embedding or entrainment of newly received inputs into the established flow-stream of the system is reflected in infinite series. Let f21/T1 = g21 define a throughflow-specific dimensionless flow intensity. In effect g21 is a probability, thus its powers form a convergent infinite series: (1 + g21 + g(221)+-----g(m1 + —). The parenthesized superscripts denote coefficients derived from matrix, not scalar, multiplication. This power series maps the boundary input z1 into the portion of throughflow at 2 contributed by this source: T21 = (1 + g21 + g (221) + —h g 21l)+ l) z1. The first term of the series brings the input into the system: 1-z1. The second term represents the "direct" flow over the link of length 1: g21z1. All other terms represent indirect flows associated with pathways of all lengths 2, 3,..., m,... as m — <». The throughflow component T21 accordingly contains a plethora of indirect flows: (g 221)+ —l- g If + —)z. This is one of three elements in the throughflow at compartment 2: T2 = T21 + T22 + T23. At compartment 1 the throughflow is similarly decomposable: T1 = T11 + T12 + T13, and at compartment 3: T3 = T31 + T32 + T33. Each term in these sums has a similar infinite series decomposition to that just given for T21. With this, one can now appreciate there is more than that meets the eye in Figure 5.3. The focal flow f21 in question has a decomposition into enfolded elements as follows:
=g2i [(i+gii++L+gir)+l) zi+(i+gi2+gi(2)+L+gm+-)z2 + (i + gi3 + gi(3) + L + gi(3m) + l) Z3 ]
This is what an ecologist measuring f21 would measure empirically and consider a "direct" flow. One can see, however, that the entire system is embodied in this measurement. This is network enfolding. It gives a strong message about the inherent holism one can expect to be expressed in natural systems and, as stated above, its broad realization is likely to influence many areas of ecology.
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