Detection of local geometry in random graphs: information-theoretic and computational limits

Abstract

We study the problem of detecting local geometry in random graphs. We introduce a model $\mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $\mathbb{S}^{d-1}$, while all remaining edges follow the Erdős--Rényi model $\mathcal{G}(n, p)$. The random geometric graph is generated by thresholding inner products of latent vectors on $\mathbb{S}^{d-1}$, with each edge having marginal probability equal to $p$. This implies that $\mathcal{G}(n, p, d, k)$ and $\mathcal{G}(n, p)$ are indistinguishable at the level of the marginals, and the signal lies entirely in the edge dependencies induced by the local geometry. We investigate both the information-theoretic and computational limits of detection. On the information-theoretic side, our upper bounds follow from three tests based on signed triangle counts: a global test, a scan test, and a constrained scan test; our lower bounds follow from two complementary methods: truncated second moment via Wishart--GOE comparison, and tensorization of KL divergence. These results together settle the detection threshold at $d = \widetildeΘ(k^2 \vee k^6/n^3)$ for fixed $p$, and extend the state-of-the-art bounds from the full model (i.e., $k = n$) for vanishing $p$. On the computational side, we identify a computational--statistical gap and provide evidence via the low-degree polynomial framework, as well as the suboptimality of signed cycle counts of length $\ell \geq 4$.

Figures (2)

• Figure 1: Two drawings of the same graph sampled from ${\mathcal{G}}(n, p, d, k)$ with $n=20$, $p=0.18$, $k=7$; we set $d=2$ for visualization purposes. (a) Vertices are positioned to reflect the latent geometry: the $k=7$ community vertices (circled, teal) have latent vectors drawn from $\mathcal{U}(\mathbb{S}^1)$, so they lie on a circle in the latent space. The orange edges---based on geometric proximity---reveal the resulting cycle-rich structure induced by the local geometry. (b) Vertices are positioned randomly, and the planted community becomes visually indistinguishable from the Erdős--Rényi background.
• Figure 2: Phase diagram for detection, for (a) $\alpha = 0$ and (b) $\alpha = 1/3$. Note that the two plots are in different scales. In the possible & easy phase (green), strong detection can be done by an efficient test statistic. In the possible & hard phase (yellow), strong detection can be done by an inefficient test statistic, and weak detection is impossible for low-degree polynomial algorithms. In the unknown & hard phase (grey), it is open whether strong detection is possible, but weak detection is impossible for low-degree polynomial algorithms. Finally, in the impossible phase (magenta), weak detection is impossible. The impossible phases extend to all values of $\beta > 0$ beyond those presented in the plots.

Theorems & Definitions (41)

• Definition 1.1: Random graphs with local high-dimensional geometry
• Definition 1.2
• Theorem 1.3: Informal
• Theorem 2.1: Detection via global test
• Theorem 2.2: Detection via scan test
• Theorem 2.3: Detection via constrained scan test
• Remark 2.4: Comparison between tests
• Theorem 2.5: Lower bound via truncated second moment
• Theorem 2.6: Lower bound via tensorization
• Remark 2.7: Comparison between lower bounds