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arXiv · 1401.1170

The Asymptotics of Large Constrained Graphs

Charles Radin·Kui Ren·Lorenzo Sadun
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Abstract

We show, through local estimates and simulation, that if one constrains simple graphs by their densities $\varepsilon$ of edges and $τ$ of triangles, then asymptotically (in the number of vertices) for over $95\%$ of the possible range of those densities there is a well-defined typical graph, and it has a very simple structure: the vertices are decomposed into two subsets $V_1$ and $V_2$ of fixed relative size $c$ and $1-c$, and there are well-defined probabilities of edges, $g_{jk}$, between $v_j\in V_j$, and $v_k\in V_k$. Furthermore the four parameters $c, g_{11}, g_{22}$ and $g_{12}$ are smooth functions of $(\varepsilon,τ)$ except at two smooth `phase transition' curves.

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Charles Radin, Kui Ren, Lorenzo Sadun. 2014-03-01. The Asymptotics of Large Constrained Graphs. https://doi.org/10.1088/1751-8113%2F47%2F17%2F175001

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