Attractor stability in nonuniform Boolean networks

The attractor graph of a Boolean network F has the attractors of F   as vertices. There is a directed edge between attractors AiAi and AjAj if there is a state x∈Aix∈Ai and an index k such that x when perturbed in the k  th coordinate yields a state x′x′ in the basin of attractor AjAj. Kauffman (2007) proposed to use the strongly connected components of the attractor graph (called ergodic sets) to model possible targets for cell differentiation. However, computational follow-up work by him and others has revealed that most Boolean networks have a single strongly connected component in an overwhelming number of cases, and two strongly connected components in rare cases.As a proof of concept, we explicitly construct classes of hierarchical, nonuniform Boolean networks where the attractor graph may have an arbitrary number of ergodic sets. The construction uses threshold functions and networks that are combinations of connected cliques. Earlier work does not address the structure of ergodic sets. Building on the previous construction we show how to generate Boolean networks where ergodic sets possess binary hyper-cubes Q2m as sub-graphs.Although the focus of this paper is to demonstrate that there are Boolean networks with a large number of ergodic sets and that ergodic sets can be arbitrarily large; we remark that networks we construct have community structure and the results may therefore be relevant to application areas ranging from biology to social science. The results also provide a starting point for additional studies into the attractor structure of Boolean networks, their basins of attractions, and conditions for stability under state noise.