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Latent geometry organizes higher-order interactions

Berné L. Nortier, Jasper van der Kolk, Robert Jankowski, Simon Dobson, M. Ángeles Serrano, Federico Battiston, Marián Boguñá

arXiv:2609.07906v1physics.soc-phcs.SI

TL;DR

Nestedness is widespread in higher-order systems, yet its generative basis remains unclear. The paper introduces a geometric model that couples interactions across orders through a latent space, finding a transition in nestedness and reproducing empirical profiles with one coupling parameter.

  • Problem

    Although nestedness is widespread and dynamically relevant, the generative principle behind its organization across interaction orders remains largely unexplored.

    Source: INTRODUCTION · p. 1

  • Method

    The paper develops an analytically tractable latent-space geometric model that couples hyperedges of different orders through a shared similarity space.

    Source: INTRODUCTION · p. 1

  • Results

    Nestedness transitions between a regime finite in the thermodynamic limit and one vanishing with system size; a weakly geometric range decays slowly, and one coupling reproduces real-world profiles.

    Source: INTRODUCTION · p. 2

  • Takeaways & Limitations

    Latent geometry is a simple organizing principle for nested higher-order interactions, and the model reproduces profiles across real-world domains with a single geometric coupling parameter.

    Source: INTRODUCTION · p. 2

    The analysis assumes all vertex-nodes are active in every layer with perfectly correlated angular coordinates, and derives analytical results for homogeneous hidden-degree distributions.

    Source: CONCLUSIONS · p. 7

Abstract

from arXiv · show

Higher-order structures offer a natural representation of complex systems that involve interactions between groups of different sizes. A widespread feature of their higher-order structure is nestedness, whereby interactions involving smaller groups are contained within larger ones. Yet, why interactions of different orders organise into nested structures remains largely unexplained. Here, we introduce an analytically tractable geometric model of higher-order networks in which a single latent geometric space couples interactions across orders, leading to the spontaneous emergence of nestedness. We show analytically and numerically that nestedness undergoes a transition between a nested geometric regime, where it remains finite in the thermodynamic limit, and a regime where it vanishes with system size. In this regime, we uncover a weakly geometric range characterized by an anomalously slow finite-size decay, allowing substantial nestedness to persist in finite systems even when its asymptotic value vanishes. Finally, with a single geometric coupling parameter, the model reproduces the nestedness profiles observed in real-world hypergraphs across different domains. Our results reveal latent geometry as a simple organising principle underlying the nested organisation of higher-order interactions.

INTRODUCTION

Nestedness is widespread in higher-order systems, but its emergence remains largely unexplained by existing generative approaches. This section introduces latent geometry as a mechanism for organizing interactions across orders and examines its relation to empirical nestedness.

INTRODUCTION

  • Although nestedness is widespread and dynamically relevant, existing generative models either do not address cross-order nestedness or prescribe it rather than explaining its emergence.Nestedness is linked to collective transitions, explosive-transition pathways, hysteresis, and compressed representations that retain structural properties.Source: INTRODUCTION · p. 1
  • Nestedness transitions from a finite thermodynamic-limit regime to vanishing with system size, while a weakly geometric range retains substantial finite-size nestedness and one coupling reproduces real-world profiles.The weakly geometric range has an anomalously slow finite-size decay even though its asymptotic nestedness vanishes.Source: INTRODUCTION · p. 2

NESTEDNESS IN REAL-WORLD SYSTEMS

  • A smaller hyperedge is nested within a larger one when all its nodes are contained in it; the measure counts observed smaller interactions nested within larger ones.The order-resolved measure is normalized by the observed number of smaller hyperedges, and global nestedness aggregates across orders.Source: NESTEDNESS IN REAL-WORLD SYSTEMS · p. 2
  • Across four empirical hypergraphs, nestedness profiles differ, while HOCM randomizations show systematically weaker nestedness, especially at larger orders, so degree constraints alone do not explain them.The HOCM preserves vertex-degree and hyperedge-order distributions, as well as inter-order degree correlations.Source: NESTEDNESS IN REAL-WORLD SYSTEMS · pp. 2–3, Figure 1 · p. 2

A HIGHER-ORDER MODEL WITH GEOMETRY

  • HO-S1 maps order-specific hypergraph layers through bipartite S1 models and couples them in a shared latent similarity space, allowing geometry to organize interactions across orders.The diagram shows the layers mapped into similarity space and combined into one latent representation.Source: A HIGHER-ORDER MODEL WITH GEOMETRY · p. 3, Figure 2 · p. 3

The higher-order S1 model

The HO-S1 model couples interactions across orders through shared latent geometry, which can generate nestedness spontaneously. Its analysis identifies finite-size regimes and shows that a single coupling parameter can reproduce empirical nestedness profiles while preserving observed degrees.

