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REVIEW 3 major objections 5 minor 55 references

Misinformation spreading on correlated multiplex networks

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that a heterogeneous edge-based compartmental theory accurately predicts misinformation outbreak sizes and thresholds on correlated multiplex networks, with positive inter-layer correlation lowering the threshold.

desk verdict Solid incremental multiplex-spreading theory with an oversold threshold: the Jacobian condition gives a beta_c that depends on lambda and gamma separately, not a single network-intrinsic value. read the letter →

arxiv 1909.00397 v1 pith:LYRFBZCM submitted 2019-09-01 physics.soc-ph cs.SI

Jiajun Xian , Dan Yang , Liming Pan , Wei Wang , Zhen Wang This is my paper · ORCID
classification physics.soc-phcs.SI
keywords misinformationspreadingmultiplexnetworksinter-layercorrelationedge-basedcompartmentaltheoryoutbreakthresholdignorant-spreader-recoveredmodeldegreeheterogeneitystabilityanalysis
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

This paper tries to predict how misinformation spreads when each person is active on two online platforms at once and the two platform networks are correlated in who is well connected on each. It proposes an ignorant-spreader-recovered model on a two-layer multiplex network and derives a heterogeneous edge-based compartmental theory for the final outbreak size $R(\infty)$. The theory also yields an outbreak threshold $\beta_c$ from a stability analysis of the trivial fixed point. The central result is that a larger average degree, stronger degree heterogeneity, or a more positive inter-layer correlation lowers $\beta_c$, so misinformation breaks out more easily, and these same factors enlarge $R(\infty)$ for weak transmission but shrink it for strong transmission. The theoretical predictions agree with stochastic simulations on configuration-model duplex networks.

What carries the argument

The central object is the heterogeneous edge-based compartmental theory, an extension of edge-based compartmental methods that tracks, separately for each degree pair $(k_a,k_b)$, the probability $\theta_a(k_a,k_b,t)$ that a node in layer $a$ has not transmitted misinformation along a given edge by time $t$. The theory closes the dynamics with the cavity assumption: the recipient of a transmission is treated as unable to transmit, so the remaining neighbors of the sender are independent. This yields the ignorance probability $I(\vec{k},t)=\theta_1(t)^{k_1}\theta_2(t)^{k_2}$, the outbreak size $R(\infty)=1-\sum_{\vec{k}}p(\vec{k})I(\vec{k},\infty)$, and a Jacobian matrix whose leading eigenvalue locates $\beta_c$.

What would settle it

Take a duplex configuration-model network with the same joint degree distribution used in the paper but with many triangles added inside each layer, run the ignorant-spreader-recovered process, and compare $R(\infty)$ versus $\beta$ to Eqs. (12)–(14). A systematic deviation beyond simulation error would show that the neighbour-independence closure, not just parameter error, is the failing step.

Watch

Extended reading notes

Core claim

On a duplex network in which each individual is represented by one replica node in each layer, the probability that a node with degree pair $(k_1,k_2)$ is still ignorant at time $t$ is written as $I(\vec{k},t)=\theta_1(t)^{k_1}\theta_2(t)^{k_2}$, where $\theta_a(t)$ is the average probability that a randomly chosen edge in layer $a$ has not yet transmitted the misinformation. The evolution of the degree-dependent transmission probabilities $\theta_a(k_a,k_b,t)$ is closed by assuming the neighbor reached along an edge is in a cavity state, which makes the states of the other neighbors statistically independent. The outbreak threshold is located by linearizing the resulting dynamical system around the fixed point $\theta_a=1$, $\xi_a^R=0$ and finding where the leading eigenvalue $\omega$ of the Jacobian leaves zero. The paper's claim is that this theory reproduces, for random and scale-free duplex networks, the simulation results for both the final outbreak size $R(\infty)$ and the threshold $\beta_c$.

