where AMC is the "average mutual constraint" known in information theory as the average mutual information (Rutledge et al., 1976).
To illustrate how an increase in AMC actually tracks the "pruning" process, the reader is referred to the three hypothetical configurations in Figure B4.2. In configuration (a) where medium from any one compartment will next flow is maximally indeterminate. AMC is identically zero. The possibilities in network (b) are somewhat more constrained. Flow exiting any compartment can proceed to only two other compartments, and the AMC rises accordingly. Finally, flow in schema (c) is maximally constrained, and the AMC assumes its maximal value for a network of dimension 4.
One notes in the formula for AMC that the scalar constant, k, has been retained. We recall that although autocatalysis is a unitary process, one can discern two separate effects: (a) an extensive effect whereby the activity, T, of the system increases, and (b) an intensive aspect whereby constraint is growing. We can readily unify these two aspects into one measure simply by making the scalar constant k represent the level of system activity, T. That is, we set k = T, and we name the resulting product the system Ascendency, A, where
Figure B4.2 Three configurations of processes illustrating how autocatalytic "pruning" serves to increase overall system constraint. (a) A maximally indeterminate four-component system with 96 units of flow. (b) The system in (a) after constraints have arisen that channel flow to only two other compartments. (c) The maximally constrained system with each compartment obligated to support only one other component.
Referring again to the Cone Spring ecosystem network in Figure B4.1, we notice that each flow in the diagram generates exactly one and only one term in the indicated sums. Hence, we see that the ascendency consists of the 18 terms:
T T 1011
I Tki I Tc
I Tk 21T
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