Robert E. Ulanowicz (2002) The balance between adaptability and adaptation. BioSystems. paper
What is Ascendency?
Ascendency is a measure of "organization", "non-randomness", or "constraints" within a system, which consists of components and exchange/transport/flow of something between components.Information in a system with components and flows
The quantity measured as Shannon entropy, H, here is the uncertainty (randomness = equal probability) in functions that a system can possibly operate. For example, in the system below, the flow between all the pairs of components are equal. This can be interpreted as the options available for system reconfiguration (the combination of edges activated, which occurs according to the probability proportional to the flow on each edge) are equally likely. (p = 10/12 for each edge to be activated, and activation of each every different edge is associated with a unique function of the system.)Therefore, a larger uncertainty (H) indicates the activation of edges are closer to equal, and hence the options of functions that the system can randomly pick are more variable, "diverse functions and flexibility of system".
How to measure ascendency
Shannon entropy seems to be a good measure of the diversity of system functions. However, there is a chance that not all of the diversity is available; there may some innate constraints between activations of edges due to the dependence between the random variables, p_i. The average reduction in uncertainty of a random variable given knowledge of another random variable can be measured by mutual information, I. This reduction in uncertainty essentially measures the constraints in the system, i.e., organization or non-randomness. Therefore, this mutual information should correspond to Ascendency. Practically, ascendency is(Ascendency) = (total flow in the system) ⨉ (mutual information between flows*)
*mutual information when flow/(total flow) for each edge is considered as a random variable. In the above network, there are 12 random variables.
Mathematical extension to incorporate the weights on nodes (or biomass in the ecosystem) is available in this link, and an extension to open systems is in this paper.
Examples of applying ascendency
A measure of the developmental status of an ecosystem: applicable to both cases, the same ecosystem between different time points, or different ecosystems.Adaptability/Ecological persistence: With H and I, the true diversity/uncertainty/randomness available for the system (adaptability), 𝜱, is 𝜱 = H - I.
By using ascendancy with biomass, one can investigate which element (resource flowing from node i to j) is limiting the activity of certain taxa (biomass) by looking at the sensitivity of ascendancy to the flow.
