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REVIEW 4 major objections 5 minor 70 references

An effective method for profiling core-periphery structures in complex networks

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Ranking nodes by coreness then in-core centrality yields denser cores and, often, the maximum clique.

desk verdict The density gain over rich-core mostly comes from the coreness pre-sort rather than the proposed d+ boundary, but the clique heuristic is a genuinely useful practical offshoot. read the letter →

arxiv 2502.07624 v2 pith:NOZFPZMB submitted 2025-02-11 physics.soc-ph physics.data-an

Jiaqi Nie , Qi Xuan , Dehong Gao , Zhongyuan Ruan This is my paper · ORCID
classification physics.soc-phphysics.data-an PACS 89.75.Hc
keywords core-peripherystructurek-coredecompositionrich-coremethodnetworkcoredensitycliquedetectionmultiplexnetworksnodecentralitycomplex
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to make core-periphery profiling produce structurally cohesive cores without adding parameters. It proposes ranking nodes first by their $k$-core coreness and then by a centrality measure computed inside that core, and taking the rank at which the number of links to higher-ranked nodes is maximal as the core boundary. Across twelve real-world networks, this approach generally yields smaller and denser cores than the degree-based rich-core method, with several cores becoming fully connected. The same ordering also gives a direct route to clique detection: the top-ranked nodes that sit exactly on the line $d^+(r)=r-1$ form a clique, and after expansion this clique matches the true maximum clique in eight of the twelve networks. The method extends to multiplex networks, where layer-averaged rankings identify dense multiplex cores.

What carries the argument

The load-bearing object is the $d^+(r)$ curve together with its upper bound. For each node ordered by richness, $d^+(r)$ is the number of links to nodes ranked higher; the turning point $r^*$ where this count is maximal defines the core-periphery boundary, with all earlier nodes in the core. The curve's upper bound, $d^+(r)=r-1$, captures the fact that the node at rank $r$ can connect to at most all $r-1$ nodes before it, so any sequence of top-ranked nodes lying exactly on that bound must be mutually connected and hence a clique. The ordering that makes the construction work is the richness tuple $\mu_i=(k_i, e_{m_i})$, which puts structurally deep nodes ahead of merely high-degree ones; the paper explains the boundary choice through random-walk persistence: as the core expands, the probability that a random walker remains inside it grows fastest around the inflection of $d^+(r)$, where the second derivative vanishes.

What would settle it

Build a network whose largest clique consists of low-coreness nodes attached to a sparse periphery, run the proposed MCC-E method, and check whether the expanded clique is smaller than the maximum clique; a failure on such a graph would show the heuristic is not universal. A second check is to search real networks for cases where the MCC core density falls below the rich-core density, which would contradict the paper's general-density claim.

Watch

Extended reading notes

Core claim

The central claim is that ordering nodes by coreness before any centrality measure makes the core boundary found by the rich-core construction sharper and the resulting core genuinely dense. Concretely, each node receives a richness tuple $\mu_i=(k_i, e_{m_i})$, where $k_i$ is its $k$-core coreness and $e_{m_i}$ is its degree or eigenvector centrality within the $k$-core; the variants MCC-E and MCC-D use eigenvector and degree centrality respectively. Nodes are sorted by this tuple and, reading down the ranking, $d^+(r)$ counts how many links the node at rank $r$ sends to earlier nodes. The boundary of the core is the rank $r^*$ at which $d^+(r)$ reaches its maximum, and the paper shows that on the twelve test networks these cores are denser, and usually smaller, than the rich-core baseline. Because every node at rank $r$ can have at most $r-1$ links to earlier nodes, the curve $d^+(r)=r-1$ is an upper bound, and consecutive top-ranked nodes that lie exactly on it are pairwise connected and therefore form a clique; expanding that seed clique by adding later nodes connected to all of its members yields a clique that equals the maximum clique in eight networks and falls only slightly short in the remaining four.

