Attractors

If a RBN evolves according to one of the update schemes described in Classification, sooner or later the network gets trapped in an attractor. An attractor is defined as a sequence of node-states that the network repeatedly visits.

In practice, the  sequence of node-states in an attractor and the length of this sequence, the attractor length, are of particular interest. An attractor can either be a point attractor or a cycle attractor, where point attractors have - by definition- attractor length one. Not all update modes exhibit both types of attractors (ARBN for example do not have cycle attractors due to the asynchronicity of their update scheme).

Depending on the initial state, a network evolves into the direction of a specific attractor.
All possible node-states and the attractors of a network can be represented by a landscape and described by the notion of stability/instability. A ball that is placed in a landscape with hills and valleys, will finally (due to gravity) end up at the bottom of a valley, which is a stable position in the system. A network does similarly: it develops through node states (descending the hills) into an attractor (bottom of the valley). All node-states that lead to a specific attractor are said to be in the basin of attraction of this attractor. Attractors are interesting because they represent the total number of alternative and stable long-term behaviours of the system. It might be interesting to study, how minimal changes in the topology (connections, node-states, rules) affect the location and size of an attractor.