Higher order trade-offs in hypergraph community detection

TL;DR

This work extends community detection from dyadic graphs to non-uniform hypergraphs under the Hypergraph Stochastic Block Model, introducing a unified signal-to-noise framework for higher-order interactions. It develops a Bethe Hessian operator for non-uniform hypergraphs enabling scalable spectral clustering with principled model selection, and derives corresponding detectability thresholds that relate to belief propagation. The key finding is that spectral methods achieve the same detectability as BP in uniform hypergraphs but fall short in non-uniform cases, revealing intrinsic trade-offs tied to hyperedge order and shape. The results illuminate practical biases toward certain higher-order configurations and demonstrate the approach on synthetic data and real-world hypergraphs, underscoring the relevance of higher-order detectability trade-offs for interpreting complex systems.

Abstract

Extending community detection from pairwise networks to hypergraphs introduces fundamental theoretical challenges. Hypergraphs exhibit structural heterogeneity with no direct graph analogue: hyperedges of varying orders can connect nodes across communities in diverse configurations, introducing new trade-offs in defining and detecting community structure. We address these challenges by developing a unified framework for community detection in non-uniform hypergraphs under the Hypergraph Stochastic Block Model. We introduce a general signal-to-noise ratio that enables a quantitative analysis of trade-offs unique to higher-order networks, such as which hypergedges we choose to split across communities and how we choose to split them. Building on this framework, we derive a Bethe Hessian operator for non-uniform hypergraphs that provides efficient spectral clustering with principled model selection. We characterize the resulting spectral detectability threshold and compare it to belief propagation limits, showing the methods coincide for uniform hypergraphs but diverge in non-uniform settings. Synthetic experiments confirm our analytical predictions and reveal systematic biases toward preserving higher-order and balanced-shape hyperedges. Application to empirical data demonstrates the practical relevance of these higher-order detectability trade-offs in real-world systems.

Figures (18)

• Figure 1: Different configurations of community structure in graphs and hypergraphs. (a) In dyadic networks, there is a single type of edge connecting pairs of nodes and edges can only be split across communities in one way. (b) In hypergraphs, hyperedges of different orders connect varying numbers of nodes and we must choose which order of hyperedge to split across communities, e.g., do we prefer to preserve the higher order 4-hyperedges or the lower order 2-hyperedges? (c) Hyperedges can connect more than two nodes and can be split across communities in multiple ways. Here, we consider two different shapes of a 4-hyperedge, a balanced shape and an imbalanced shape.
• Figure 2: Spectral methods exhibit a weaker detectability limit in non-uniform hypergraphs. An experiment with nonuniform hypergraphs with $n=30,000$ nodes, $q=3$ communities, $\mathcal{K}=\left\{2, 3\right\}$ hyperedge orders and $d=10$ mean node degree. For each value of $\epsilon$, we run our Bethe Hessian spectral clustering 100 times and belief propagation for 5 times and calculate the mean AMI with the planted community partition. The yellow vertical dash line shows the critical $\epsilon_{\rm BH}^*$, the red vertical dash line shows the critical $\epsilon_{\rm BP}^*$.
• Figure 3: Hyperedge shape preferences in community detection. (a) An illustration of the planted community structure in hypergraphs used in experiments to explore the effect of hyperedge shape. All the hyperedges are 4-order and connect 2 of the 4 communities. The balanced hyperedges are represented by rectangle shapes between communities $\left\{0, 1\right\}$ or $\left\{2, 3\right\}$. The imbalanced hyperedges are represented by triangle shapes between communities $\left\{0, 2\right\}$ or $\left\{1, 3\right\}$. (b) Experimental results with $n=8000$ nodes, $d=10$ mean node degree, $\kappa=4$ and $\kappa^*=4$ order hyperedges. We see the $\mathrm{AMI}_{01;23}$ and $\mathrm{AMI}_{02;13}$ as we vary $\rho$ the ratio of imbalanced to balanced hyperedges. The plot shows that when we have equal number of hyperedges the partition $01;23$ is detected indicating that splitting balanced hypedges and preserving imbalanced hyperedges is preferred.
• Figure 4: Hyperedge order preferences in community detection. (a) Synthetic hypergraphs with 4 planted communities and all the hyperedges between communities $\left\{0, 1\right\}$ or $\left\{2, 3\right\}$ are of order $\kappa^*$, the hyperedges between communities $\left\{0, 2\right\}$ or $\left\{1, 3\right\}$ are of order $\kappa$. All hypergraphs have $n=8000$ nodes. The average degree is either $d=50$ (b-c) or $d=10$ (d-e). We see the $\mathrm{AMI}_{01;23}$ and $\mathrm{AMI}_{02;13}$ as we vary $\rho$ the ratio of high-order to low-order hyperedges. We observe a preference for preserving higher-order $\kappa^*$-hyperedges and splitting lower-order $\kappa$-hyperedges.
• Figure 5: Comparison between detected communities and class structure in human contact hypergraphs. Confusion matrices of communities detected and school classes in hypergraphs of human contact interactions in (a) the primary school and (b) the high school. The rows are normalized such that each entry represents the proportion of individuals in the class assigned to each community.