Abstract
Dynamical networks constitute a very wide class of complex and adaptive systems. Examples range from ecological prey–predator networks to the gene expression and protein networks constituting the basis of all living creatures as we know it. 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 sophisticated neural network. On a social level we interact through social networks, to give a further example – networks are ubiquitous through the domain of all living creatures.
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Notes
- 1.
Mathematicians generally prefer the somewhat more abstract term “graph” instead of “network”.
- 2.
The reader without prior experience with Green’s functions may skip the following derivation and pass directly to the result, namely to Eq. (1.13).
- 3.
Taking the principal part signifies that one has to consider the positive and the negative contributions to the \(1/\lambda\) divergences carefully.
Further Reading
For further studies several books (Watts, 1999; Dorogovtsev and Mendes, 2003; Caldarelli, 2007) and review articles (Albert and Barabási, 2002; Dorogovtsev and Mendes, 2002) on general network theory are recommended.
The interested reader might delve into some of the original literature on, e.g. the original Watts and Strogatz(1998) small-world model, the Newman and Watts (1999) model, the mean-field solution of the preferential attachment model (Barabási et al. 1999), the formulation of the concept of clique percolation (Derenyi et al., 2005), an early study of the WWW (Albert et al., 1999), a recent study of the time evolution of the Wikipedia network (Capocci et al., 2006), a study regarding the community structure of real-world networks (Palla et al., 2005), the notion of assortative mixing in networks (Newman, 2002) or the mathematical basis of graph theory (Erdös and Rényi, 1959). A good starting point is Milgram’s (1967) account of his by now famous experiment, which led to the law of “six degrees of separation” (Guare, 1990).
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.
Moukarzel, C.F. 1999 Spreading and shortest paths in systems with sparse long-range connections. Physics Review E 60, 6263–6266.
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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Exercises
Exercises
1.1.1 Bipartite Networks
Consider \(i=1,\dots,9\) managers sitting on the boards of six companies with (1,9), (1,2,3), (4,5,9), (2,4,6,7), (2,3,6) and (4,5,6,8) being the respective board compositions. Draw the graphs for the managers and companies, by eliminating from the bipartite manager/companies graph one type of nodes. Evaluate for both networks the average degree z, the clustering coefficient C and the graph diameter D.
1.1.2 Degree Distribution
Online network databases can be found on the Internet. Write a program and evaluate for a network of your choice the degree distribution p k , the clustering coefficient C and compare it with the expression (1.27) for a generalized random net with the same p k .
1.1.3 Ensemble Fluctuations
Derive Eq. (1.7) for the distribution of ensemble fluctuations. In the case of difficulties Albert and Barabási (2002) can be consulted. Alternatively, check Eq. (1.7) numerically.
1.1.4 Self-Retracing Path Approximation
Look at Brinkman and Rice (1970) and prove Eq. (1.12). This derivation is only suitable for readers with a solid training in physics.
1.1.5 Probability Generating Functions
Prove that the variance σ 2 of a probability distribution p k with a generating functional \(G_0(x)=\sum_k p_k\,x^k\) and average \(\langle k\rangle\) is given by \(\sigma^2=G_0^{\prime\prime}(1)+\langle k\rangle -\langle k\rangle^2\). Consider now a cummulative process, compare Eq. (1.41), generated by \(G_C(x)=G_N(G_0(x))\). Calculate the mean and the variance of the cummulative process and discuss the result.
1.1.6 Clustering Coefficient
Prove Eq. (1.61) for the clustering coefficient of one-dimensional lattice graphs. Facultatively, generalize this formula to a d-dimensional lattice with links along the main axis.
1.1.7 Scale-Free Graphs
Write a program that implements preferential attachments and calculate the resulting degree distribution p k . If you are adventurous, try alternative functional dependencies for the attachment probability \(\varPi(k_i)\) instead of the linear assumption (1.63).
1.1.8 Epidemic Spreading in Scale-Free Networks
Consult “R. Pastor-Satorras and A. Vespigiani, Epidemic spreading in scale-free networks, Physical Review Letters, Vol. 86, 3200 (2001)”, and solve a simple molecular-field approach to the SIS model for the spreading of diseases in scale-free networks by using the excess degree distribution discussed in Sect. 1.2.1, where S and I stand for susceptible and infective individuals respectively.
1.1.9 Epidemic Outbreak in the Configurational Model
Consult “M.E.J. Newman, Spread of epidemic disease on networks, Physical Review E, Vol. 66, 16128 (2002)”, and solve the SIR model for the spreading of diseases in social networks by a generalization of the techniques discussed in Sect. 1.3, where S, I and R stand for susceptible, infective and removed individuals respectively.
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Gros, C. (2011). Graph Theory and Small-World Networks. In: Complex and Adaptive Dynamical Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04706-0_1
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