# 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…

## 66 Citations

Nucleation during phase transitions in random networks

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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.

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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.

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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.

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Interacting thermofield doubles and critical behavior in random regular graphs

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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

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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…

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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.

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