## Ascendency and Development Capacity

Ascendency, being the product of TST and AMI, takes the following mathematical form:

Assessing ecosystem growth and development can be done comparing ascendency with its maximum and minimum limits. Ascendency, deriving from TST and AMI, shows a minimum value of 0 (when output and input flow probabilities are completely independent; see Figure 3 a), while the upper boundary is defined as development capacity.

For each ecosystem, the development capacity depends on the constraints established by real network topology. When number of compartments (n) and TST are assigned, the highest development capacity is associated to a wholly connected and balanced network, decreasing when flows j j j i=1

 A 4 B
 D C 4

TST = 48 AMI = 1.144 611 A = 54.931 35 C = 137.058 65

 A 12 B
12

D

C

Figure 3 Hypothetical networks with four compartments and TST = 48 energy units: (a) the more unarticulated topology with minimum AMI and ascendency values (both equal to 0) and maximum development capacity (192); (b) and (c) intermediate configurations showing increasing AMI and A; C is lower than that observed for the first network; (d) a closed and linear chain (maximally articulated flows) with highest AMI (2) and ascendency equal to development capacity (96).

 Network topology Development capacity (C) Fully connected (A) 192 Intermediate I (B) 144 Intermediate II (C) 137 Linear and closed chain (D) 96

become more articulated (its minimum value is achieved with closed linear chain topology, when it corresponds to system ascendency) (see Figure 3 and Table 1).

In what follows, minimum and maximum ascendency limits are explained through a probabilistic approach.

The lower ascendency value rises with fully connected topology, that is when input and output flow probabilities are completely independent:

Substituting this relation into eqn [9] yields

Conversely, under minimum uncertainty conditions, ascendency can be inferred setting each inflow and outflow probabilities as mutually determined (to the output a coincides, exclusively, the input bj):

and the consequent development capacity is n+2 / \ n+2

or, in terms of energy (or matter) transfers (with K = TST), n+2 n+2

The more articulated topology (Figure 3d), with receiving node always determined when the compartment from which the flow is exiting is known, implies that C = A. In general,

From the strictly positive difference between development capacity and ascendency is estimated the system overhead (\$), measuring the degree of freedom in flow organization preserved by an ecosystem:

Ascendency and system overhead are usually scaled with development capacity to define them as percentage of the theoretical upper bound on organization:

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