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arxiv: 2605.18420 · v1 · pith:4JWZDUU4new · submitted 2026-05-18 · ⚛️ physics.soc-ph

The nestedness of higher-order networks

Timothy LaRock , Yanting Zhang , Jean-Gabriel Young , Nicole Eikmeier , Renaud Lambiotte , Nicholas W. Landry This is my paper

Pith reviewed 2026-05-19 23:38 UTC · model grok-4.3

classification ⚛️ physics.soc-ph
keywords higher-order networksnestednesssimplicialityencapsulationinclusiondirected acyclic graphsocial networksmesoscale structure
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The pith

Higher-order networks display nested containment that unifies several measures via encapsulation graphs

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how subgroups embed within larger groups in higher-order interactions, a pattern seen in ecology, sociology, and other fields under names like nestedness and simpliciality. It unifies these concepts by showing each can be expressed as a property of the encapsulation directed acyclic graph. The authors apply this to social data and find nestedness is widespread, with each measure adding unique insight. Notably, the lack of such nesting can signal important features in how the system is organized at intermediate scales.

Core claim

By defining the encapsulation directed acyclic graph to capture containment relations, the paper shows that existing measures of nestedness, simpliciality, encapsulation, and inclusion are all functions of this graph's properties. Analysis of empirical higher-order networks from social domains demonstrates the prevalence of nested structures and the distinct information each measure provides. The absence of nestedness emerges as a signal of mesoscale organization in these systems.

What carries the argument

The encapsulation directed acyclic graph, a mathematical object that encodes containment among higher-order interactions and serves as the basis for expressing all related measures.

If this is right

  • Nested structure is prevalent in social systems across several domains.
  • Different measures capture complementary aspects of this structure.
  • The absence of nestedness can itself be a powerful indicator of the mesoscale organization of a system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Researchers could use this unified framework to compare results across studies that previously used incompatible measures.
  • The approach might extend to other domains like biological or technological networks where containment occurs.
  • Absence of nestedness could be used as a diagnostic tool for detecting specific types of community structures.

Load-bearing premise

The encapsulation directed acyclic graph can formulate each existing measure as a function of its properties without significant loss or distortion of the original definitions.

What would settle it

A concrete counterexample would be a higher-order network dataset where one of the original measures of nestedness or simpliciality differs substantially from the value computed via the encapsulation directed acyclic graph.

Figures

Figures reproduced from arXiv: 2605.18420 by Jean-Gabriel Young, Nicholas W. Landry, Nicole Eikmeier, Renaud Lambiotte, Timothy LaRock, Yanting Zhang.

Figure 1.1
Figure 1.1. Figure 1.1: An illustration of our unified nestedness framework based on the encapsulation DAG. The left panel illustrates a hypergraph with its as￾sociated encapsulation DAG, which captures the hierarchical subset relations between hyperedges. In the latter, the number of nodes (hyperedges) is |E| and the number of directed relations is |D|. Hyperedges are arranged vertically by their size. To interpret any measure… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Nestedness in random models of hypergraphs. The first two columns show two models that explicitly control for the nestedness of the hyper￾graphs, while the last three columns show models capturing different properties, like the degree sequence of the community structure. Example hypergraphs are shown at the top (with fewer nodes for clarity), while the subsequent rows plot the various nestedness statisti… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Nestedness Statistics for the coauth-mag-geology dataset. Panel (a) plots the out-degree distribution of the DAG, panel (b) plots the distribution of the out-degree, normalized by the maximum possible value for an edge of that size, panel (c) plots the distribution of path lengths through the transitively reduced DAG (Hasse diagram), and lastly, panel (d) plots the simplicial matrix, where we visualize t… view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: Nestedness in proximity-based contact networks. In both plots, panel (a) plots the out-degree distribution of the DAG, panel (b) plots the distribution of the out-degree, normalized by the maximum possible value for an edge of that size, panel (c) plots the distribution of path lengths through the transitively reduced DAG (Hasse diagram), and lastly, panel (d) plots the simplicial matrix, where we visual… view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: Nestedness in political co-sponsorship networks. In both plots, panel (a) plots the out-degree distribution of the DAG, panel (b) plots the distribution of the out-degree, normalized by the maximum possible value for an edge of that size, panel (c) plots the distribution of path lengths through the transitively reduced DAG (Hasse diagram), and lastly, panel (d) plots the simplicial matrix, where we visua… view at source ↗
read the original abstract

In contrast to dyadic interactions, higher-order interactions may contain one another, with subgroups naturally embedded within larger groups. These containment patterns arise empirically in ecology, sociology, computer science and the science of science, and have been studied under the names nestedness, simpliciality, encapsulation, and inclusion. In this chapter, we review each of these measures and unify them through a mathematical object known as the encapsulation directed acyclic graph, formulating each measure as a function of its properties. We demonstrate that nested structure is prevalent in social systems across several domains, show that different measures capture complementary aspects of this structure, and find that the absence of nestedness can itself be a powerful indicator of the mesoscale organization of a system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reviews measures of nested structure in higher-order networks (nestedness, simpliciality, encapsulation, and inclusion) and unifies them by formulating each as a function of properties of the encapsulation directed acyclic graph (EDAG). It claims that such nested structure is prevalent in social systems across domains, that the different measures capture complementary aspects of the structure, and that the absence of nestedness can indicate mesoscale organization.

Significance. If the unification is shown to be exact and lossless, the work offers a useful synthesis across ecology, sociology, computer science, and the science of science, with empirical support for prevalence and the indicator role of nestedness. Credit is given for the broad review of prior measures and the attempt to ground multiple concepts in a single mathematical object.

major comments (2)
  1. [§3] §3 (Encapsulation DAG and unification): the claim that simpliciality (originally a closure property on the face lattice of a simplicial complex) can be exactly recovered as a function of EDAG reachability or topological order must be verified explicitly. Pairwise containment in the DAG may omit higher-dimensional subset constraints that are not recoverable from the reformulation, which would undermine the subsequent claims that the measures are complementary and that absence of nestedness signals mesoscale organization.
  2. [§4–5] §4–5 (Formulation of inclusion and encapsulation): if the mapping from original definitions to EDAG properties is not bijective, the demonstrations that different measures capture complementary aspects rest on an altered object rather than the source concepts. A concrete check (e.g., an example simplicial complex where the EDAG-derived simpliciality score differs from the lattice-based definition) is needed.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'in this chapter' should be clarified for readers who encounter the manuscript as a standalone arXiv preprint.
  2. [Notation] Notation: introduce the acronym 'EDAG' at first use and ensure consistent terminology for 'containment' versus 'inclusion' across sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the unification.

read point-by-point responses

  1. Referee: [§3] §3 (Encapsulation DAG and unification): the claim that simpliciality (originally a closure property on the face lattice of a simplicial complex) can be exactly recovered as a function of EDAG reachability or topological order must be verified explicitly. Pairwise containment in the DAG may omit higher-dimensional subset constraints that are not recoverable from the reformulation, which would undermine the subsequent claims that the measures are complementary and that absence of nestedness signals mesoscale organization.

    Authors: We appreciate the referee's emphasis on explicit verification. The EDAG is the Hasse diagram of the inclusion poset: nodes are the higher-order sets and a directed edge exists from A to B precisely when B properly contains A with no intermediate set. Reachability in this DAG therefore recovers the full transitive closure of the subset relation, which encodes all higher-order inclusions (including multi-level subset constraints) via chains rather than direct pairwise links alone. Simpliciality, as downward closure under subsets, is recovered by verifying that every node has all its lower-reachable nodes present in the collection. We will add an explicit proof of this equivalence to the revised §3, showing that the topological order and reachability preserve the original lattice properties without loss. This supports rather than undermines the claims of complementarity and the indicator role of nestedness absence. revision: yes

  2. Referee: [§4–5] §4–5 (Formulation of inclusion and encapsulation): if the mapping from original definitions to EDAG properties is not bijective, the demonstrations that different measures capture complementary aspects rest on an altered object rather than the source concepts. A concrete check (e.g., an example simplicial complex where the EDAG-derived simpliciality score differs from the lattice-based definition) is needed.