Load-bearing premise

The load-bearing assumption is that the rank at which the number of links to higher-ranked nodes peaks marks the true core boundary once nodes are ordered by coreness; the paper justifies this with a heuristic random-walk argument rather than a proof that the chosen core is the densest possible one.

Editorial extensions

If this is right

  • Core-periphery profiling becomes one parameter-free pass: $k$-core decomposition, one centrality calculation, and a scan for the maximum of $d^+(r)$.
  • Because the resulting cores are smaller and denser, studies of influence, rich-club ordering, or contagion will be working with a more tightly interconnected set of nodes.
  • The clique-identification procedure gives a fast approximation to the NP-hard maximum-clique problem, matching it exactly in eight of the twelve tested networks.
  • On multiplex networks, the layer-averaged richness ranking finds denser multiplex cores than the existing multiplex rich-core baseline on all five datasets tested.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension the paper leaves open is whether the choice of centrality inside the $k$-core matters systematically; betweenness or local clustering may outperform degree and eigenvector centrality on networks with bottlenecks.
  • The clique heuristic should break down when the true maximum clique has low coreness, since such nodes would be ranked late; planting a large clique weakly attached to a sparse periphery is a direct way to probe that limit.
  • The random-walk story for $r^*$ is a heuristic; comparing the core boundary against the maximum of the persistence-probability second derivative on synthetic planted-core networks would show how often the two coincide.
  • Because the method is built on static pairwise links, a time-resolved or group-interaction version is a natural next step for dynamic and higher-order interaction data.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes an improved method for profiling core-periphery structure, called MCC-E and MCC-D, in which nodes are ranked first by k-core coreness and then by a centrality measure (eigenvector or degree) within the k-core. The core boundary is set at the rank r* that maximizes the number of links to higher-ranked nodes, d+(r), following the rich-core framework of Ma and Mondragón. The authors report that on twelve real-world single-layer networks the identified cores are generally denser than those from the rich-core method, that the top-ranked nodes lying exactly on the upper bound d+(r)=r-1 plus a greedy expansion yield cliques that match the maximum clique in 8 of 12 networks, and that a multiplex extension gives dense multiplex cores on five datasets. The paper includes complexity analysis and discussion of the random-walk interpretation of the core boundary.

Significance. If the claims hold, the method is a practically valuable, parameter-free heuristic for dense core and approximate maximum-clique detection in large networks, with linear-logarithmic complexity. The paper's strengths include the clarity of the algorithmic definitions, the breadth of the empirical validation across networks spanning several orders of magnitude in size, and the explicit comparison with the established rich-core method. The multiplex extension addresses a timely problem. However, the central claim that the new ranking and boundary procedure, rather than the coreness pre-sort alone, produces the density improvement is not yet supported by an ablation study, and the theoretical justification for the boundary r* remains heuristic and depends on an unverified monotonicity condition.

