Complex Networks by Tomer Kalisky, Reuven Cohen, Daniel ben-Avraham, Shlomo

By Tomer Kalisky, Reuven Cohen, Daniel ben-Avraham, Shlomo Havlin (auth.), Eli Ben-Naim, Hans Frauenfelder, Zoltan Toroczkai (eds.)

This quantity is dedicated to purposes of the concepts of statistical physics to the characterization and modeling of complicated networks. the 1st components of the ebook obstacle concept and modeling of networks, whereas the final components survey functions to a large choice of ordinary and synthetic networks. the academic reports that include this e-book are meant for college kids and beginners to the sector, and so they represent a latest and finished reference for specialists. To this goal, all contributions were conscientiously peer-reviewed not just for medical content material but additionally for self-consistency and clarity.

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Mokryn and Y. Shavitt, IEEE INFOCOM’03 (2003). 49. -I. Goh, B. Kahng and D. Kim, Phys. Rev. Lett. 87, 278701 (2001). 50. -I. Goh, E. Oh, H. Jeong, B. Kahng and D. Kim Proc. Natl. Acad. Sci. USA 99, 12583 (2002). 51. R. Pastor-Sattoras and A. Vespignani, Phys. Rev. E. 63, 066117 (2001). 52. R. Pastor-Satorras, and A. Vespignani Phys. Rev. E 65, 036104 (2002). 53. Z. -L. Barabasi, Phys. Rev. E 65, 055103 (R) (2002). 54. R. Cohen, D. ben-Avraham and S. Havlin, Phys. Rev. Lett. 91, 247901 (2003). Spectral Analysis of Random Networks Sergei N.

The authors wish to thank Danny Dolev, Osnat Mokryn, Yuval Shavitt and Keren Erez for useful discussions. References 1. R. Albert, H. Jeong, and A. L. Barab´ asi, Nature, 406, 6794, 378 (2000). 2. R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. 85, 4626 (2000). 3. D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. Lett. 85, 5468 (2000).. 4. R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. 86, 3682 (2001). 5. M. E. J. Newman, S. H. Strogatz, and D.

The cutoff determines the upper boundary of eigenvalues: λ < 1/2 kcut . This result agrees with an estimation of the largest eigenvalue of sparse random graphs obtained in [34]. 7 Spectrum of a Transition Matrix We will consider now the case of random walks on a graph with the transition probability 1/kv of moving from a vertex v to any one of its neighbors. The transition matrix P then satisfies, Pw,v = aw,v /kv . (30) For each vertex v, Pw,v obeys the normalization condition: P is related with the Laplacian of the graph by, Lv,w = 1√ if v = u , −av,w / kv kw otherwise w Pw,v = 1.

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