# State-space and basins of attraction

 For a network size n, an example of one of its states B might be 1010...0110. State-space is made up of all 2n states, the space of all possible bitstrings or patterns. Part of a trajectory in state-space, where C is a successor of B, and A is a predecessor, pre-image, of B, according to the dynamics on the network. The state B may have other pre-images besides A, the total is the in-degree. The pre-image states may have their own pre-images or none. States without pre-images are known as garden-of-Eden states. Any trajectory must sooner or later encounter a state that occurred previously - it has entered an attractor cycle. The trajectory leading to the attractor is a transient. The period of the attractor is the number of states in its cycle, which may be just one - a point attractor. Take a state on the attractor, find its pre-images (excluding the pre-image on the attractor). Now find the pre-images of each pre-image, and so on, until all garden-of-Eden states are reached. The graph of linked states is a transient tree rooted on the attractor state. Part of the transient tree is a subtree defined by its root. Construct each transient tree (if any) from each attractor state. The complete graph is the basin of attraction. Some basins of attraction have no transient trees, just the bare ``attractor''. Now find every attractor cycle in state-space and construct its basin of attraction. This is the basin of attraction field containing all 2n states in state-space, but now linked according to the dynamics on the network. Each discrete dynamical network imposes a particular basin of attraction field on state-space.

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