Multipodal Structure and Phase Transitions in Large Constrained Graphs

TL;DR

This paper analyzes large constrained graphs, revealing they tend to form a finite number of vertex groups with specific connection probabilities, and maps out the possible graph structures and phase transitions for various subgraph density constraints.

Contribution

It proves the multipodal structure of large constrained graphs and characterizes the phase space and nonuniqueness of entropy maximizers for these models.

Findings

Graphs are asymptotically multipodal with finitely many vertex subsets.

The phase space of achievable edge and $k$-star densities is fully determined for $2 \,\leq\, k \leq 30$.

Nonuniqueness of entropy maximizers occurs at certain boundary points and extends into the interior for the 2-star model.

Abstract

We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of $k$-star subgraphs, $k\ge 2$ fixed. We prove that under such constraints graphs are "multipodal": asymptotically in the number of vertices there is a partition of the vertices into $M < \infty$ subsets $V_1, V_2, \ldots, V_M$, and a set of well-defined probabilities $g_{ij}$ of an edge between any $v_i \in V_i$ and $v_j \in V_j$. For $2\le k\le 30$ we determine the phase space: the combinations of edge and $k$-star densities achievable asymptotically. For these models there are special points on the boundary of the phase space with nonunique asymptotic (graphon) structure; for the 2-star model we prove that the nonuniqueness extends to entropy maximizers in the interior of the phase space.