Diameter of the Spike-Flow Graphs of Geometrical Neural Networks
Average path length is recognised as one of the vital characteristics of random graphs and complex networks. Despite a rather sparse structure, some cases were reported to have a relatively short lengths between every pair of nodes, making the whole network available in just several hops. This small-worldliness was reported in metabolic, social or linguistic networks and recently in the Internet. In this paper we present results concerning path length distribution and the diameter of the spike-flow graph obtained from dynamics of geometrically embedded neural networks. Numerical results confirm both short diameter and average path length of resulting activity graph. In addition to numerical results, we also discuss means of running simulations in a concurrent environment.
Keywordsgeometrical neural networks path length distribution graph diameter small-worldliness
Unable to display preview. Download preview PDF.
- 1.Albert, R., Jeong, H., Barabasi, A.L.: Diameter of the World-Wide Web. Nature 401 (September 9, 1999)Google Scholar
- 2.Albert, R., Barabasi, A.L.: Statistical mechanics of complex networks. Reviews of Modern Physics 74 (January 2002)Google Scholar
- 3.Bassett, D.S., Bullmore, E.: Small-World Brain Networks. The Neuroscientist 12(6) (2006)Google Scholar
- 4.Bullmore, E., Sporns, O.: Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews, Neuroscience 10 (March 2009)Google Scholar
- 5.Chung, F., Lu, L.: Complex graphs and networks. In: Conference Board of the Mathematical Sciences. American Mathematical Society (2006)Google Scholar
- 6.Csermely, P.: Weak links: the universal key to the stability of networks and complex systems. Springer, Heidelberg (2009)Google Scholar
- 8.Piekniewski, F.: Spontaneous scale-free structures in spike flow graphs for recurrent neural networks. Ph.D. dissertation, Warsaw University, Warsaw, Poland (2008)Google Scholar
- 11.Piersa, J., Piekniewski, F., Schreiber, T.: Theoretical model for mesoscopic-level scale-free self-organization of functional brain networks. IEEE Transactions on Neural Networks 21(11) (November 2010)Google Scholar
- 12.Piersa, J., Schreiber, T.: Scale-free degree distribution in information-flow graphs of geometrical neural networks. Simulations in concurren environment (in Polish). Accepted for Mathematical Methods in Modeling and Analysis of Concurrent Systems — Postproceedings, Poland (July 2010)Google Scholar
- 13.Schreiber, T.: Spectra of winner-take-all stochastic neural networks, arXiv 3193(0810), pp. 1–21 (October 2008), http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.3193v2.pdf