Abstract
Dynamical and adaptive networks are the backbone of many complex systems. Examples range from ecological prey–predator networks to the gene expression and protein providing the grounding of all living creatures The brain is probably the most complex of all adaptive dynamical systems and is at the basis of our own identity, in the form of a highly sophisticated neural network. On a social level we interact through social and technical networks like the Internet. Networks are ubiquitous through the domain of all living creatures. A good understanding of network theory is therefore of basic importance for complex system theory. In this chapter we will discuss the most important notions of graph theory, like clustering and degree distributions, together with basic network realizations. Central concepts like percolation, the robustness of networks with regard to failure and attacks, and the “rich-get-richer” phenomenon in evolving social networks will be treated.
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- 1.
The reader without prior experience with Green’s functions may skip the following derivation and pass directly to the result, namely to Eq. (1.15).
- 2.
Taking the principal part signifies that one has to consider the positive and the negative contributions to the 1∕λ divergences carefully.
References
Albert, R., Barabási, A.-L. 2002 Statistical mechanics of complex networks. Review of Modern Physics 74, 47–97.
Albert, R., Jeong, H., Barabási, A.-L. 1999 Diameter of the world-wide web. Nature 401, 130–131.
Barabasi, A.L., Albert, R., Jeong, H. 1999 Mean-field theory for scale-free random networks. Physica A 272, 173–187.
Brinkman, W.F., Rice, T.M. 1970 Single-particle excitations in magnetic insulators. Physical Review B 2, 1324–1338.
Caldarelli, G. 2007 Scale-Free Networks: Complex Webs in Nature and Technology. Oxford University Press, Oxford.
Capocci, A. et al. 2006 Preferential attachment in the growth of social networks: The internet encyclopedia Wikipedia. Physical Review E 74, 036116.
Derenyi, I., Palla, G., Vicsek, T. 2005 Clique percolation in random networks. Physical Review Letters 94, 160202.
Dorogovtsev, S.N., Mendes, J.F.F. 2002 Evolution of networks. Advances in Physics 51, 1079–1187.
Dorogovtsev, S.N., Mendes, J.F.F. 2003 Evolution of Networks. From Biological Nets to the Internet and WWW. Oxford University Press, Oxford.
Erdös, P., Rényi, A. 1959 On random graphs. Publications Mathematicae 6, 290–297.
Guare, J. 1990 Six Degrees of Separation: A play. Vintage, New York.
Milgram, S. 1967 The small world problem. Psychology Today 2, 60–67.
Newman, M.E.J. 2002a Random Graphs as Models of Networks. http://arxiv.org/abs/cond-mat/0202208.
Newman, M.E.J. 2002b Assortative mixing in networks. Physical Review Letters 89, 208701.
Newman, M.E.J., Strogatz, S.H., Watts, D.J. 2001 Random graphs with arbitrary degree distributions and their applications. Physical Review E 64, 026118.
Newman, M.E.J., Watts, D.J. 1999 Renormalization group analysis of the small world network model. Physics Letters A 263, 341–346.
Palla, G., Derenyi, I., Farkas, I., Vicsek, T. 2005 Uncovering the overlapping community structure of complex networks in nature and society. Nature 435, 814–818.
Watts, D.J. 1999 Small Worlds: The Dynamics of Networks Between Order and Randomness. Princeton University Press, Princeton.
Watts, D.J., Strogatz, S.H. 1998 Collective dynamics of small world networks. Nature 393, 440–442.
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Gros, C. (2015). Graph Theory and Small-World Networks. In: Complex and Adaptive Dynamical Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-16265-2_1
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DOI: https://doi.org/10.1007/978-3-319-16265-2_1
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