major comments (4)
  1. [Section V, Eq. (5)] The reported MCC-E cores coincide with the highest k-core shell in 7 of 12 networks (Karate 5/4, Les Misérables 10/9, Jazz 30/29, Netscience 20/19, DBLP 115/114, Citeseer 87/86, In-2004 489/488), all with density φ=1.0. Since a subgraph of size k+1 within a k-core is necessarily a clique, these results could be produced by the coreness pre-sort alone, without the d+ turning-point rule or the within-core centrality tie-breaker. The paper does not ablate these components (e.g., by comparing against simply taking the maximum k-core shell, or by randomizing the within-shell order), so the central claim that MCC improves on rich-core cannot be attributed to the proposed mechanism.
  2. [Section IV, Eqs. (2)-(3)] The random-walk justification assumes that the persistence probability α_c is increasing in the core size c, but when a newly added node has d+=0, the numerator of Eq. (5) is unchanged while the denominator increases, so α_c can decrease. The statement that the boundary corresponds to α''_c=0, and hence to an extremum of d+(r), relies on a monotonicity condition that is not established for the coreness-based ordering. This weakens the theoretical grounding of the central boundary rule r*.
  3. [Section III.B, Fig. 3] The multiplex extension aggregates layer-wise coreness and centrality into a single richness value per node [Eq. (2)], but computes d+ layer by layer by counting neighbors with higher richness [Eq. (3)]. The paper acknowledges that the two conventions differ when richness values are tied, but it does not analyze how tie-breaking affects the multiplex core boundary or the densities reported in Table II. In addition, the equal-weight aggregation in Eq. (2) is an untested assumption; without sensitivity analysis over layer-weighting schemes, the multiplex contribution is less supported than the single-layer results.
  4. [Section III.B, Fig. 3] The clique identification is a greedy expansion whose behavior is illustrated only by the specific examples in Fig. 3. The headline empirical claim that the identified clique equals the maximum clique in 8 of 12 networks would be stronger if the paper reported the number of expansion steps per network and compared against a simpler baseline, such as the clique obtained from the maximum k-core shell without any centrality ordering. As written, the reader cannot tell how much of the clique-finding success comes from the proposed ranking versus from the k-core decomposition itself.
minor comments (5)
  1. [Table I] The footnote suggests weighting richness by maximum coreness and maximum centrality when layers differ in link density, but it is not stated whether this weighting is used in the experiments in Table II; please clarify.
  2. [Section II] The rich-core column reports clique sizes Sc, but the text does not state explicitly whether the same greedy expansion procedure was applied to the rich-core ranking for all networks; a sentence in the caption or in Section III.B would remove ambiguity.
  3. [Section III.B] The degree centrality within the k-core is defined as ed/(en-1), but the normalization of eigenvector centrality is not specified. Since the ranking depends on the centrality values only through their order, a brief note on normalization would improve reproducibility.
  4. [Section V] Because tied richness values are broken randomly, the results in Table I and Fig. 2(e) may vary across realizations. Reporting the variability (e.g., across several random tie-breaking runs) for at least a few networks would help gauge the stability of the reported density and clique sizes.
  5. [Section V] The notation α''_c in Eq. (5) uses a continuous second derivative while c is an integer core size; a discrete formulation or an explicit statement of the continuous approximation would avoid confusion.

Circularity Check

1 steps flagged · score 2.0 of 10

No load-bearing circularity in the main empirical claims; one explanatory step in the Discussion restates the definition of r* as the maximum of d+ rather than independently deriving it.

  1. self definitional [Section V (Discussion), paragraph after Eq. (5)]
    "This condition requires, to a first approximation, the second derivative of g(c) ≡ Σ_{i=1}^c d+_i = ∫_1^c d+_i(y)dy to be 0, indicating that d+_i(y) has an extremum (maximum), which we refer to as the core-periphery boundary."

    In Section II, r* is explicitly defined as the rank corresponding to the maximal value of d+(r), and this r* is stated to determine the core-periphery boundary. The Discussion's random-walk argument concludes that the boundary is where d+ has an extremum (maximum). That conclusion is therefore identical to the original definition; the random-walk derivation gives no independent characterization of the boundary. The intermediate step from α''_c=0 to g''(c)=0 is asserted rather than derived, so the 'physical interpretation' is a post-hoc restatement of the construction. This circularity is explanatory only: the dense-core and clique results are measured afterward and compared with independent quantities, so the central claims do not reduce to this step.

full rationale

The main derivation chain is not circular. The MCC method ranks nodes by (coreness, within-k-core centrality), defines r* as the first maximizer of d+(r), and then measures the density phi of the resulting core and the size Sc of the clique found on the d+(r)=r-1 line. Both phi and Sc are compared with quantities (rich-core density, Bron-Kerbosch maximum clique size) that are not used to fit any parameter. The fact that in several networks the dense core is exactly the maximum k-core shell is a property of the coreness input; it weakens the attribution of the result to the new d+ or centrality machinery, but that is an ablation/evidence gap rather than circularity. The self-citations (Refs. [41] and [51]) are illustrative and not load-bearing. The only circularity identified is in the Discussion's random-walk rationalization, which restates the definition of r* as the maximum of d+ rather than independently justifying it.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The method introduces no new physical or network entities; it combines existing coreness and centrality measures. The only ad hoc elements are the ranking formula and the multiplex averaging. No free parameters are fitted to data, but the choice of centrality is empirical.