    Authors: We agree that a concrete check strengthens the argument. In the revised manuscript we will insert, in §4 or §5, a worked example of a small simplicial complex. For this example we will compute the simpliciality score both from the original face-lattice definition and from the corresponding EDAG reachability properties, demonstrating numerical agreement. This will also serve as an explicit test of bijectivity for the relevant cases. Should any non-bijective edge cases appear, we will delineate the precise conditions under which the mapping remains faithful. Because the EDAG is constructed directly from the inclusion relations that underlie the source definitions, we expect the scores to match, but the added example will make this transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity in unification of nestedness measures

full rationale

The paper reviews prior definitions of nestedness, simpliciality, encapsulation, and inclusion, then introduces the encapsulation directed acyclic graph as a unifying object and expresses each measure as a function of DAG properties. This is a standard review-and-reformulation contribution whose central claims rest on independent grounding in the original combinatorial definitions and empirical data rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. No equation or step reduces the unification result to its own inputs by construction; concerns about possible distortion in the mapping are questions of correctness or fidelity, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper introduces the encapsulation DAG as the central unifying object and relies on standard mathematical properties of DAGs and containment relations; no free parameters or invented physical entities are indicated in the abstract.

axioms (1)
  • standard math Directed acyclic graphs can represent containment hierarchies without cycles
    Invoked to model nested group structures in higher-order networks.
invented entities (1)
  • encapsulation directed acyclic graph no independent evidence
    purpose: To unify and formulate measures of nestedness as functions of its properties
    New mathematical object introduced to connect previously separate concepts; no independent empirical evidence provided in abstract.

pith-pipeline@v0.9.0 · 5659 in / 1165 out tokens · 33380 ms · 2026-05-19T23:38:27.358896+00:00 · methodology

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Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Measuring and Model- ing Bipartite Graphs with Community Structure

    Sinan G. Aksoy, Tamara G. Kolda, and Ali Pinar. “Measuring and Model- ing Bipartite Graphs with Community Structure”. In:Journal of Complex Networks5.4 (Aug. 2017), pp. 581–603.issn: 2051-1310.doi: 10.1093/ comnet/cnx001.url: https://doi.org/10.1093/comnet/cnx001

    work page doi:10.1093/comnet/cnx001 2017

  2. [2]

    Hypernetwork Science via High-Order Hypergraph Walks

    Sinan G. Aksoy et al. “Hypernetwork Science via High-Order Hypergraph Walks”. In:EPJ Data Science9.1 (Dec. 2020), p. 16.issn: 2193-1127.doi: 10.1140/epjds/s13688-020-00231-0.url: https://doi.org/10.1140/epjds/ s13688-020-00231-0

    work page doi:10.1140/epjds/s13688-020-00231-0.url: 2020

  3. [3]

    The Measure of Order and Disorder in the Distribution of Species in Fragmented Habitat

    Wirt Atmar and Bruce D. Patterson. “The Measure of Order and Disorder in the Distribution of Species in Fragmented Habitat”. In:Oecologia96.3 (Dec. 1993), pp. 373–382.issn: 1432-1939.doi: 10 . 1007 / BF00317508. url: https://doi.org/10.1007/BF00317508

    work page doi:10.1007/bf00317508 1993

  4. [4]

    Weighted Sim- plicial Complexes and Their Representation Power of Higher-Order Net- work Data and Topology

    Federica Baccini, Filippo Geraci, and Ginestra Bianconi. “Weighted Sim- plicial Complexes and Their Representation Power of Higher-Order Net- work Data and Topology”. In:Physical Review E106.3 (Sept. 2022), 22 T. LaRock, Y. Zhang, J.-G. Young, N. Eikmeier, R. Lambiotte, and N. Landry p. 034319.doi: 10.1103/PhysRevE.106.034319.url: https://doi.org/10. 1103...

    work page doi:10.1103/physreve.106.034319.url: 2022

  5. [5]

    Emergence of Scaling in Ran- dom Networks

    Albert-L´ aszl´ o Barab´ asi and R´ eka Albert. “Emergence of Scaling in Ran- dom Networks”. In:Science286.5439 (Oct. 1999), pp. 509–512.doi: 10. 1126/science.286.5439.509.url: https://doi.org/10.1126/science.286. 5439.509

    work page doi:10.1126/science.286 1999

  6. [6]

    Counting Simplicial Pairs in Hypergraphs

    Jordan Barrett et al. “Counting Simplicial Pairs in Hypergraphs”. In: Journal of Complex Networks13.4 (Aug. 2025), cnaf021.issn: 2051-1329. doi: 10.1093/comnet/cnaf021.url: https://doi.org/10.1093/comnet/ cnaf021

    work page doi:10.1093/comnet/cnaf021.url: 2025

  7. [7]