The higher-order S1 model

  • HO-S1 models each interaction order as a bipartite S1 layer, with shared vertex coordinates coupling layers and βℓ controlling each layer’s dependence on geometric similarity.The analytically tractable model uses a soft ensemble with expected degrees fixed; empirical comparisons use a hard ensemble preserving observed degrees exactly.Source: The higher-order S1 model · pp. 3–4

nestedness

  • At zero geometric coupling the model reduces to the higher-order soft configuration model, where nestedness disappears at large system size; at extreme coupling it reaches maximal nestedness.At extreme coupling, vertex participation is determined by angular distance.Source: nestedness · p. 4
  • Nestedness remains finite in the thermodynamic limit when both coupling parameters exceed one; below the critical manifold it vanishes, with weak geometric coupling allowing anomalously slow decay.In the fully non-nested regime it decays as N^-1, while the weakly nested regime can retain substantial nestedness in finite systems.Source: nestedness · p. 5
  • Simulations for orders 3 and 4 show appreciable nestedness when both layers are geometric, strong suppression below the critical manifold, and substantial finite-size nestedness in the weakly nested regime.Along βs = βb, numerical scaling estimates approach the analytical prediction as system size increases; the weakly nested range begins at β = 0.75.Source: nestedness · p. 5

MODELLING NESTEDNESS IN REAL-WORLD HYPERGRAPHS

  • With one coupling parameter across orders, the model recovers the main order-dependent structure of empirical nestedness profiles and improves on the higher-order configuration model.The coupling is selected by matching global nestedness; the order-resolved profile is not fitted independently.Source: MODELLING NESTEDNESS IN REAL-WORLD… · p. 6

CONCLUSIONS

  • The framework’s analytical results assume homogeneous hidden-degree distributions, all nodes active in every layer, and perfectly correlated angular coordinates across orders.The conclusion identifies heterogeneous degrees, order-dependent activity, partial interlayer correlations, and partial overlaps as directions for extending the framework.Source: CONCLUSIONS · p. 7

Randomising real-world datasets

  • The real-world null model preserves higher-order degree distributions and cross-order degree correlations while destroying overlap between different orders.It randomizes the larger-order hyperedges while keeping the smaller layer fixed, preserving lower-order same-size overlap.Source: Randomising real-world datasets · p. 7

The S1 family

The S1 family models connections through latent similarity and popularity, extending the circle-based geometry to higher-order interactions. The section defines global nestedness and overlap measures, then describes a degree-preserving geometric randomisation for bipartite networks.

The S1-model of simple graphs

  • In the S1 model, nodes occupy a circle and connect according to latent similarity and hidden popularity, with β controlling clustering and μ fixing average degree.The model also extends to bipartite networks and hypergraph layers with layer-specific hidden sizes and geometric couplings.Source: The S1-model of simple graphs · pp. 7–8
  • The higher-order S1 model represents hypergraphs as bipartite vertex–hyperedge networks, with layer-specific geometry and hyperedge sizes fixed to their order.Its minimal setting assumes vertices occur in every layer; the model can also be extended to non-overlapping layer support.Source: The S1-model of simple graphs · p. 8

Global nestedness

  • Global nestedness is a weighted average of local nestedness across order pairs, with each layer weighted by its share of hyperedges.When the number of hyperedges is equal across layers, the weights are uniform.Source: Global nestedness · p. 8

The overlap-size term

  • In the bipartite representation, overlap between hyperedges counts shared vertex-neighbours; its expected value depends on their angular separation and layer pair.Independent membership links make the probability of a shared neighbour the product of the two link probabilities.Source: The overlap-size term · pp. 8–9

