by a bitstring of length d : [56]. This work performed simulations on the base graph G (2) 12 for (t l , t u ) = (1, 10) for different values of influx I starting with an empty base graph.…”

Self-Tolerance in a Minimal Model of the Idiotypic Network

“…Whereas the preceding work by Brede and Behn (58), Schmidtchen and Behn (59), Schmidtchen et al (60), and Schmidtchen and Behn (61) considered the autonomous idiotypic network, i.e., the network of B-lymphocytes and their antibodies without foreign or self antigen, we investigate here the evolution of the idiotypic network, in the presence of self, toward an architecture where the expansion of autoreactive clones is controlled by idiotypic interactions. Self is modeled by permanently present idiotypes which influence the evolution of the network but are themselves not affected by idiotypic interactions.…”

“…(58) which describes the evolution toward complex, functional architectures. The model uses a discrete shape space spanned by bitstrings which represent idiotypes.…”

Immune networks: multi-tasking capabilities at medium load

“…This description complements the more standard approaches, which tend to be phrased in the language of dynamical systems [11][12][13][14], network theory [15][16][17][18][19][20] or multi-scale mathematical biology [21][22][23].…”