First structural vector Q, eccentricities and measure of complexity

Statements formulated in the previous part allow the proposal of a variant of radical modification (changes) of structural complexity estimation Ψ(K), which is based on results of the Polyhedral Dynamics procedure. Speaking about simplicial complex K, we imply that this corresponds to complex as well. A total number of vector Q elements is equal to (dimK+1) — they can be denoted as , where ,

Q= dim K, is the number of connectivity components at the dimensional level q — if two simplices are connected by a q-chain, then they are also (q-1)-,…,1-, 0-connected in complex K.

According to the definition, a simplicial complex K is a collection of simplices and their faces, therefore the local properties of individual simplices in the analysis of a system’s structure are also important. In order to specify a measure of simplex’s integrity in the complex, as well as to define its role in connecting the other simplices (grouping them together in a sense of the Polyhedral Dynamics procedure) at different dimensional levels, an eccentricity characteristic was proposed. The formula for calculating eccentricity of simplex

ecc(σ) =

requires the dimensionality of simplex σ dim(σ) and the largest value of q for which simplex σis connected to some other simplex in complex K(q). If simplex σ gets into the same connectivity component with at least one other simplex at dimensional level q=dim(σ), then ecc(σ)=0, and this simplex is classified as «well (or strongly) integrated» in the complex K. Assuming all simplices to form one connectivity component at the dimensional level -1 (model K represents a system), a simplexσ isolated at the level q = 0 is characterized by the infinite value (∞) of eccentricity since q= −1.

Vector Q can be considered as a global characteristic of simplicial complex K, but it does not characterize the complex uniquely. The latter means that two different complexes may have the same vector Q (Earl 1981), although a distinction between complexes can be revealed by studying chains of connectivity. At the same time, the axiomatically introduced estimation ΨK is calculated exactly on the basis of values as follows:

, N=dimK

If the results of Q-analysis for two different simplicial complexesare the same, then the numerical estimations Ψ()and Ψ() are equal as well. In this case, the fact that estimations coincide should be regarded as a minor drawback, because the main problem is concealed in coincidence of first structural vectors Q (as well as in some cases), and hence it results in «identification» of structures under study. For example, assume matrix representations of two arbitrary relations λ≤ and – Figure 1 shows matrices A and B in which elements of sets and denote the rows. Equivalent geometric forms of corresponding complexes ( and ( (in a short notation, and ) are outwardly very similar, the only exception being that 0-simplex in complex is a face of simplex , while in the complex one 0-simplex is a face of .

2. Make a written translation of the following passage.