Geometric Randomisation

  • B-GR extends geometric randomisation to bipartite networks by assigning random angular positions and rewiring links with Metropolis–Hastings; it works for any β > 0 while preserving empirical degree distributions.This includes non-geometric, weakly geometric, and geometric parameter regions.Source: Geometric Randomisation · p. 9

I. REAL-WORLD HYPERGRAPHS

The empirical collection spans eight domains, including social, scientific, pharmacological, ecological, culinary, political, and literary hypergraphs. Dataset statistics and hyperedge-order distributions are reported to characterize the collection and calculate nestedness.

I. REAL-WORLD HYPERGRAPHS

  • The collection covers eight domains, with four representative datasets in the main text and 18 additional hypergraphs described in this section.The four representative examples are email, flavour-compound, coauthorship, and face-to-face proximity networks.Source: §I · p. 12
  • Social datasets represent face-to-face groups or online interactions, while coauthorship hyperedges group authors across scientific fields.Online examples include email exchanges and MathOverflow answerers in question categories.Source: §I · p. 12
  • The collection also includes pharmacological datasets linking drugs or substances, ecological plant–pollinator groups, and culinary ingredient–recipe or ingredient–compound relations.The pharmacological examples are DAWN and NDC Substances.Source: §I · p. 12
  • Political hyperedges group legislators cosponsoring bills, while the literary dataset links characters appearing together in books.Source: §I · p. 12
  • All datasets except Friends & Family are publicly available; its hypergraph uses the construction introduced in Ref..Source: §I · p. 12
  • Table S1 reports each dataset’s category, node count N, hyperedge count E, and largest hyperedge order M; Fig. S1 plots hyperedge-order distributions with logarithmic counts.The distributions are used to calculate nestedness I.Source: §I · p. 12, Table S1 · p. 13, Figure S1 · p. 13

II. SIGNIFICANCE OF NESTEDNESS IN REAL-WORLD SYSTEMS

The supplementary analysis compares empirical nestedness profiles with a configuration-model null that preserves degree and hyperedge-order distributions and inter-order degree correlations. Average and cell-wise z-scores show that empirical nestedness generally exceeds the null expectation.

II. SIGNIFICANCE OF NESTEDNESS IN REAL-WORLD SYSTEMS

  • The null model preserves vertex degrees, hyperedge orders, and inter-order degree correlations while randomizing the remaining structure.The analysis samples 100 randomized hypergraphs and calculates cell-wise z-scores against their elementwise means and standard deviations.Source: §II · p. 14
  • Averaged profile z-scores exceed 2 for every dataset, showing substantial deviation from randomized counterparts.The statistic averages cell-wise z-scores across each nestedness profile.Source: §II · p. 14, Figure S2 · p. 14
  • With few exceptions, cell-wise comparisons show higher nestedness than expected at random; some null ensembles produce no nestedness at higher orders.Figure S3 compares empirical and average randomized profiles and displays the associated cell-wise z-scores.Source: §II · p. 14, Figure S3 · p. 15

III. ANALYTICAL DERIVATION OF HYPEREDGE NESTEDNESS

The derivation represents hyperedges as sets of vertex memberships and calculates the probability that a smaller hyperedge is contained in a larger one. It then averages over hyperedge sizes and hidden variables to obtain expected nestedness.

III. ANALYTICAL DERIVATION OF HYPEREDGE NESTEDNESS

  • A smaller hyperedge is nested in a larger one when all its vertex-neighbors are also neighbors of the larger hyperedge.The derivation treats nesting between orders s and b, with s < b.Source: §III · p. 16
  • Membership vectors encode which vertices belong to each hyperedge, with vertex inclusions modeled as independent Bernoulli trials.The derivation uses these vectors to form the joint probability of a hyperedge's size and its shared neighbors with another hyperedge.Source: §III · p. 16
  • The nesting probability sums over possible hyperedge sizes, requires complete containment, and is averaged across smaller-layer hyperedges to obtain expected nestedness.The probability of nesting within a larger layer is defined as the probability of containment in any hyperedge at that layer.Source: §III · p. 17
  • The derivation states that nestedness vanishes in the thermodynamic limit for the case considered.The supplied passage does not specify the parameter conditions for this case.Source: §III · p. 18

IV. SCALING BEHAVIOUR OF NESTEDNESS WITH SYSTEM SIZE

The scaling analysis starts with homogeneous degree distributions within each layer and partitions angular distances to evaluate how overlap terms scale with system size.

