How the result of graph clustering methods depends on the construction of the graph

We study the scenario of graph-based clustering algorithms such as spectral clustering. Given a set of data points, one first has to construct a graph on the data points and then apply a graph clustering algorithm to find a suitable partition of the graph. Our main question is if and how the construction of the graph (choice of the graph, choice of parameters, choice of weights) influences the outcome of the final clustering result. To this end we study the convergence of cluster quality measures such as the normalized cut or the Cheeger cut on various kinds of random geometric graphs as the sample size tends to infinity. It turns out that the limit values of the same objective function are systematically different on different types of graphs. This implies that clustering results systematically depend on the graph and can be very different for different types of graph. We provide examples to illustrate the implications on spectral clustering.

💡 Research Summary

The paper investigates a fundamental yet often overlooked aspect of graph‑based clustering methods such as spectral clustering: how the construction of the underlying graph influences the final clustering outcome. Starting from a set of i.i.d. data points drawn from a bounded density p on ℝⁿ, the authors consider three families of random geometric graphs – the k‑nearest‑neighbor (kNN) graph, the ε‑radius (r‑graph), and the complete graph equipped with Gaussian edge weights. For each graph type they define the cut between two vertex sets, the volume of a set, and the two standard quality measures, Normalized Cut (NCut) and Cheeger Cut.

The core theoretical contribution is a set of almost‑sure convergence results for the cut and volume as the sample size n → ∞. By introducing appropriate scaling sequences s_cutⁿ and s_volⁿ that depend on n, the graph parameters (k, r, σ) and the graph type, they show that s_cutⁿ·cutⁿ converges to a deterministic limit C_utLim and s_volⁿ·volⁿ converges to V_olLim. Crucially, these limits differ across graph families. For unweighted r‑graphs the limit involves an integral of p(s)^{1‑1/d} over the separating hyperplane, while for weighted graphs the relationship between the bandwidth σ and the radius r determines whether the limit scales like σ^{-d} or like n²σ². The complete Gaussian graph’s limit depends only on σ and the squared density. Similar distinctions hold for the volume terms, with kNN and r‑graphs yielding limits proportional to ∫_H p(x)dx or ∫_H p(x)²dx, and the Gaussian weighting introducing a σ^{-d} factor.

Using these limits the authors define asymptotic versions of NCut and Cheeger Cut (N_CutLim and Cheeger_CutLim) and prove that the empirical NCut_n and CheegerCut_n converge almost surely to these quantities for each graph construction. They also provide optimal convergence rates under the various regimes (σ dominated by r, r dominated by σ, etc.).

Beyond the asymptotic analysis, the paper demonstrates practical implications. Synthetic experiments with mixtures of Gaussians illustrate that the optimal partition for a kNN graph can be different from that for an r‑graph or a Gaussian‑weighted complete graph. Even for moderate sample sizes, spectral clustering that minimizes NCut yields markedly different cluster boundaries depending on the underlying graph. This confirms that the theoretical differences in limit functionals manifest in finite‑sample behavior.

The authors conclude that graph construction is not a mere implementation detail but a critical design choice that directly affects statistical consistency and the objective being optimized. In unsupervised settings where cross‑validation is unavailable, the selection of k, r, and σ must be guided by theory or adaptive methods rather than heuristic “gut feeling.” The paper suggests future work on data‑driven parameter selection, multi‑graph ensembles, or meta‑learning strategies to mitigate the sensitivity of graph‑based clustering to graph construction.

Overall, the work provides a rigorous foundation showing that the type of geometric graph and its parameters systematically alter the limiting values of common clustering quality measures, thereby leading to potentially divergent clustering results. This insight is essential for both theoreticians studying consistency of graph‑based methods and practitioners applying spectral clustering to real‑world data.