free parameters (1)
  • choice of centrality measure (degree vs eigenvector)
    The paper tests two variants, MCC-D and MCC-E, and reports that performance varies by network (e.g., IPv4 favors MCC-D, Dolphins favors MCC-E). This is a selection of method variant based on empirical results, not a fitted parameter per se, but it introduces flexibility in reporting best outcomes.
assumptions (3)
  • domain assumption The core boundary is defined as the rank maximizing d+(r), following the rich-core method.
    The paper inherits this definition from Ref. [21] and applies it to the new ranking. The theoretical justification in the Discussion is a heuristic approximation and is not proven for the new ranking.
  • standard math The k-core decomposition and centrality measures are computed on the subgraph of nodes with coreness at least k.
    Standard network analysis tools; the paper uses them as defined.
  • ad hoc to paper For multiplex networks, averaging richness across layers (Eq. 2) is a meaningful aggregation.
    The paper states this is the 'most simple case' and notes weighting as an alternative (footnote in Sec. IV), but no justification is provided for the averaging choice.

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Pith. "Pith review of An effective method for profiling core-periphery structures in complex networks." pith.science (2026). https://pith.science/paper/NOZFPZMB

@misc{pith2026250207624,
  author       = {Pith},
  title        = {Pith review of: An effective method for profiling core-periphery structures in complex networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NOZFPZMB}},
  note         = {Machine review of arXiv:2502.07624}
}
abstract

Profiling core-periphery structures in networks has attracted significant attention, leading to the development of various methods. Among these, the rich-core method is distinguished for being entirely parameter-free and scalable to large networks. However, the cores it identifies are not always structurally cohesive, as they may lack high link density. Here, we propose an improved method building upon the rich-core framework. Instead of relying on node degree, our approach incorporates both the node's coreness $k$ and its centrality within the $k$-core. We apply the approach to twelve real-world networks, and find that the cores identified are generally denser compared to those derived from the rich-core method. Additionally, we demonstrate that the proposed method provides a natural way for identifying an exceptionally dense core, i.e., a clique, which often approximates or even matches the maximum clique in many real-world networks. Furthermore, we extend the method to multiplex networks, and show its effectiveness in identifying dense multiplex cores across several well-studied datasets. Our study may offer valuable insights into exploring the meso-scale properties of complex networks.

Figures

Figures reproduced from arXiv: 2502.07624 by the authors.

Figure 1
Figure 1. FIG. 1: (a) An illustrative example for ranking nodes based on their [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Number of links [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages

  1. [1]

    This is a widely studied social network in network science

    Karate Club [26] . This is a widely studied social network in network science. Nodes in the network represent the members of the karate club, and links denote social ties or friendships between pairs of members

    work page

  2. [2]

    This dataset describes the social interac- tions within a group of dolphins, illustrating their behavioral connections

    Dolphins [27]. This dataset describes the social interac- tions within a group of dolphins, illustrating their behavioral connections

    work page

  3. [3]

    This is a social network which describes the co-appearance of characters in Les Mis ´erables, the novel by Victor Hugo

    Les Mis ´erables [28] . This is a social network which describes the co-appearance of characters in Les Mis ´erables, the novel by Victor Hugo

    work page

  4. [4]

    This dataset portrays relationships among jazz musicians who belong to the same band

    Jazz [29]. This dataset portrays relationships among jazz musicians who belong to the same band

    work page

  5. [5]

    circles” (or “friend lists

    Facebook [30]. This dataset consists of “circles” (or “friend lists”) for a subset of users on Facebook

    work page

  6. [6]

    This is an email communication network at Rovira i Virgili University in southern Catalonia, Spain

