Mathematical Foundations of Complex Networked Information by P.R. Kumar, Martin J. Wainwright, Riccardo Zecchina, Fabio

By P.R. Kumar, Martin J. Wainwright, Riccardo Zecchina, Fabio Fagnani, Sophie M. Fosson, Chiara Ravazzi

Introducing the reader to the maths past complicated networked structures, those lecture notes examine graph thought, graphical types, and techniques from statistical physics. complicated networked structures play a basic function in our society, either in way of life and in medical study, with purposes starting from physics and biology to economics and finance.

The booklet is self-contained, and calls for basically an undergraduate mathematical background.

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Extra resources for Mathematical Foundations of Complex Networked Information Systems: Politecnico di Torino, Verrès, Italy 2009

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Müuller, On the giant component of random hyperbolic graphs, in Proceedings of the 7th European Conference on Combinatorics, Graph Theory and Applications, EUROCOMB’13 (2013), pp. 425–429 3. B. Bollobás, Random Graphs. Cambridge Studies in Advanced Mathematics, vol. 73, 2nd edn. (Cambridge University Press, Cambridge, 2001) 4. S. Chatterjee, P. Diaconis, A. Sly, Random graphs with a given degree sequence. Ann. Appl. Probab. 21(4), 1215–1625 (2011) 5. R. Durrett, Random Graph Dynamics. Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, New York, 2007) 6.

Therefore, the quantity ! c/ N (9) should vanish for c D c . / and be strictly negative otherwise. In the following, we shall compute ! c/ and thus obtain information not only on the typical number of clusters but also on the large deviations. 32 C. Baldassi et al. q/ D max c log q C ! c/ : (11) 0ÄcÄ1 fQ and ! are conjugated Legendre transforms and it will turn out that a direct computation of fQ is easier. 1 The Potts Model Representation We proceed by computing the properties of random graphs by using a mapping of the generating function of the cluster probability to the so-called Potts model.

JCx j 2. If < c, Ângj D 1/ ! 1 as n ! max jCx j x2Vn ˇn/ ! 0 as n ! 1: In Fig. 0; 10/ averaged over 50 independent samples of the graph. Let c be the lowest connectivity at which the fraction of vertices in the largest cluster is greater than 0 in the macroscopic limit. It should be noted that the critical value is sharply defined and is around c D 4:52. The proof of Theorem 5 requires considerable effort (see [16, Chapters 10–11]). Some simpler proofs have further been provided for subcases of these results.

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