Second moment

The formulas for higher moments are more interesting. If R is a certain product of matrices, G the template for admissible boundary conditions, then the square of the number of ancestors represented by R is

However the relation

( designates the tensor product [12]), yields

Identities for the tensor product allow transforming this expression into

from which the first term can often be extracted as a constant factor from a sum of traces. If the sum in Eq. 15 is now carried out for all values of R, it is possible to recognize the expansion of

 

This sum has to be evaluated in each particular case; the simple closed form of the general average was due to an algebraic identity which does not apply to the higher moments. Nevertheless, even when it has to be analyzed numerically, it constitutes one single matrix encompassing the behavior of all the ancestors. Particularly interesting is the fact that it is just the connectivity matrix of the pair diagram introduced in Sec 3.4.

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Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx