Topology of networks

Graph theory Describes topology of networks and subnetworks, based on quantification of the number of nodes (signaling components) and links between them. Dynamic properties of networks through Boolean analysis. Network analysis based on probabilities (Markov chain and Bayesian) to identify paths and relationships between different nodes in the network. (75-81)... 

The interested reader can find more information on the topology of networks in a recent review, with more details on what we now know about nets [33],... 

W. Scheider [1975] Theory of the Frequency Dispersion of Electrode Polarization Topology of Networks with Fractional Power Frequency Dependence, J. Phys. Chem. 79, 127-136. 

The Kohonen network or self-organizing map (SOM) was developed by Teuvo Kohonen [11]. It can be used to classify a set of input vectors according to their similarity. The result of such a network is usually a two-dimensional map. Thus, the Kohonen network is a method for projecting objects from a multidimensional space into a two-dimensional space. This projection keeps the topology of the multidimensional space, i.e., points which are close to one another in the multidimensional space are neighbors in the two-dimensional space as well. An advantage of this method is that the results of such a mapping can easily be visualized. 

Tt provides unsupervised (Kohonen network) and supervised (counter-propagation network) learning techniques with planar and toroidal topology of the network. 

In this approach, connectivity indices were used as the principle descriptor of the topology of the repeat unit of a polymer. The connectivity indices of various polymers were first correlated directly with the experimental data for six different physical properties. The six properties were Van der Waals volume (Vw), molar volume (V), heat capacity (Cp), solubility parameter (5), glass transition temperature Tfj, and cohesive energies ( coh) for the 45 different polymers. Available data were used to establish the dependence of these properties on the topological indices. All the experimental data for these properties were trained simultaneously in the proposed neural network model in order to develop an overall cause-effect relationship for all six properties. 

Shen W, Zhang KC, Komfield JA et al (2006) Tuning the erosion rate of artificial protein hydrogels through control of network topology. Nat Mater 5 153-158... 

A pipeline network is a collection of elements such as pipes, compressors, pumps, valves, regulators, heaters, tanks, and reservoirs interconnected in a specific way. The behavior of the network is governed by two factors (i) the specific characteristics of the elements and (ii) how the elements are connected together. The first factor is determined by the physical laws and the second by the topology of the network. 

The mathematical abstraction of the topology of a pipeline network is called a graph which consists of a set of vertices (sometimes also referred to as nodes, junctions, or points)... 

More recently, Cheng (C7) showed that to apply tearing effectively one must take into consideration the topology of the network and the nature of problem specifications. For instance, if the network consists of a number of cyclic subnetworks imbedded in an acyclic framework and if all the external flows and one reference pressure are specified, the flows and the pressures external to the cyclic subnetworks may be computed sequentially and only... 

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. 

The growing cell structure algorithm is a variant of a Kohonen network, so the GCS displays several similarities with the SOM. The most distinctive feature of the GCS is that the topology is self-adaptive, adjusting as the algorithm learns about classes in the data. So, unlike the SOM, in which the layout of nodes is regular and predefined, the GCS is not constrained in advance to a particular size of network or a certain lattice geometry. 

The finished network automatically reflects the characteristics of the data domain Not only do the network weights evolve so that they describe the data as fully as possible, but so also does the network geometry. The size of the network is not chosen in advance and as topology is determined by the algorithm and the dataset in combination, it is more likely to be appropriate than the geometry used for a SOM, especially in the hands of an inexperienced user, who might find it difficult to choose an appropriate size of network or suitable values for the adjustable parameters in the SOM. 

Equation (29) shows that the modulus is proportional to the cycle rank , and that no other structural parameters contribute to the modulus. The number of entanglements trapped in the network structure does not change the cycle rank. Possible contributions of these trapped entanglements to the modulus therefore cannot originate from the topology of the phantom network. 

We have investigated the static and dynamic mechanical properties of networks of different chemical and topological structures ( 19,20). In a previous paper, we reported results obtained on networks with crosslink functionality four (21). In the present study, we investigated the effect of the structure of junctions on the mechanical behaviour of PDMS. Rather uncommon networks with comb-like crosslinks were employed, intending that these would be most challenging to theoretical predictions. 

Dl) A molecular unit is an aggregate of atoms that is linked by a topologically connected network of covalent bonds equivalently, an electronic distribution that links a collection of nuclei by a contiguous network of covalent bonds. 

A considerable improvement over purely graph-based approaches is the analysis of metabolic networks in terms of their stoichiometric matrix. Stoichiometric analysis has a long history in chemical and biochemical sciences [59 62], considerably pre-dating the recent interest in the topology of large-scale cellular networks. In particular, the stoichiometry of a metabolic network is often available, even when detailed information about kinetic parameters or rate equations is lacking. Exploiting the flux balance equation, stoichiometric analysis makes explicit use of the specific structural properties of metabolic networks and allows us to put constraints on the functional capabilities of metabolic networks [61,63 69]. 

---