IV. SCALING BEHAVIOUR OF NESTEDNESS WITH SYSTEM SIZE

  • The analysis assumes homogeneous degree distributions within each layer, while allowing mean degrees to differ between layers.It evaluates the scaling separately over intervals partitioning angular distance from 0 to π.Source: §IV · p. 20
  • For the scaling comparison, the couplings are ordered as βs < βb, which implies ci < cj under the stated large-system scaling.The passage gives c ∼ N^-max(1,1/β) and derives the ordering from βs < βb.Source: §IV · p. 20
  • The average overlap m(s,b)(∆ϕij) is split into two integrals and evaluated across cases determined by angular relations.The passage organizes the cases according to the relationship between θ and ∆ϕij.Source: §IV · p. 20

A = 1, B = 1, C = 1

The analysis derives system-size scaling for nestedness by evaluating overlap contributions across angular-distance intervals and coupling regimes. It also relates overlap to hidden-degree parameters and characterizes the smoothness of the transition.

A = 1, B = 1, C = 1

  • The expected nestedness scaling is derived by combining angular-interval contributions across coupling regimes.The derivation assigns scaling exponents to interval contributions to the overlap, then combines them into an expression for expected nestedness.Source: A = 1, B = 1, C = 1 · pp. 24–25
  • For βs, βb > 1, the overlap scales as ζiζj times a positive constant independent of the coupling parameters.The stated result follows from integrating the leading-order overlap term over the angular domain.Source: A = 1, B = 1, C = 1 · p. 26
  • The transition is described as hypersmooth at (1, 1), while away from that point along the boundary min(βs, βb) = 1 its order is 1.5.The passage identifies the boundary case as distinct from the transition point (1, 1).Source: A = 1, B = 1, C = 1 · p. 26

VI. REPRODUCING THE NESTEDNESS IN REAL-WORLD SYSTEMS

With empirical degree distributions held fixed, the model reproduces or exceeds observed nestedness for most tested datasets, but fails on three of 22. The results also show a transition at β = 1 and identify simplifying assumptions that may contribute to some mismatches.

VI. REPRODUCING THE NESTEDNESS IN REAL-WORLD SYSTEMS

  • The analysis fixes each dataset’s observed degree distributions and uses geometric randomisation to sweep β for nestedness between layers 3 and 4 with β3 = β4.In Figure S4, white and purple regions indicate underestimation and overestimation; β* marks the value matching average empirical global nestedness, averaged over 100 realizations.Source: §VI · p. 27, Figure S4 · p. 27
  • The model matches or exceeds observed nestedness for most datasets, but fails on 3 of 22: Coauthorship (CVPR), Coauthorship (KDD), and Cora.Figure S4 shows the model’s nestedness comparisons across increasing β values.Source: §VI · p. 27, Figure S4 · p. 27
  • All datasets exhibit a transition in nestedness at β = 1, as predicted analytically.For the three datasets not reproduced, nestedness appears to saturate at a value of I(3,4).Source: §VI · p. 27
  • The model’s simplicity—one hidden variable per node and hyperedge plus an inter-layer coupling parameter—limits expectations that it will reproduce real networks’ full structural variety.The authors nevertheless note rich behaviour in this minimal model.Source: §VI · p. 27
  • Two assumptions may contribute to mismatches: all nodes can participate in both layers despite partial empirical overlap, and node angles are perfectly correlated across layers.Partial overlap may noticeably limit nestedness in finite networks; the authors expect no effect in the large-N limit.Source: §VI · p. 27
  • Figures S5 and S6 show nestedness fingerprints across β for datasets 1–22, illustrating how nestedness increases with dependence on the underlying geometry.The two figures cover datasets 1–15 and 16–22, respectively.Source: §VI · p. 27, Figure S5 · p. 28, Figure S6 · p. 29
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