    Email [31]. This is an email communication network at Rovira i Virgili University in southern Catalonia, Spain

    work page

  7. [7]

    This is a co-author network composed of scientists engaged in network theory and experiments

    Netscience [32]. This is a co-author network composed of scientists engaged in network theory and experiments. Nodes represent researchers, and a link between two nodes indicates that the corresponding researchers have co-authored at least one paper together

    work page

  8. [8]

    This is a co-author network extracted from the DBLP (Digital Bibliography & Library Project) computer science bibliography

    CoAuthorsDBLP [33]. This is a co-author network extracted from the DBLP (Digital Bibliography & Library Project) computer science bibliography

    work page

Show all 70 references
  1. [9]

    This is a co-author network 3 derived from the Citeseer database

    CoAuthorsCiteseer [33]. This is a co-author network 3 derived from the Citeseer database

  2. [10]

    This is a communication network representing the structure of IPv4 routing

    IPv4 [34]. This is a communication network representing the structure of IPv4 routing. Nodes represent Autonomous Systems (AS), and edges represent the connectivity between them

  3. [11]

    This is a hyperlink network of the .in domain in India

    In-2004 [35]. This is a hyperlink network of the .in domain in India. Nodes represent individual web pages using the .in domain, and edges represent hyperlinks between them

    2004

  4. [12]

    This is a communication network of the English Wikipedia

    Wiki-Talk [35]. This is a communication network of the English Wikipedia. Nodes represent users, and an edge indicates that one user has posted a message on another user’s talk page. The basic properties of these networks are summarized in Table I, including the number of node...

    2004

  5. [13]

    This network consists of three layers of relationships among 7th graders in Victoria, Australia: getting along, best friends, and preferred work part- ners

    Vickers-Chan-7thGraders [46]. This network consists of three layers of relationships among 7th graders in Victoria, Australia: getting along, best friends, and preferred work part- ners

  6. [14]

    This network represents three types of relationships between partners and associates of a corporate law firm: coworkers, friendship, and advice

    Lazega-Law-Firm [47]. This network represents three types of relationships between partners and associates of a corporate law firm: coworkers, friendship, and advice

  7. [15]

    This network comprises three layers of relationships within an international crisis group: trust, oper- ational interactions, and communication

    Noordin-Top [48]. This network comprises three layers of relationships within an international crisis group: trust, oper- ational interactions, and communication

  8. [16]

    ElectrJ”), chemical monadic (“MonoSyn

    Celegans-Connectome [49]. This network includes three types of synaptic connections in the Caenorhabditis ele- gans connectome: electric (“ElectrJ”), chemical monadic (“MonoSyn”), and polyadic (“PolySyn”)

  9. [17]

    This network contains three layers of social relationships among Twitter users during the Sanremo Music Festival Final: retweets, mentions, and replies

    Sanremo-2016 [50]. This network contains three layers of social relationships among Twitter users during the Sanremo Music Festival Final: retweets, mentions, and replies. The properties of the networks, including the number of nodes, links and the maximum coreness value in ea...

    2016

  10. [18]

    Detecting community structure in net- works,

    M. E. J. Newman, “Detecting community structure in net- works,” Eur. Phys. J. B38, 321-330 (2004)

    2004

  11. [19]

    Modularity and community structure in net- works,

    M. E. J. Newman, “Modularity and community structure in net- works,” Proc. Natl. Acad. Sci. USA 103, 8577-8582 (2006)

    2006

  12. [20]

    20 years of network commu- nity detection,

    S. Fortunato, M. E. J. Newman, “20 years of network commu- nity detection,”, Nat. Phys. 18, 848-850 (2022)

    2022

  13. [21]

    Community detection in networks: A user guide,

    S. Fortunato and D. Hric, “Community detection in networks: A user guide,” Phys. Rep. 659, 1-44 (2016)

    2016

  14. [22]

    Shang, Generalized k-core percolation in networks with community structure, SIAM J

