Dense and nondense limits for uniform random intersection graphs

Abstract

We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we prove a.s. convergence to an explicit deterministic graphon. Uniform unit interval graphs are nondense and we prove convergence in the sense of Gromov-Prokhorov after normalization of the distances: the limiting object is the interval $[0,1]$ endowed with a random metric defined through a Brownian excursion. Asymptotic results for the number of cliques of size $k$ ($k$ fixed) in a uniform random graph in each of these three families are also given. In all three cases, an important ingredient of the proof is that, for indecomposable graphs in each class (where the notion of indecomposability depends on the class), the combinatorial object defining the graph (permutation, matching, or intervals) is essentially unique.

Figures (13)

• Figure 1: Illustration of the three families studied in this article. From top to bottom on the right: a permutation graph, a circle graph, a unit interval graph. In each case one of its representatives $(s_1,s_2,\dots,s_n)$ is shown on the left.
• Figure 2: Left: The adjacency matrix of the permutation graph $G_{\sigma_n}$ of a uniform random permutation $\sigma_n$ of size $n=1000$, where vertices are ordered by decreasing degrees. Note that $G_{\sigma_n}$ is not a uniform permutation graph but \ref{['prop:cv_Gsigma']} ensures that this is a fair approximation of the graphon $W^{(\mathrm{perm})}$. Right: The average of $50$ independent adjacency matrices of graphs $G_{\sigma_n}$ for $n=1000$, all ordered by decreasing degrees.
• Figure 3: Left: The adjacency matrix of the intersection graph $G_{M_n}$ of a uniform matching $M_n$ of size $n=1000$, where vertices are ordered by decreasing degrees. Note that $G_{M_n}$ is not a uniform circle graph but \ref{['prop:gcross']} ensures that this is a fair approximation of the graphon $W^{(\mathrm{circle})}$. Right: The average of $50$ independent adjacency matrices of graphs $G_{M_n}$ for $n=1000$, all ordered by decreasing degrees.
• Figure 4: Two samples of uniform connected unit interval graphs with $n=150$ vertices. Plots of adjacency matrices are not relevant for nondense graphs so we rather show geometric embeddings of graphs (they were obtained with the python library networkx). These graph drawings illustrate the fact that the limiting object is one-dimensional with a variable density of vertices.
• Figure 5: Left: The circular representation of a $k$-decomposable matching for $k=\#\left\{c,d,f,g\right\}=4$. Right: the corresponding intersection graph with the cut $\left\{a,b,e\right\}\uplus\left\{c,d,f,g\right\}$ induced by $C_1,\dots ,C_4$. The corresponding cut-set $\left\{b,e\right\}\times\left\{c,d,g\right\}$ is a complete bipartite graph and the split is depicted with the dashed line.

Theorems & Definitions (79)

• Remark 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Definition 3.4