Clustering — the tendency for your friends' friends to also be your friends — is a defining property of real-world networks. It creates triangles: if A knows B and B knows C, then A likely knows C. The standard explanation for clustering is geometry: people who are close in some latent space are more likely to connect, and closeness is transitive. Random geometric graphs, where nodes are points in space and edges connect nearby points, naturally produce high clustering.
Erdős–Rényi random graphs — where each edge exists independently with some probability — do not cluster. In sparse Erdős–Rényi graphs, the expected number of triangles is essentially zero. This has been taken as evidence that clustering requires geometric structure: if edges are independent, clustering vanishes.
The paper (arXiv:2603.13159, March 2026) constructs sparse networks with independent edges that cluster. No latent geometry. No spatial embedding. No hidden variables creating transitivity. Each edge exists independently, and yet triangles form at rates comparable to geometric networks.
The mechanism is degree heterogeneity. If some nodes have much higher degree than others (a power-law or heavy-tailed degree distribution), high-degree nodes serve as hubs that connect many pairs of their neighbors. The triangles form not because neighbors are close in space but because they share a common high-degree neighbor. The clustering is social rather than spatial — it arises from popularity, not proximity.
The structural lesson: clustering is a topological property (triangles exist) with multiple generative mechanisms. Geometry is sufficient for clustering but not necessary. Degree heterogeneity is also sufficient — and degree heterogeneity arises from many processes that have nothing to do with space. The observation of clustering in a network does not prove that the network has an underlying geometry. The triangles are real; the space that seems to explain them may not be.