    Y . Shang, Generalized k-core percolation in networks with community structure, SIAM J. Appl. Math. 80, 1272 (2020)

    2020

  15. [23]

    Core- Periphery Structure in Networks (Revisited),

    P. Rombach, M. A. Porter, J. H. Fowler, and P. J. Mucha, “Core- Periphery Structure in Networks (Revisited),” SIAM Review 59, 619 (2017)

    2017

  16. [24]

    Models of core/periphery structures,

    S. P. Borgatti and M. G. Everett, “Models of core/periphery structures,” Social networks 21, 375-395 (2000)

    2000

  17. [25]

    Core-periphery organization of complex networks,

    P. Holme, “Core-periphery organization of complex networks,” Phys. Rev. E 72, 046111 (2005)

    2005

  18. [26]

    Core- periphery structure in directed networks,

    A. Elliott, A. Chiu, M. Bazzi, G. Reinert, M. Cucuringu, “Core- periphery structure in directed networks,” Proc. R. Soc. A 476, 20190783 (2020)

    2020

  19. [27]

    A clarified typology of core-periphery structure in networks,

    R. J. Gallagher, J. G. Young, B. F. Welles, “A clarified typology of core-periphery structure in networks,” Sci. Adv.7, eabc9800 (2021)

    2021

  20. [28]

    Hierarchical core-periphery structure in networks,

    A. Polanco, M. E. J. Newman, “Hierarchical core-periphery structure in networks,” Phys. Rev. E108, 024311 (2023)

    2023

  21. [29]

    Revealing the structure of the world airline network,

    T. Verma, N. A. Ara ´ujo, and H. J. Herrmann, “Revealing the structure of the world airline network,” Sci. Rep. 4, 5638 (2014)

    2014

  22. [30]

    Emergence of core–peripheries in networks,

    T. Verma, F. Russmann, N. A. M. Ara ´ujo, J. Nagler, H. J. Herrmann, “Emergence of core–peripheries in networks,” Nat. Comm. 7, 10441 (2016)

    2016

  23. [31]

    The evolution of the world trade web: a weighted-network analysis,

    G. Fagiolo, J. Reyes, and S. Schiavo, “The evolution of the world trade web: a weighted-network analysis,” J. Evol. Econ. 20, 479-514 (2010)

    2010

  24. [32]

    Profiling core- periphery network structure by random walkers,

    F. D. Rossa, F. Dercole, and C. Piccardi, “Profiling core- periphery network structure by random walkers,” Sci. Rep. 3, 1467 (2013)

    2013

  25. [33]

    Financial networks and contagion,

    M. Elliott, B. Golub, and M. O. Jackson, “Financial networks and contagion,” Am. Econ. Rev.104, 3115-3153 (2014)

    2014

  26. [34]

    Vip-club phenomenon: Emergence of elites and masterminds in social networks,

    N. Masuda and N. Konno, “Vip-club phenomenon: Emergence of elites and masterminds in social networks,” PLoS One 28, 297-309 (2006). 7

    2006

  27. [35]

    Density-based and transport-based core-periphery structures in networks,

    S. H. Lee, M. Cucuringu, and M. A. Porter, “Density-based and transport-based core-periphery structures in networks,” Phys. Rev. E 89, 032810 (2014)

    2014

  28. [36]

    Finding multiple core-periphery pairs in networks,

    S. Kojaku and N. Masuda, “Finding multiple core-periphery pairs in networks,” Phys. Rev. E96, 052313 (2017)

    2017

  29. [37]

    Identification of core-periphery structure in networks,

    X. Zhang, T. Martin, M. E. J. Newman, “Identification of core-periphery structure in networks,” Phys. Rev. E91, 032803 (2015)

    2015

  30. [38]

    Rich-cores in networks,

    A. Ma and R. J. Mondrag ´on, “Rich-cores in networks,” PLoS One 10, e0119678 (2015)

    2015

  31. [39]

