The lower tail of triangle counts in sparse random graphs
Dong Huang, Shuangping Li · arXiv: 2610.32879v1 · Summary page
Why cover this story this week?
How a random graph loses its triangles
Suppose a random network contains far fewer triangles than expected. Has it simply become uniformly sparser, or has it reorganized? A new preprint answers this for sparse Erdős–Rényi graphs in the stated probability range. The typical conditioned graph resembles either another Erdős–Rényi graph or a symmetric block model with equally sized groups. A first order phase transition separates these alternatives at the threshold 0.009217… for the retained fraction of the expected triangle count.
Earlier work reduced the probability and structure of this rare event to an optimization problem over graphons, continuous objects that describe large networks. The new paper solves that problem throughout the relevant range, showing that every minimizer is constant or has the specified block structure. It resolves a conjecture and an open problem about triangles, illustrating a broader lesson in probability: an unusually low count can reflect a change in organization, not merely a reduction in overall density.
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