Modelling topological network

The description of a network structure is based on such parameters as chemical crosslink density and functionality, average chain length between crosslinks and length distribution of these chains, concentration of elastically active chains and structural defects like unreacted ends and elastically inactive cycles. However, many properties of a network depend not only on the above-mentioned characteristics but also on the order of the chemical crosslink connection — the network topology. So, the complete description of a network structure should include all these parameters. It is difficult to measure many of these characteristics experimentally and we must have an appropriate theory which could describe all these structural parameters on the basis of a physical model of network formation. At present, there are only two types of theoretical approaches which can describe the growth of network structures up to late post-gel stages of cure. One is based on tree-like models as developed by Dusek7 I0-26,1 The other uses computer-simulation of network structure on a lattice this model was developed by Topolkaraev, Berlin, Oshmyan 9,3l) (a review of the theoretical models may be found in Ref.7) and in this volume by Dusek). Both approaches are statistical and correlate well with experiments 6,7 9 10 13,26,31). They differ mainly mathematically. However, each of them emphasizes some different details of a network structure. 

In this approach only static topological properties of networks can be investigated such as the number and length of the paths between two molecules, the number and length of cycles, the enumeration of occurrences of particular subnetwork structures, and how those properties scale with the size of the network. This type of modelling we shall refer to as topological network modelling. 

As the first example, we shall discuss two notions of modules often used in topological models of networks (Milo et al. 2002 Ravasz et al. 2002 Itzkovitz et al. 2003 Clauset et al. 2004 Newman 2006). The first notion is that of a cluster on the basis of the nodes within the cluster having more interactions among themselves than with nodes outside of the cluster. Clusters can be quantitatively defined on the basis of a cluster coefficient. Escherichia coli has been shown to be clustered in a hierarchical fashion (Ravasz et al. 2002). Another definition is that of a network motif (Mho et al. 2002 Shen-Orr et al. 2002 Kashtan et al. 2004 Yeger-Lotem et al. 2004). These authors... 

The 802.3 CSMA/CD model defines a bus topology network that uses a 50-ohm coaxial baseband cable and carries transmissions at 10Mbps. This standard groups data bits into frames and uses the Carrier Sense Multiple Access with Collision Detection (CSMA/CD) cable access method to put data on the cable. 

The model of network polymers considered next describes quantitatively the formation of structure (and, hence, properties) and is based on fundamental principles of physics. The initial parameters used for crosslinked systems are the v value (topological characteristics) and some generally accepted molecular parameters. Since the chemical crosslinking processes are complex, the model makes use of the simplest scheme of structure formation, which does not take into account, for example, the decrease in the rate of structure formation with time when one structure element starts to influence the conditions of formation of an adjacent element. An example of taking account of effects of this type has been reported [117]. Although this scheme is simple, it provides a quantitative description of the processes considered and the structures formed. 

A considerable number of experimental studies, as well as theoretical developments, have been done on the equilibrium elastic properties of regular model silicone networks in absence of pendant chains. The goal of most of these studies has been to test quantitatively the molecular basis of the theory of rubber elasticity. One of the major concerns has been the influence of topological interactions between chains on elastic properties of the networks. However, despite the considerable amount of experimental work, there is still considerable debate concerning the validity and applicability of different models. 

Recently, several QSPR solubility prediction models based on a fairly large and diverse data set were generated. Huuskonen developed the models using MLRA and back-propagation neural networks (BPG) on a data set of 1297 diverse compoimds [22]. The compounds were described by 24 atom-type E-state indices and six other topological indices. For the 413 compoimds in the test set, MLRA gave = 0.88 and s = 0.71 and neural network provided... 

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. 

We shall start out with elementary general topological considerations of flow by studying network flow. We shall follow this by a variety of models from operations research that illustrate analytical methods and problems. No illustrations of statistical methods will be given here because statistics, a fundamental tool of science, is abundantly discussed in the literature of science. 

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