4. a LM simplex

Selection of order parameters for complex systems is sometimes more of an art than a science. Here as in the selection of order-parameters for simpler (albeit still-complex) thermodynamic systems (Sethna2006), we seek a measure based on information available with minimal perturbation.

For inputs, we begin with six normalized positive numbers fi representing the fraction of an organism's effort allocated to each of the 6 subsystem correlation-layers i.e. which look in/out from skin, family and culture. For visualization-purposes these six positive normalized fi values allow us to map the layer-focus of organisms to individual points within the equilateral 5-simplex between unit-vertices in 6-space (cf. Fig. \ref{Fig2}), just as ternary-diagrams map any three normalized positive-numbers onto an equilateral triangle or 2-simplex in 3-space.

To inventory order we then define a single metazoan-individual's niche-network layer-multiplicity m as the behavior-defined effective-number of correlation-buffering choices, expressed as an entropy-exponential in terms of that organism's set of 6 fractional-attention values {f}: 

where Σi=1,6fi = 1 i.e. sums to one over the level-index i=1,6.

This multiplicity measure can also be expressed in terms of the number of bits of surprisal or state-uncertainty S in bits about which correlation layer (e.g. self, friends, family, job, culture, profession) they are working on at any given time, i.e. S = ln2[m] = Σi=1,6filn2[1/fi]. However use of #choices instead of #bits probably makes more sense here since the numbers are so small.

Population-averages i.e. normalized-sums over all N community members (say using index j=1,N) will be denoted with angle-brackets like ⟨ ⟩. Thus the population-average individual-multiplicity is ⟨m⟩ = (1/N)Σj=1,Nmj. The population-average value for attention-fraction fi is ⟨fi⟩ = (1/N)Σj=1,Nfij where fij is the jth individual's layer-i attention-fraction.

We'll use {⟨f⟩} to refer to the set of all i=1,6 attention-fraction population-averages. This allows us to define a center-of-mass multiplicity Mcm = Πi=1,6(1/⟨fi⟩)⟨fi⟩, representing the spread in attention-focus for the community as a whole.

We may also want to consider population average-surprisal or entropy ⟨S⟩ = (1/N)Σj=1,NSj. This leads simply to the geometric-average individual-multiplicity, defined as Mgeom = 2⟨S⟩ = (Πj=1,Nmj)1/N for which it is easy to show that Mgeom ≤ Mcm. Because of this organic relation to the center-of-mass value, we'll use Mgeom as our indicator of the spread in attention-focus for individual organisms with the community. 

Finally, this inequality also lets us define organism and community specialization-indices, whose logarithms are KL-divergences, which decrease in value toward 1 only as the spread of individual foci begins to match that of the community as a whole. For the community specialization index R, we use 1 ≤ R ≡ Mcm/Mgeom ≤ Mcm. For the jth individual organism the corresponding specialization index rj obeys 1 ≤ R ≡ mj*/mj ≤ N, where the individual center-of-mass multiplicity is defined as mj* ≡ Πi=1,6(1/⟨fi⟩)fij. 

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