    Network partition via a bound of the spectral radius,

    R. J. Mondrag ´on, “Network partition via a bound of the spectral radius,” J. Complex Netw.5, 513-526 (2017)

    2017

  32. [40]

    Anatomy of funded research in science,

    A. Ma, R. J. Mondrag ´on, and V . Latora, “Anatomy of funded research in science,” Proc. Natl Acad. Sci. USA 112, 14760- 14765 (2015)

    2015

  33. [41]

    Multiplex core–periphery organization of the human connectome,

    F. Battiston, J. Guillon, M. Chavez, V . Latora, and F. De Vico Fallani, “Multiplex core–periphery organization of the human connectome,” J. R. Soc. Interface 15, 20180514 (2018)

    2018

  34. [42]

    Identification of influential spreaders in complex networks,

    M. Kitsak, L. K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H. E. Stanley, and H. A. Makse, “Identification of influential spreaders in complex networks,” Nat. Phys. 6, 888 (2010)

    2010

  35. [43]

    An information flow model for conflict and fission in small groups,

    W. W. Zachary, “An information flow model for conflict and fission in small groups,” J. Anthropol. Res.33, 452-473 (1977)

    1977

  36. [44]

    The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting as- sociations: can geographic isolation explain this unique trait?

    D. Lusseau, K. Schneider, O. J. Boisseau, P. Haase, E. Slooten, and S. M. Dawson, “The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting as- sociations: can geographic isolation explain this unique trait?” Behav. Ecol. Sociobiol. 54, 39...

    2003

  37. [45]

    D. E. Knuth, The Stanford GraphBase: a platform for combina- torial computing, (AcM Press, New York, 1993)

    1993

  38. [46]

    Community structure in jazz,

    P. M. Gleiser and L. Danon, “Community structure in jazz,” Adv. Complex Syst. 6, 565-573 (2003)

    2003

  39. [47]

    On the evolution of user interaction in facebook,

    B. Viswanath, A. Mislove, M. Cha, and K. P. Gummadi, “On the evolution of user interaction in facebook,” In Proceedings of the 2nd ACM workshop on Online social networks (ACM, 2009), pp. 37-42

    2009

  40. [48]

    Self-similar community structure in a network of human interactions,

    R. Guimera, L. Danon, A. Diaz-Guilera, F. Giralt, and A. Are- nas, “Self-similar community structure in a network of human interactions,” Phys. Rev. E68, 065103 (2003)

    2003

  41. [49]

    Finding community structure in networks us- ing the eigenvectors of matrices,

    M. E. Newman, “Finding community structure in networks us- ing the eigenvectors of matrices,” Phys. Rev. E 74, 036104 (2006)

    2006

  42. [50]

    The network data repository with interactive graph analytics and visualization,

    R. A. Rossi and N. K. Ahmed, “The network data repository with interactive graph analytics and visualization,” In Proceed- ings of 29th AAAI Conference on Artificial Intelligence (ACM, 2015), pp. 4292-4293

    2015

  43. [51]

    See https://catalog.caida.org/dataset/as relationships serial 1 for information about the AS relationship network

  44. [52]

    KONECT — The Koblenz Network Collection

    J. Kunegis. “KONECT — The Koblenz Network Collection.” In Proceedings of the 22nd International Conference on World Wide Web Companion (ACM, 2013), pp. 1343-1350

    2013

  45. [53]

    Algorithm 457: finding all cliques of an undirected graph,

    C. Bron and J. Kerbosch, “Algorithm 457: finding all cliques of an undirected graph,” Commun. ACM 16, 575-577 (1973)

    1973

  46. [54]

    The worst-case time complexity for generating all maximal cliques and computa- tional experiments,

    E. Tomita, A. Tanaka, and H. Takahashi, “The worst-case time complexity for generating all maximal cliques and computa- tional experiments,” Theor. Comput. Sci.363, 28-42 (2006)

    2006

  47. [55]

    A note on the problem of reporting maximal cliques,

    F. Cazals and C. Karande, “A note on the problem of reporting maximal cliques,” Theor. Comput. Sci.407, 564-568 (2008)

    2008

  48. [56]