    The Nested Assembly of Plant–Animal Mutu- alistic Networks

    Jordi Bascompte et al. “The Nested Assembly of Plant–Animal Mutu- alistic Networks”. In:Proceedings of the National Academy of Sciences 100.16 (Aug. 2003), pp. 9383–9387.doi: 10.1073/pnas.1633576100.url: https://doi.org/10.1073/pnas.1633576100

    work page doi:10.1073/pnas.1633576100.url: 2003

  8. [8]

    Networks beyond Pairwise Interactions: Struc- ture and Dynamics

    Federico Battiston et al. “Networks beyond Pairwise Interactions: Struc- ture and Dynamics”. In:Physics Reports. Networks beyond Pairwise In- teractions: Structure and Dynamics 874 (Aug. 2020), pp. 1–92.issn: 0370- 1573.doi: 10.1016/j.physrep.2020.05.004.url: https://doi.org/10.1016/ j.physrep.2020.05.004

    work page doi:10.1016/j.physrep.2020.05.004.url: 2020

  9. [9]

    Deep learning for early warning signals of tipping points

    Austin R. Benson et al. “Simplicial Closure and Higher-Order Link Predic- tion”. In:Proceedings of the National Academy of Sciences115.48 (Nov. 2018), E11221–E11230.issn: 0027-8424, 1091-6490.doi: 10.1073/pnas. 1800683115.url: https://doi.org/10.1073/pnas.1800683115

    work page doi:10.1073/pnas 2018

  10. [10]

    Triadic Approximation Reveals the Role of Interaction Overlap on the Spread of Complex Conta- gions on Higher-Order Networks

    Giulio Burgio, Sergio G´ omez, and Alex Arenas. “Triadic Approximation Reveals the Role of Interaction Overlap on the Spread of Complex Conta- gions on Higher-Order Networks”. In:Physical Review Letters132.7 (Feb. 2024), p. 077401.doi: 10 . 1103 / PhysRevLett . 132 . 077401.url: https : //doi.org/10.1103/PhysRevLett.132.077401

    work page doi:10.1103/physrevlett.132.077401 2024

  11. [11]

    Configuration Models of Random Hypergraphs

    Philip S Chodrow. “Configuration Models of Random Hypergraphs”. In: Journal of Complex Networks8.cnaa018 (June 2020).issn: 2051-1329. doi: 10.1093/comnet/cnaa018.url: https://doi.org/10.1093/comnet/ cnaa018

    work page doi:10.1093/comnet/cnaa018.url: 2020

  12. [12]

    Generative Hy- pergraph Clustering: From Blockmodels to Modularity

    Philip S. Chodrow, Nate Veldt, and Austin R. Benson. “Generative Hy- pergraph Clustering: From Blockmodels to Modularity”. In:Science Ad- vances7.28 (July 2021), eabh1303.doi: 10.1126/sciadv.abh1303.url: https://doi.org/10.1126/sciadv.abh1303

    work page doi:10.1126/sciadv.abh1303.url: 2021

  13. [13]

    A Comparison of Relict versus Equilibrium Models for Insular Mammals of the Gulf of Maine

    Kenneth L. Crowell. “A Comparison of Relict versus Equilibrium Models for Insular Mammals of the Gulf of Maine”. In:Biological Journal of the Linnean Society28.1-2 (May 1986), pp. 37–64.issn: 0024-4066.doi: 10. 1111/j.1095-8312.1986.tb01748.x.url: https://doi.org/10.1111/j.1095- 8312.1986.tb01748.x

    work page doi:10.1111/j.1095- 1986

  14. [14]

    Hypergraph Reconstruction from Dynamics

    Robin Delabays et al. “Hypergraph Reconstruction from Dynamics”. In: Nature Communications16.1 (Mar. 2025), p. 2691.issn: 2041-1723.doi: 10.1038/s41467-025-57664-2.url: https://doi.org/10.1038/s41467-025- 57664-2. 1 The nestedness of higher-order networks 23

    work page doi:10.1038/s41467-025-57664-2.url: 2025

  15. [15]

    Line Graphs, Link Partitions, and Over- lapping Communities

    T. S. Evans and R. Lambiotte. “Line Graphs, Link Partitions, and Over- lapping Communities”. In:Physical Review E80.1 (July 2009), p. 016105. doi: 10 . 1103 / PhysRevE . 80 . 016105.url: https : / / doi . org / 10 . 1103 / PhysRevE.80.016105

    work page 2009

  16. [16]

