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… 

Figures from this paper

Nucleation during phase transitions in random networks
TLDR
By use of a natural edge flip dynamics, nucleation barriers as a random network crosses the transition are determined, in analogy to the process a material undergoes when frozen or melted, and some of the stochastic properties of the network nucleation are characterized.
Phase Transitions in Finite Random Networks
TLDR
Finite size effects in ensembles where the number of nodes is between 30 and 66 are studied, finding that phase transitions are clearly visible and the structure of a typical graph in each phase is very similar to the graphon that describes the system as n diverges.
The phases of large networks with edge and triangle constraints
TLDR
Based on numerical simulation and local stability analysis, the structure of the phase space of the edge/triangle model of random graphs is described, and changes in symmetry are related to discontinuities at these transitions.
A symmetry breaking transition in the edge/triangle network model
TLDR
The main result is the analysis of a sharp transition between two phases with different symmetries, analogous to the transition between a fluid and a crystalline solid.
A Detailed Investigation into Near Degenerate Exponential Random Graphs
TLDR
This work dives deeper into this near degenerate tendency of the exponential family of random graphs and gives an explicit characterization of the asymptotic graph structure as a function of the parameters.
A detailed investigation into near degenerate exponential random graphs
TLDR
This work dives deeper into this near degenerate tendency of the exponential family of random graphs and gives an explicit characterization of the asymptotic graph structure as a function of the parameters.
Interacting thermofield doubles and critical behavior in random regular graphs
We discuss numerically the non-perturbative effects in exponential random graphs which are analogue of eigenvalue instantons in matrix models. The phase structure of exponential random graphs with
Phase Transitions in Edge-Weighted Exponential Random Graphs: Near-Degeneracy and Universality
Conventionally used exponential random graphs cannot directly model weighted networks as the underlying probability space consists of simple graphs only. Since many substantively important networks
Ensemble equivalence for dense graphs
TLDR
This paper considers a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles, and finds that breaking of ensemble equivalence occurs when the constraints are frustrated.
...
1
2
3
4
5
...

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
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
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 multipartite structure, separated by a phase transition from a
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.
...
1
2
3
4
5
...