Load-bearing premise

The main load-bearing assumption is the mean-field closure: all nodes with the same degree pair behave identically, and once a neighbor is placed in the cavity state, the states of the remaining neighbors are treated as statistically independent.

Editorial extensions

If this is right

  • Misinformation outbreaks in this model are continuous: $R(\infty)$ rises smoothly from zero once $\beta$ passes $\beta_c$, so there is no hysteresis or bistability.
  • Because larger average degree lowers $\beta_c$, any intervention that reduces connection density in either platform raises the amount of transmission needed for an outbreak.
  • Positive inter-layer correlation makes outbreaks harder to prevent: when the same individuals are hubs on both platforms, the leading eigenvalue of the Jacobian crosses zero at a smaller $\beta$.
  • At large effective transmission rates the picture reverses: stronger heterogeneity and positive correlation leave more low-degree margin nodes ignorant, so the final outbreak size is smaller than in homogeneous or negatively correlated networks.

Reading between the lines

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

  • A consequence the authors leave implicit is that hub-pairs, users who are central on both platforms, are the natural targets for intervention: reducing their activity on either layer should push the leading eigenvalue back below zero.
  • The reinforced recovery rule is doing important work in the theory; re-running the same analysis with constant recovery probability would show whether the reported crossover in $R(\infty)$ is a property of reinforcement or of the multiplex structure itself.
  • On empirical multi-platform data, the paper's crossover prediction is testable directly: measure the rank correlation of degree rankings across platforms and check whether platforms with more positive correlation show smaller rumor cascades at high exposure but larger ones at low exposure.
  • If the model is extended to more than two layers, the same formulas generalize by replacing the product $\theta_1^{k_1}\theta_2^{k_2}$ with a product over layers, which would make the predicted threshold shift even more sensitive to positive inter-layer correlation.
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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

3 major / 5 minor

Summary. This paper studies misinformation spreading on two-layer multiplex networks with tunable inter-layer degree correlations. The authors propose an ignorant-spreader-recovered (ISR) model in which a spreader's recovery probability grows with the number of its spreader/recovered neighbors, develop a heterogeneous edge-based compartmental theory for the final outbreak size R(∞), and derive a threshold from a Jacobian stability analysis of the mean-field equations. They then compare theory with stochastic simulations on Erdős–Rényi and scale-free multiplex networks. The central claims are that R(∞) grows continuously with the effective transmission probability β = λ/γ above an outbreak threshold β_c, and that larger average degree, stronger degree heterogeneity, and more positive inter-layer correlation reduce β_c, while heterogeneity and correlation enlarge (reduce) R(∞) for small (large) β.

Significance. If the threshold analysis is corrected, the paper would offer a useful extension of edge-based compartmental theory to multiplex networks with a nonlinear, neighbor-dependent recovery rule. Strengths include a self-contained derivation that is compared with stochastic simulations without any fitted parameters, and coverage of both homogeneous and heterogeneous multiplex networks with positive and negative inter-layer correlations. The circularity burden is low because the equations are derived from the model and the threshold is not tuned to the simulation data. However, the current definition of β_c is technically flawed: the eigenvalue condition in Section III.B yields a threshold that depends on λ and γ separately, not on the ratio β alone, which undermines the paper's main quantitative claim as stated.

