Network geometry

Once the network geometry and the type of activation function have been chosen, the network will be ready to use as soon as the connection weights have been determined, which requires a period of training. 

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. 

The water cluster examples in Table 6 are meant to illustrate a somewhat more difficult minimization problem. Optimal cluster-network geometries are more difficult to locate not only because of the difficulty in constructing good starting points the optimal configurations are dominated by long-range nonbonded forces that are computationally not feasible to consider in preconditioners. 

An alternate approach to the analysis of the network geometry from the viewpoint of multiplicity and stability is due to Beretta and his coworkers (1979, 1981) the latter approach is based on knot theory. The Schlosser-Feinberg theory reproduces some of the Beretta-type theory results concerning stability from a somewhat different viewpoint. 

Fig. 23.5. Controlling enzymatic reactions by network geometry. Fluorescent micrographs (A-C, E-G, I-K, M-N) and normalized fluorescence intensities vs. time (D, H, L, and O), representing product formation from an enzymatic reaction in networks with different geometries. Initially, the enzyme-filled vesicles are vesicle 1 in images A-C, E-G, and I-K and vesicles 1 and 5 in M-N. All other vesicles are substrate filled. The broken lines in graphs D, H, L, and O show the theoretical fit to the experimentally measured product formation. The scale bars in the images represent fOpm. Pictures are reproduced with permission [26]...

Fig. 23.7. Dynamics of an enzymatic reaction in lipid nanotube networks with variable topology numeric calculations (bottom)/fluorescence intensity of the reaction product (top) vs. time for three differently chosen network geometries, (a) Reference experiment a static four-vesicle network. The product concentration displays a cascade-like behavior in time and space, (b) Linear-to-circular topology change in the four-vesicle network (c) A model study of the effect of product inhibition as the linear four-vesicle network (top panel) undergoes the same change in structure (bottom panel) as the network in the reference experiment ([28], reprinted with permission)...

Abstract A methodology for quantifying the contributions of hydro-mechanical processes to fractured rock hydraulic property distributions has been developed and tested. Simulations have been carried out on discrete fracture networks to study the sensitivity of hydraulic properties to mechanical properties, stress changes with depth, mechanical boundary conditions, initial mechanical apertures and fracture network geometry. The results indicate that the most important (and uncertain) parameters for modelling HM processes in fractured rock are fracture density and rock/fracture mechanical properties. Aperture variation with depth below ground surface is found to be of second order importance. 

Input of the fracture network geometry into 3DEC... 

A specific command of RESOBLOK is used to generate a 3DEC file including all information regarding fracture network geometry. 

The complex geometry of flow padis in fractured rock results, primarily, fivm rock discontinuities that are present on aU scales, extending from the microscale of microfissures (ammig the mineral components of the rock) to the macroscale of various types of joints and fruits (29, 30). The complexity of the fracture-network geometry can cause either divergence or convergence of localized and nonuniform flow paths in different parts of fractured media, as well as cq>illary barrier effects at the intersection of flow paths. 

Typical input data for a moment tensor inversion consist of the network geometry (coordinates of the sensors) knowledge of sensor polarity (i.e. whether an upwards deflection at the sensor indicates a compression or dilatation) sensor orientation source coordinates P and/or S-wave displacement amplitudes recorded at each sensor (time-domain inversion) or P and/or S-wave spectral amplitudes (frequency-domain inversion) and the polarities of the wave phases. 

Like the other imbibition methods, SII is dependent on both surface chemistry and pore network geometry. It works best for rocks that imbibe only one fluid, but could be modified to a form like the Amott-Harvey RDI. It also has the potential, by its formulation, to evaluate wettability alteration by surfactants in reservoirs. 

A simple extension of transition metal or metal cluster coordination geometry by exploiting linear bifunctional ligands. In such cases the metal moiety acts as a node that defines the overall network geometry of the coordination polymer. Such a strategy would be expected to afford any of the architectures illustrated in Fig. 1. Indeed, all of these motifs have been realized in recent years. 

Most percolation studies rely on Monte Carlo simulations, whose properties of interest are critical cluster size, bond densities (i.e. number of bonds that lead to network per site), particle volume in the matrix, particle orientation and network geometry at the threshold. In most cases simulations treat the percolation in a statistical manner which means that the particles are randomly distributed in the matrix and network pathways are formed simply by increasing the particle volume fraction in the composite. Although such an approach has merit and provides valuable insight on the network formation, it is far away from reality, especially when polymers are concerned. Addition of particles in a polymer matrix is mainly performed via solution or melt mixing, which means that both particles and polymer chains are in motion and interact with each other. More advanced theoretical approaches do take into consideration the thermodynamic interactiOTis between the composite constituents (particle-particle, particle-polymer and... 

Robustness Parameter insensitivity. External perturbations, such as the effects of thermal disorder, or modifications to network geometry have little effect on the overall efficiency of the light harvesting process. 

Optimality of excitation transfer network. Even though natural light harvesting systems appear to be optimized in terms of the details of their network geometry, this is probably not a high priority constraint for artificial light harvesting systems. 

Benkstein KD, KopidaMs N, van de Lagemaat J, Frank AJ (2003) Influence of the percolation network geometry on electron transport in dye-sensitized titanium dioxide solar cells. J Phys Chem B 107 7759-7767... 

The error budget in seismic location problems is traditionally described as the combination of measurement and model errors. The effect of errors due to ignoring the higher-order terms in the Taylor expansion in Geiger s method, except for some highly degenerate network geometries, is typically second order compared to the model and measurement errors. 

---