Topological Classification of Networks

First of all we observe that the network (6.26) contains an A2-element as integrated circuit for the autocatalytic production of X. In addition to this autocatalytic loop there is a second loop formed by the reaction R2. From the topological point of view, there is a strikingly close resemblance between the Brusselator in (6.26) and our autocatalytic excitation model in (6.7). Despite this topological resemblance, the two networks behave fundamentally differently, namely multiple steady states in (6.7) and a limit cycle in (6.26). This shows that a purely network topological classification of types of models will find its limits or is at least a pretentious program. 

From the structural viewpoint there is much to commend the classification of problems based on the topology of the pipeline network— single branch pipelines, tree networks, and cyclic networks. However, since some methods are applicable to more than one category, rigorous adherence to this classification will lead to unnecessary duplication and overlaps. 

Edwards (300) was able to make some progress on the problem by showing that topological classification can serve only to raise the modulus. As a simple example, consider all strand pairs in a network which have, within some small tolerance, a specified set of internal junction coordinates. Suppose there are B such pairs, and that the strands of each pair, labeled 1 and 2, have co1 and co2 distinguishable configurations each as free strands, and fractions (gt)0 and (g2)o respectively which have the end-to-end distances specified by the equilibrium junction coordinates. If the crosslinks were formed in the system at equilibrium, then the total number of configurations for each strand of the pair is cj1(g1)0 and o)2(02h> and the number available to the pair is (o1(o2(gl)0(g2)0. 

Therefore, the principal difficulty connected with the application of Eq. (12) is due to the incompleteness of the Gauss invariant. So, the use of the Gauss invariant for adequate classification of topologically different states in many-chain systems is very problematic. Nevertheless, that approach was used repeatedly for consideration of such physically important question as the high-elasticity of polymer networks with topological constraints [15]. Unfortunately,... 

To date, two different OCR methods were implemented in CLiDE. The first one used a back-propagation neural network for classification of the characters. The character features used as input to the neural network are determined by template matching (Venczel 1993). The second OCR implementation in CLiDE is based on topological and geometrical feature analysis, and it uses a filtering technique for the classification of characters (Simon 1996). 

FIGURE 4.15 Scheme of a multilayer Kohonen network including input layer, output layer, and a topological map layer for classification of input data. The dimension of input and output vectors is equal to the dimension of the corresponding neurons in the network. 

In general, it appears that the fraction of configurations in the various topological classes can be determined for models in which one of the elements is a fixed curve and the other is a random coil. The detailed calculations are intricate and difficult, however, and some simple generalizations are needed which could be used as a step towards building classification effects into the network theories. Classification for the case of two random coils and for self-entanglement are unsolved problems at the present time. 

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