Large random intersection graphs inside the critical window and triangle counts

TL;DR

This work determines the scaling limits of random intersection graphs in their critical window, revealing a dichotomy: in the light and heavy clustering regimes the large components converge to the continuum Erdős–Rényi graph, while the moderate regime yields a two-parameter family of limit objects $\mathcal{G}^{\mathrm{RIG}}(\lambda,\theta)$. The authors develop a unified framework based on depth-first exploration of the underlying bipartite graph, encoding spanning forests as tilted Bienaymé trees, and connecting component structure to excursion/height processes. They also establish limit theorems for triangle counts and surplus edges, showing convergence to Poisson point measures and excursion lengths, and provide a functional Gromov–Hausdorff–Prokhorov convergence result for the limiting objects. The results yield a detailed probabilistic picture of clustering effects in critical random graphs and pave the way for precise geometric and combinatorial functionals, with potential applications to understanding triangles in clustered networks. The methodology offers a robust bridge between discrete random graphs with clustering and their continuum limits, enriching the theory of scaling limits for inhomogeneous and structured random graphs.

Abstract

We identify the scaling limit of random intersection graphs inside their critical windows. The limit graphs vary according to the clustering regimes, and coincide with the continuum Erdos--Renyi graph in two out of the three regimes. Our approach to the scaling limit relies upon the close connection of random intersection graphs with binomial bipartite graphs, as well as a graph exploration algorithm on the latter. This further allows us to prove limit theorems for the number of triangles in the large connected components of the graphs.

Figures (2)

• Figure 1: On the left, an example of $B(n, m, p)$ with $n=6$, $m=8$. On the right, the induced intersection graph $G(n, m, p)$. Observe that the red edges and the adjacent vertices on the left, which depict a community with 4 members, induce a $K_{4}$ as subgraph (in red) on the right.
• Figure 2: Upper left: an example of $B(n, m, p)$ and its exploration; dashed edges are those not in the spanning forest. Upper right: the corresponding $S^{n, m}$. Lower left: the corresponding $R^{n, m}$. Lower right: the corresponding $H^{n, m}$. The successive values of $\#\mathcal{N}_{k}$ are $3, 2, 0, 0, 0, 0, 1, 2, 0, 0$; the values of $\#\mathcal{M}_{k}$ are resp. $2, 1, 0, 0, 0, 0, 1, 2, 0, 0$. The values of $\#\mathcal{A}_{k}$ are resp. $2, 3, 2, 1, 0, 0, 0, 1, 0, 0$.

Theorems & Definitions (77)

• Theorem 2.1: Scaling limit in the moderate clustering regime
• Theorem 2.2: Scaling limit in the light clustering regime
• Theorem 2.3: Scaling limit in the heavy clustering regime
• Proposition 2.4
• Lemma 2.6
• Proposition 2.7: Convergence of the exploration processes in the light clustering regime
• Proposition 2.8: Convergence of the exploration processes in the moderate clustering regime
• Corollary 2.9: Convergence of the exploration processes in the heavy clustering regime
• Lemma 2.10
• proof