    The structure and dynamics of multilayer networks,

    S. Boccaletti, G. Bianconi, R. Criado, C.I. del Genio, J. G´omez- Garde˜nes, M. Romance, I. Sendi ˜na-Nadal, Z. Wang, and M. Zanin, “The structure and dynamics of multilayer networks,” Phys. Rep. 544, 1-122 (2014)

    2014

  49. [57]

    Towards real-world com- plexity: an introduction to multiplex networks,

    K. M. Lee, B. Min, and K. I. Goh, “Towards real-world com- plexity: an introduction to multiplex networks,” Eur. Phys. J. B 88, 1-20 (2015)

    2015

  50. [58]

    Totally asymmetric simple exclusion process on multiplex networks,

    G. Shen, X. Fan, and Z. Ruan, “Totally asymmetric simple exclusion process on multiplex networks,” Chao 30, 023103 (2020)

    2020

  51. [59]

    Integrated travel network model for studying epidemics: Interplay between jour- neys and epidemic,

    Z. Ruan, C. Wang, P. M. Hui, and Z. Liu, “Integrated travel network model for studying epidemics: Interplay between jour- neys and epidemic,” Sci. Rep. 5, 11401 (2015)

    2015

  52. [60]

    Shang, Generalized k-core percolation on correlated and uncorrelated multiplex networks, Phys

    Y . Shang, Generalized k-core percolation on correlated and uncorrelated multiplex networks, Phys. Rev. E 101, 042306 (2020)

    2020

  53. [61]

    Shang, Feature-enriched core percolation in multiplex net- works, Phys

    Y . Shang, Feature-enriched core percolation in multiplex net- works, Phys. Rev. E 106, 054314 (2022)

    2022

  54. [62]

    To address this imbalance, one approach is to weight richness by the maximum coreness and maximum centrality value in each layer

    In some cases, different layers may differ significantly in link density. To address this imbalance, one approach is to weight richness by the maximum coreness and maximum centrality value in each layer

  55. [63]

    Representing classroom social struc- ture,

    M. Vickers and S. Chan, “Representing classroom social struc- ture,” Victoria Institute of Secondary Education, Melbourne (1981)

    1981

  56. [64]

    Lazega, The collegial phenomenon: The social mechanisms of cooperation among peers in a corporate law partnership (Ox- ford University Press, USA, 2001)

    E. Lazega, The collegial phenomenon: The social mechanisms of cooperation among peers in a corporate law partnership (Ox- ford University Press, USA, 2001)

    2001

  57. [65]

    Structural measures for multiplex networks,

    F. Battiston, V . Nicosia, V Latora, “Structural measures for multiplex networks,” Phys. Rev. E89, 032804 (2014)

    2014

  58. [66]

    Wiring opti- mization can relate neuronal structure and function,

    B. L. Chen, D. H. Hall, and D. B. Chklovskii, “Wiring opti- mization can relate neuronal structure and function,” Proc. Natl Acad. Sci. USA 103, 4723-4728 (2006)

    2006

  59. [67]

    Unraveling the origin of social bursts in collective attention,

    M. De Domenico and E. G. Altmann, “Unraveling the origin of social bursts in collective attention,” Sci. Rep.10, 4629 (2020)

    2020

  60. [68]

    Z. Ruan, B. Yu, X. Shu, Q. Zhang, and Q. Xuan, The impact of malicious nodes on the spreading of false information, Chaos 30, 083101 (2020)

    2020

  61. [69]

    Holme and J

    P. Holme and J. Saramaki, Temporal networks, Phys. Rep. 519, 97 (2012)

    2012

  62. [70]

    Battiston, G

    F. Battiston, G. Cencetti, I. Iacopini, V . Latora, M. Lucas, A. Patania, J. G. Young, and G. Petri, Networks beyond pairwise interactions: structure and dynamics, Phys. Rep. 874, 1 (2020)

    2020

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.