Structural reducibility of hypergraphs

TL;DR

The paper introduces an information-theoretic framework to quantify and exploit structural redundancies in hypergraphs by identifying a minimal set of representative higher-order layers that preserve the essential higher-order structure. By modeling the transmission of a hypergraph via representative layers and overlaps across orders, it defines a multiscale reducibility measure $\eta$ that ranges from 0 to 1 and can be optimized greedily to handle large systems. Across synthetic models with tunable nestedness and real-world networks, the method reveals substantial compressibility where layer overlaps exist, while dynamical measures often fail to capture this redundancy. Importantly, reduced hypergraphs retain key global, mesoscale, and local structural and dynamical properties, enabling more efficient analyses without substantial loss of fidelity.

Abstract

Higher-order interactions provide a nuanced understanding of the relational structure of complex systems beyond traditional pairwise interactions. However, higher-order network analyses also incur more cumbersome interpretations and greater computational demands than their pairwise counterparts. Here we present an information-theoretic framework for determining the extent to which a hypergraph representation of a networked system is structurally redundant, and for identifying its most critical higher orders of interaction that allow us to remove these redundancies while preserving essential higher-order structure.

Figures (12)

• Figure 1: Structural reducibility. (a) Hypergraph containing layers $\mathcal{L}=\{2,3,4,5\}$ of size $E^{(\ell)}$, which is reducible to an optimal representative layer set $\mathcal{R}^*=\{3,5\}$ with reducibility $\eta=0.34$ (Eq. (\ref{['eq:reducibility']})). (b) Reducibility of a noisy nested hypergraph, with noise parameter $\epsilon$ determining the fraction of randomized hyperedges, for various hypergraph dimensions $\ell_{\textrm{max}}$.
• Figure 2: Comparison of structural and dynamical lucas2024functional reducibility measures. (a) Pairwise layer similarity matrices of block-nested hypergraphs at increasing levels of noise $\epsilon$. Hypergraph layers $\ell=2$, $4$, and $6$ are fully nested within $\ell=3$, $5$, and $7$, respectively, and the similarity is lost as $\epsilon$ increases. (b) Structural and dynamical reducibility measures against all $\epsilon$ values. The dynamical reducibility does not detect any compressibility between layers. The structural reducibility uncovers both the structural redundancies and the planted, optimal representative layers $\mathcal{R}^*=\{3,5,7\}$.
• Figure S1: Multiscale structural reducibility. Multiscale reducibility (Eq. (S6)) versus mixing parameter $p$ determining the expected fraction of nodes in each hyperedge that belong to the majority community within the hyperedge. As $p$ increases the generated hypergraphs become less clustered with respect to the input partition $\bm{b}$. As the number of communities $B$ in $\bm{b}$ increases, the hypergraph becomes less reducible with respect to $\bm{b}$, approaching the standard reducibility (Eq. (\ref{['eq:reducibility']}), blue).
• Figure S2: Impact of node partition on multiscale reducibility. The multiscale reducibility of synthetic hypergraphs with community structure generated using partition $\bm{b}$ is plotted against the amount of noise (number of pairwise shuffles) applied to $\bm{b}$ prior to computing the multiscale reducibility. As the partition we use for computing the multiscale reducibility becomes less correlated with the underlying community structure of the graph, we see reducibility drop. Experiments are repeated over ten trials and error bars represent two standard errors in the mean.
• Figure S3: Reducibility $\eta$ against noise parameter $\epsilon$ for synthetic hypergraphs with tunable nestedness. Heatmaps on the left illustrate the pairwise layer similarity at four values of $\epsilon$, while the plots on the right-side show the structural reducibility $\eta$ against all $\epsilon\in[0,1]$. (a) Model S1 where layers $\ell=2$, 4, and 6 are generated from $\ell=3$, 5, and 7, respectively, and all but layers 2 and 3 are rewired. The optimal representatives $\mathcal{R}^*=\{3,5,7\}$ are precisely obtained at $\epsilon=0$, with the reducibility $\eta$ decreasing with $\epsilon$ but never reaching zero, as the redundancy between nested layers 2 and 3 is kept intact. (b) Model S2, which is equivalent to S1, but layers $\ell=2$, 3, 4, and 5 are kept fixed (i.e. only $\ell=6$ and 7 are rewired with probability $\epsilon$). Reducibility drops only slightly from its initial value because the layer redundancies are preserved through most layers being kept fixed. (c) Model S3, where layers $\ell=5$, 6, and 7 generate $\ell=2$, 3, and 4, respectively. All layers are rewired with probability $\epsilon$. As noise levels increase, the reducibility $\eta$ decreases and eventually reaches zero because all layers are fully rewired at $\epsilon=1$. (d) Model S4, which is equivalent to S3, but layers 4 and 7 are kept fixed while all the others are rewired. Reducibility is decreased but remains strictly positive due to left-over layer redundancies. (e) Model S5, which is equivalent to S3, but layers $\ell=3$, 4, 6, and 7 are kept fixed. Reducibility is decreased but remains close to its original value because of the remaining layer redundancies. (f) Model S6, where layers $\ell=4$, 6, and 7 are nested within layers 3, 2, and 5, respectively. Reducibility $\eta$ vanishes completely because all layers are fully rewired at $\epsilon=1$.