    Connecting the Congress: A Study of Cosponsorship Networks

    James H. Fowler. “Connecting the Congress: A Study of Cosponsorship Networks”. In:Political Analysis14.4 (2006), pp. 456–487.issn: 1047- 1987, 1476-4989.doi: 10.1093/pan/mpl002.url: https://doi.org/10. 1093/pan/mpl002

    work page doi:10.1093/pan/mpl002.url: 2006

  17. [17]

    Legislative Cosponsorship Networks in the US House and Senate

    James H. Fowler. “Legislative Cosponsorship Networks in the US House and Senate”. In:Social Networks28.4 (Oct. 2006), pp. 454–465.issn: 0378-8733.doi: 10.1016/j.socnet.2005.11.003.url: https://doi.org/10. 1016/j.socnet.2005.11.003

    work page doi:10.1016/j.socnet.2005.11.003.url: 2006

  18. [18]

    Improving the Analyses of Nestedness for Large Sets of Matrices

    Paulo R. Guimar˜ aes and Paulo Guimar˜ aes. “Improving the Analyses of Nestedness for Large Sets of Matrices”. In:Environmental Modelling & Software21.10 (Oct. 2006), pp. 1512–1513.issn: 1364-8152.doi: 10.1016/ j.envsoft.2006.04.002.url: https://doi.org/10.1016/j.envsoft.2006.04.002

    work page doi:10.1016/j.envsoft.2006.04.002 2006

  19. [19]

    The Temporal Dy- namics of Group Interactions in Higher-Order Social Networks

    Iacopo Iacopini, M´ arton Karsai, and Alain Barrat. “The Temporal Dy- namics of Group Interactions in Higher-Order Social Networks”. In:Na- ture Communications15.1 (Aug. 2024), p. 7391.issn: 2041-1723.doi: 10.1038/s41467-024-50918-5.url: https://doi.org/10.1038/s41467-024- 50918-5

    work page doi:10.1038/s41467-024-50918-5.url: 2024

  20. [20]

    Contagion Dynamics on Hy- pergraphs with Nested Hyperedges

    Jihye Kim, Deok-Sun Lee, and K.-I. Goh. “Contagion Dynamics on Hy- pergraphs with Nested Hyperedges”. In:Physical Review E108.3 (Sept. 2023), p. 034313.doi: 10.1103/PhysRevE.108.034313.url: https://doi. org/10.1103/PhysRevE.108.034313

    work page doi:10.1103/physreve.108.034313.url: 2023

  21. [21]

    From Networks to Optimal Higher-Order Models of Complex Systems

    Renaud Lambiotte, Martin Rosvall, and Ingo Scholtes. “From Networks to Optimal Higher-Order Models of Complex Systems”. In:Nature Physics 15.4 (Apr. 2019), pp. 313–320.issn: 1745-2481.doi: 10.1038/s41567-019- 0459-y.url: https://doi.org/10.1038/s41567-019-0459-y

    work page doi:10.1038/s41567-019- 2019

  22. [22]

    Hypergraph Assortativity: A Dynamical Systems Perspective

    Nicholas W. Landry and Juan G. Restrepo. “Hypergraph Assortativity: A Dynamical Systems Perspective”. In:Chaos: An Interdisciplinary Journal of Nonlinear Science32.5 (May 2022), p. 053113.issn: 1054-1500.doi: 10.1063/5.0086905.url: https://doi.org/10.1063/5.0086905

    work page doi:10.1063/5.0086905.url: 2022

  23. [23]

    The Sim- pliciality of Higher-Order Networks

    Nicholas W. Landry, Jean-Gabriel Young, and Nicole Eikmeier. “The Sim- pliciality of Higher-Order Networks”. In:EPJ Data Science13.1 (Dec. 2024), pp. 1–20.issn: 2193-1127.doi: 10.1140/epjds/s13688-024-00458-1. url: https://doi.org/10.1140/epjds/s13688-024-00458-1

    work page doi:10.1140/epjds/s13688-024-00458-1 2024

  24. [24]

    Filtering Higher-Order Datasets

    Nicholas W. Landry et al. “Filtering Higher-Order Datasets”. In:Journal of Physics: Complexity5.1 (Feb. 2024), p. 015006.issn: 2632-072X.doi: 10 . 1088 / 2632 - 072X / ad253a.url: https : / / doi . org / 10 . 1088 / 2632 - 072X/ad253a

    work page 2024

  25. [25]

