Phase transitions in a complex network

@article{Radin2013PhaseTI,
  title={Phase transitions in a complex network},
  author={Charles Radin and Lorenzo A Sadun},
  journal={Journal of Physics A},
  year={2013},
  volume={46},
  pages={305002}
}
We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle density and a range of edge density, though we can only prove they are local, not global, maxima of the entropy density. With this assumption we then prove that the resulting entropy density must lose its… 
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References

SHOWING 1-10 OF 47 REFERENCES
Solution for the properties of a clustered network.
  • Juyong Park, M. Newman
  • Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2005
We study Strauss's model of a network with clustering and present an analytic mean-field solution which is exact in the limit of large network size. Previous computer simulations have revealed a…
Emergent Structures in Large Networks
We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to the emergent multipartite structure, separated by a phase…
Large Networks and Graph Limits
TLDR
Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks.
Limits of dense graph sequences
Singularities in the Entropy of Asymptotically Large Simple Graphs
We prove that the asymptotic entropy of large simple graphs, as a function of fixed edge and triangle densities, is nondifferentiable along a certain curve. We also determine the precise…
Finitely forcible graphons
A Proof of Crystallization in Two Dimensions
TLDR
This work shows rigorously that under suitable assumptions on the potential V which are compatible with the growth behavior of the Lennard-Jones potential the ground state energy per particle converges to an explicit constant E*: where E* ∈ ℝ is the minimum of a simple function on [0,∞).
Convexity in the Theory of Lattice Gases
In this book, Robert Israel considers classical and quantum lattice systems in terms of equilibrium statistical mechanics. He is especially concerned with the characterization of…
Phase transitions in exponential random graphs
We derive the full phase diagram for a large family of two-parameter exponential random graph models, each containing a first order transition curve ending in a critical point.
Networks: An Introduction
TLDR
This book brings together for the first time the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas.
...
...