Network theory is a powerful tool for describing and modeling complex systems having applications in widelydiffering areas including epidemiology [16], neuroscience [34], ecology [20] and the Internet [26]. In its beginning, one often compared an empirically given network, whose nodes are the elements of the system and whose edges represent their interactions, with an ensemble having the same number of nodes and edges, the most popular example being the random graphs introduced by Erdos and Renyi [11]. As the field matured, it became clear that the naive model above needed to be refined, due to the observation that real-world networks often differ significantly from the Erdos–Renyi random graphs in having a highly heterogenous non-Poisson degree distribution [5, 15] and in possessing a high level of clustering [33].
Methods for generating random networks with arbitrary degree distributions and for calculating their statistical properties are now well understood.
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Notes
- 1.
More accurately: ∀ε > 0 Pr(C > ε) → 0 as N →∞.
- 2.
In Section 2.2, when C > 0, we will need a similar observation; namely, that the probability to have a cycle of length four, that is not composed of two triangles, scales as N −1 and hence can also be neglected for large N.
- 3.
p is usually a function of N, p(N).
- 4.
The idea is basically as follows: find the eigenvalues of the matrix \(L = D - A\), where A is the graph adjacency matrix and D is a diagonal matrix with the degree of node j at the D jj -th entry; the multiplicity of the eigenvalue 0 is the number of connected components.
- 5.
Also known respectively as site, bond and joint site+bond percolation.
- 6.
Durret [10] gives a nice critique on the claim that “the internet is robust.. after dilution (in a certain parameters regime) we still get a GC.” In the regime referred to, “if all 6 billion people were initially connected then after the removal only 36 people can check their email.”
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Acknowledgements
MT and YB are grateful for the support of the EC (project MATHfSS 15661) and DIP (project Compositionality F 1.2). LS and YAR are grateful for the support of the James S. McDonnell Foundation and the Israeli Science Foundation.
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Berchenko, Y., Artzy-Randrup, Y., Teicher, M., Stone, L. (2009). The Big Friendly Giant: The Giant Component in Clustered Random Graphs. In: Ganguly, N., Deutsch, A., Mukherjee, A. (eds) Dynamics On and Of Complex Networks. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4751-3_14
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