    XGI: A Python Package for Higher-Order Interaction Networks

    Nicholas W. Landry et al. “XGI: A Python Package for Higher-Order Interaction Networks”. In:Journal of Open Source Software8.85 (May 2023), p. 5162.issn: 2475-9066.doi: 10.21105/joss.05162.url: https: //doi.org/10.21105/joss.05162. 24 T. LaRock, Y. Zhang, J.-G. Young, N. Eikmeier, R. Lambiotte, and N. Landry

    work page doi:10.21105/joss.05162.url: 2023

  26. [26]

    Encapsulation Structure and Dynamics in Hypergraphs

    Timothy LaRock and Renaud Lambiotte. “Encapsulation Structure and Dynamics in Hypergraphs”. In:Journal of Physics: Complexity4.4 (Dec. 2023), p. 045007.issn: 2632-072X.doi: 10.1088/2632-072X/ad0b39.url: https://doi.org/10.1088/2632-072X/ad0b39

    work page doi:10.1088/2632-072x/ad0b39.url: 2023

  27. [27]

    Higher-Order Motif Analysis in Hyper- graphs

    Quintino Francesco Lotito et al. “Higher-Order Motif Analysis in Hyper- graphs”. In:Communications Physics5.1 (Dec. 2022), p. 79.issn: 2399- 3650.doi: 10.1038/s42005-022-00858-7.url: https://doi.org/10.1038/ s42005-022-00858-7

    work page doi:10.1038/s42005-022-00858-7.url: 2022

  28. [28]

    Federico Malizia et al.Nested Hyperedges Promote the Onset of Collective Transitions but Suppress Explosive Behavior. Jan. 2026.doi: 10.48550/ arXiv.2601.10522. arXiv: 2601.10522.url: http://arxiv.org/abs/2601. 10522

    work page arXiv 2026

  29. [29]

    PLoS One18(5) (2015) https://doi.org/10.1371/journal.pone

    Rossana Mastrandrea, Julie Fournet, and Alain Barrat. “Contact Patterns in a High School: A Comparison between Data Collected Using Wear- able Sensors, Contact Diaries and Friendship Surveys”. In:PLOS ONE 10.9 (Sept. 2015), e0136497.issn: 1932-6203.doi: 10.1371/journal.pone. 0136497.url: https://doi.org/10.1371/journal.pone.0136497

    work page doi:10.1371/journal.pone 2015

  30. [30]

    InAssociation for Computing Machinery In The World Wide Web Conference (WWW ’19)

    Jan Overgoor, Austin Benson, and Johan Ugander. “Choosing to Grow a Graph: Modeling Network Formation as Discrete Choice”. In:The World Wide Web Conference. WWW ’19. New York, NY, USA: Association for Computing Machinery, May 2019, pp. 1409–1420.isbn: 978-1-4503-6674- 8.doi: 10.1145/3308558.3313662.url: https://doi.org/10.1145/3308558. 3313662

    work page doi:10.1145/3308558.3313662.url: 2019

  31. [31]

    Using Wearable Proximity Sensors to Characterize Social Contact Patterns in a Village of Rural Malawi

    Laura Ozella et al. “Using Wearable Proximity Sensors to Characterize Social Contact Patterns in a Village of Rural Malawi”. In:EPJ Data Science10.1 (Sept. 2021), p. 46.issn: 2193-1127.doi: 10.1140/epjds/ s13688-021-00302-w.url: https://doi.org/10.1140/epjds/s13688-021- 00302-w

    work page doi:10.1140/epjds/ 2021

  32. [32]

    The Shape of Collaborations

    Alice Patania, Giovanni Petri, and Francesco Vaccarino. “The Shape of Collaborations”. In:EPJ Data Science6.1 (Aug. 2017), p. 18.issn: 2193- 1127.doi: 10.1140/epjds/s13688-017-0114-8.url: https://doi.org/10. 1140/epjds/s13688-017-0114-8

    work page doi:10.1140/epjds/s13688-017-0114-8.url: 2017

  33. [33]

    Nested Subsets and the Structure of Insular Mammalian Faunas and Archipelagos

    Bruce D. Patterson and Wirt Atmar. “Nested Subsets and the Structure of Insular Mammalian Faunas and Archipelagos”. In:Biological Journal of the Linnean Society28.1-2 (1986), pp. 65–82.issn: 1095-8312.doi: 10.1111/j.1095-8312.1986.tb01749.x.url: https://doi.org/10.1111/j. 1095-8312.1986.tb01749.x

    work page doi:10.1111/j.1095-8312.1986.tb01749.x.url: 1986

  34. [34]

