Book contents
- Frontmatter
- Contents
- 1Introduction
- 2Foundations
- 3Basic results for the edge-triangle model
- 4Phases near the boundary of the parameter space
- 5The edge-2star model
- 6Edge-𝑘star models
- 7Supersaturated graphs in the edge-triangle model
- 8Undersaturated graphs in the edge-triangle model
- Appendix ANumerical investigations
- Appendix BModerate deviations
- Appendix CThe functions f(e, e') and fk(e, e')
- References
- Index
Appendix A - Numerical investigations
Published online by Cambridge University Press: aN Invalid Date NaN
Book contents
- Frontmatter
- Contents
- 1Introduction
- 2Foundations
- 3Basic results for the edge-triangle model
- 4Phases near the boundary of the parameter space
- 5The edge-2star model
- 6Edge-𝑘star models
- 7Supersaturated graphs in the edge-triangle model
- 8Undersaturated graphs in the edge-triangle model
- Appendix ANumerical investigations
- Appendix BModerate deviations
- Appendix CThe functions f(e, e') and fk(e, e')
- References
- Index
Summary
Appendix A is largely nonrigorous. It starts with a tour of the conjectured phases in the Razborov triangle and the nature of the transitions between them, recalling the numerical evidence behind the conjectured structure, and in particular the conjecture that all entropy maximizers are multipodal. We then turn to a Monte Carlo simulation of finite graphs near the A(3,0) to B(1,1) phase transition. By looking at the eigenvalues and eigenvectors of the adjacency matrices of randomly generated graphs, we can track a sharp transition with graphs as small as 30 nodes and quantify the differences between graphons, which represent graphs of infinite size, and ensembles of finite graphs.
Information
- Type
- Chapter
- Information
- Emergence and Phase Transitions in Uniform Random Graphs , pp. 206 - 217Publisher: Cambridge University PressPrint publication year: 2026
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