Small-world network

Several extensions of this statistical formalism are possible and would allow taking into account more physiological and neuroanatomic features. For example, most of neural network models consider that every neuron receives exactly the same input current from the electrode, however in the case of DBS neurons do not receive the same current depending of their position with respect to the electrode [12]. This consideration is the first element in favor of adding a spatial state variable to the population density. Another example that goes in favor of this idea is the fact that peculiar features of small-world networks should be included. This class of complex networks was recently formalized [66] and some recent studies... 

Modolo J., Garenne A., Henry J., Beuter A. Probabilistic model of the subthalamic nucleus with small-world networks properties (XXVIIIth Symposium of Computational Neuroscience, Montreal (Canada), 2006). 

Watts D.)., Strogatz S.H. Collective dynamics of small-world networks. Nature, 1998, 393,440-442. 

It is envisaged that most complex naturally occurring networks are driven by small world networks. Thus, the model developed and implemented in this work is inspired from Boolean networks (Thomas and Kauffman, 2001a 2001b Silvescu and Honavar, 2001 Albert, 2004), weight matrices (Weaver et al, 1999 and Ando and Iba, 2001) and small world phenomenon (Milgram, 1967 Watts and Strogatz, 1998 ... 

The model ensures a set of simple non-dominated networks (Pareto optimal) while minimizing the numbers of indirect links in the networks. The total number of indirect links is minimized to keep the network sparse (i.e., to keep the network simple). The model also maximizes the percentage difference between clustering coefficients to maintain one of the small world network (SWN) characteristic and minimizes the sum of squared errors to fit the experimental data. One of the constraints is included to ensure the minimum connectivity in all non dominated networks. The other constraint is included to make the L(avgshortestpath) nearly equal to log -if so that SWN characteristic can be ensured. Mathematically, the model (Ganta, 2007) is ... 

Elettreby M. F. (2005). Multiobjective bak-sneppen model on a small-world network. Chaos Solitons and Fractals. 26, pp 1009-1017. 

Newman M. E. J., Moore C. and Watts D. J. (2000). Mean-field solution of the small-world network model. Physical Review Letters. 84, pp 3201-3204. 

Large systems. If two oscillators can adjust their rhythms due to an interaction, then one can expect similar behavior in large populations of units. Among different models under investigation we outline regular oscillator lattices (which in the limit case provide a description of extended systems) [9], ensembles of globally (all-to-all) coupled elements, the main topic of the present contribution, random oscillator networks (small world networks, scale-free networks, etc), and spatially-extended systems [10, 44, 55]. 

Amaral, L. A., Scala, A., Barthelemy, M., Stanley, H. E. (2000). Classes of small-world networks. Proceeding of the National Academy of Sciences of the United States of America, 97, 11149—11152. 

Latora V. and Maichiori M., Efficient Behavior of Small-World Networks, Physical Review Letters, Vol. 87, N. 19, 2001. 

In a previous work (Gad Balev, 2008a), we have studied the two properties of SSE-IN related to the small-world model, namely their 1/Lrg and C/Crg ratios. Our results show (see Fig. 21) that nearly 60% of the proteins have SSE-IN consisting of between 100 and 500 amino acids. The small-world properties are satisfied mainly when the size of the network does not exceed 500 amino acids and there are about 15.3% small-world networks among all SSE-IN. Fig. 4 explains the reason for this low rate. One can see that although highly clustered, most SSE-IN do not satisfy the first small-world property. 

Fig. 21. Size distribution of proteins SSE-IN and small-world networks ratio.

In very dilute solutions, long flexible macromolecifles hardly see each other. It is then reasonable, at least as a first step, to consider each chain separately. Now, given the flexibihty of the chain and the stochastic featiues involved, monomers that are distant from one another along the chain s backbone may get to be close to each other in space. Such monomer pairs can be then chemically cross-linked by means of, say, irradiation. The polymer structure obtained by cross-linking in this fashion represents a reahzation of a so-called small-world network (SWN) [129-135]. 

Watts, J. and S. Strogatz, 1998. Coll, dynamics of small-world networks. Nat 393,440-442. 

Amaral LAN, Scala A, Barthelemy M, Stanley HE (2000) Classes of small-world networks. Proc Nat Acad Sci U S A 97 11149-11152... 

Very often in the literature, social networks are confused with small-world networks. After studying real social networks, we identified three categories. 

Small-world networks have been studied extensively since they were first described in Watts and Strogatz [3], who defined them as graphs with three properties power-law degree distribution, high clustering coefficient and small average shortest path. They are locally dense (sparse with dense sub-graphs.)... 

Ghoniem at al. [1] showed that NL are very poor at visualizing almost complete graphs and more generally dense networks. Conversely, NL are very well suited to visualizing almost trees networks. Since Small-World networks are locally dense, NL readability is questionable. 

Auber, D., Chiricota, Y., Jourdan, F., Melancon, G. Multiscale visualization of small world networks. In Proceedings of the IEEE Symposium on Information Visuahzation, pp. 75-81 (2003)... 

Create novel visualizations designed for locally dense networks (small-world networks) as well as associated interaction techniques ... 

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