    Hypergraph Analytics: Modeling Higher-Order Structures and Probabilities

    Ankit Sharma. “Hypergraph Analytics: Modeling Higher-Order Structures and Probabilities”. PhD thesis. 2020, p. 269.isbn: 979-8-6623-8799-7.url: https://www.proquest.com/dissertations-theses/hypergraph-analytics- modeling-higher-order/docview/2425568146/se-2

    work page arXiv 2020

  35. [35]

    Weighted Simplicial Complex: A Novel Approach for Predicting Small Group Evolution

    Ankit Sharma et al. “Weighted Simplicial Complex: A Novel Approach for Predicting Small Group Evolution”. In:Advances in Knowledge Discovery and Data Mining. Ed. by Jinho Kim et al. Vol. 10234. Cham: Springer International Publishing, 2017, pp. 511–523.isbn: 978-3-319-57454-7.doi: 1 The nestedness of higher-order networks 25 10.1007/978-3-319-57454-7 40.u...

    work page doi:10.1007/978-3-319-57454-7 2017

  36. [36]

    An Overview of Microsoft Academic Service (MAS) and Applications

    Arnab Sinha et al. “An Overview of Microsoft Academic Service (MAS) and Applications”. In:Proceedings of the 24th International Conference on World Wide Web. WWW ’15 Companion. New York, NY, USA: As- sociation for Computing Machinery, May 2015, pp. 243–246.isbn: 978-1- 4503-3473-0.doi: 10.1145/2740908.2742839.url: https://doi.org/10. 1145/2740908.2742839

    work page doi:10.1145/2740908.2742839.url: 2015

  37. [37]

    Partial Entropy Decomposition Reveals Higher- Order Information Structures in Human Brain Activity

    Thomas F. Varley et al. “Partial Entropy Decomposition Reveals Higher- Order Information Structures in Human Brain Activity”. In:Proceedings of the National Academy of Sciences120.30 (July 2023), e2300888120. doi: 10.1073/pnas.2300888120.url: https://doi.org/10.1073/pnas. 2300888120

    work page doi:10.1073/pnas.2300888120.url: 2023

  38. [38]

    Erik Teigen, P

    Vaiva Vasiliauskaite, Tim S. Evans, and Paul Expert. “Cycle Analysis of Directed Acyclic Graphs”. In:Physica A: Statistical Mechanics and its Applications596 (June 2022), p. 127097.issn: 03784371.doi: 10.1016/j. physa.2022.127097.url: https://doi.org/10.1016/j.physa.2022.127097

    work page doi:10.1016/j 2022

  39. [39]

    Community Detection in Bipar- tite Networks with Stochastic Block Models

    Tzu-Chi Yen and Daniel B. Larremore. “Community Detection in Bipar- tite Networks with Stochastic Block Models”. In:Physical Review E102.3 (Sept. 2020), p. 032309.doi: 10.1103/PhysRevE.102.032309.url: https: //doi.org/10.1103/PhysRevE.102.032309

    work page doi:10.1103/physreve.102.032309.url: 2020

  40. [40]

    Re- construction of Plant–Pollinator Networks from Observational Data

    Jean-Gabriel Young, Fernanda S. Valdovinos, and M. E. J. Newman. “Re- construction of Plant–Pollinator Networks from Observational Data”. In: Nature Communications12.1 (June 2021), p. 3911.issn: 2041-1723.doi: 10.1038/s41467-021-24149-x.url: https://doi.org/10.1038/s41467-021- 24149-x

    work page doi:10.1038/s41467-021-24149-x.url: 2021

  41. [41]

    Higher-Order Interactions Shape Collective Dynamics Differently in Hypergraphs and Simplicial Complexes

    Yuanzhao Zhang, Maxime Lucas, and Federico Battiston. “Higher-Order Interactions Shape Collective Dynamics Differently in Hypergraphs and Simplicial Complexes”. In:Nature Communications14.1 (Mar. 2023), p. 1605.issn: 2041-1723.doi: 10.1038/s41467-023-37190-9.url: https: //doi.org/10.1038/s41467-023-37190-9

    work page doi:10.1038/s41467-023-37190-9.url: 2023