Patterns in randomly evolving networks: Idiotypic networks

TL;DR

This paper models the evolution of networks on regular graphs, revealing stable and dynamic patterns influenced by parameters, with applications to immune system networks and broad implications for complex network analysis.

Contribution

It introduces a novel model for network evolution with thresholds, analyzing pattern formation, stability, and transitions, especially in the context of idiotypic immune networks.

Findings

Stable long-living patterns on low connectivity graphs

Dynamic patterns and transitions on high connectivity graphs

Application to immune system network modeling

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,t_u) 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.