major comments (3)
  1. [Section III.B, Eq. (21)] The outbreak threshold obtained from Eq. (21) is not a function of β = λ/γ alone. Writing M = [[X1−I, Y12], [Y21, X2−I]], the Jacobian has the block form J = [[λM, λI], [γ(λ−1)M, γ(λ−1)I]]. For an eigenvector (u,v) with M u = μ u, the eigenvalues of J are 0 and λμ + γ(λ−1). The nontrivial branch crosses zero at λ_c = γ/(μ_max+γ), i.e., β_c = 1/(μ_max+γ), which depends explicitly on γ. Therefore two parameter pairs (λ,γ) with the same ratio β can lie on opposite sides of the outbreak threshold, contradicting the abstract's and Section IV's statements that a single network-structure-dependent β_c governs spreading. The manuscript should present the threshold as a curve in the (λ,γ) plane and qualify the structural-effect statements to fixed γ.
  2. [Section III.B] The criterion that the leading Jacobian eigenvalue ω 'deviates from zero' at β_c is not a well-defined stability threshold and is inconsistent with Eq. (21). The vector (u,−Mu)^T is in the kernel of J for every eigenvector u of M, so J has an exact zero eigenvalue for all λ and γ. Below the threshold the leading eigenvalue is therefore exactly zero, not 'a small number very close to zero' as stated in the text. The threshold should be defined by the sign change of the nonzero branch λμ_max + γ(λ−1). In addition, the caption of Fig. 2 does not state the value of γ used for each curve, which is essential under the corrected criterion.
  3. [Section IV, Figs. 3-7] Because β_c depends on γ, the comparisons of R(∞) versus β at different γ values (Figs. 3, 5, and 6) and the threshold estimates from the peak of Δ (Figs. 3, 5, and 7) need a quantitative report of the theoretical and simulated threshold values for each γ and network structure. The manuscript currently marks the theoretical thresholds only as dotted lines and gives no numeric values, so the claimed 'good agreement' cannot be fully checked from the information provided. A table reporting β_c (or λ_c) from Eq. (21), from the Δ-peak method, and from simulations would resolve this.
minor comments (5)
  1. [Eq. (17) and surrounding text] The factor '1 − λ −1' should be typeset as 1 − λ^{-1}; the missing superscript makes the equation ambiguous and should be corrected throughout.
  2. [Eqs. (7)-(9)] The notation for the probabilities r_a^a and r_b^a is garbled in the printed text (they appear as 'ra' and 'rb'), and the subscripts and superscripts in Eqs. (8)-(9) are inconsistent. Please rewrite these equations with unambiguous labels, such as r_{a→a} and r_{b→a}.
  3. [General production quality] The manuscript contains several OCR or encoding artifacts, including the stray characters '澳' and 'ť' in the text and garbled axis labels in Figs. 2-7; these should be corrected in the production version.
  4. [Section III.A] The cavity and mean-field assumptions underlying Eqs. (4)-(6) and (12) are not stated as explicitly as they should be; a short paragraph noting that the theory targets tree-like configuration-model networks and may be less accurate for clustered or strongly correlated empirical networks would make the scope clear.
  5. [Abstract and Section I] The statement that 'no systematical theoretical study has been performed to date' is stronger than necessary; earlier edge-based and mean-field analyses of multiplex spreading should be cited in this context to avoid overclaiming novelty.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EBCM and Jacobian threshold are derived from the model assumptions and validated against independent stochastic simulations, not from fitted inputs or self-citation chains.

full rationale

The paper's central derivation is self-contained. The EBCM equations (Eqs. (3)-(14)) are constructed from the stated ISR model rules (transmission probability λ, recovery probability 1−(1−γ)^n, joint degree distribution p(k1,k2)) using the standard cavity closure, and the outbreak threshold is obtained by a Jacobian eigenvalue analysis at the trivial fixed point (Eqs. (15)-(21)). No parameter is fitted to the simulation outputs, and the threshold is not chosen to match the data; it is computed from the model equations and then compared with independently measured simulation thresholds. Although several references are self-citations (e.g., Refs. 28, 36, 37, 39, 40, 49, 50), the load-bearing edge-based compartmental framework is also grounded in the externally established works of Refs. 43-48 (Miller, Volz, Valdez, Braunstein), and the variability-peak method of Ref. 50 is used only as a simulation-side estimator of the threshold, not as an input to the theory. The simulations are stochastic realizations of the same model, so agreement is a genuine consistency check rather than a circular restatement. The possible dependence of the Jacobian eigenvalue on λ and γ separately in Eq. (21) is a mathematical consistency concern about the claimed β-only threshold, but it is not a case of a predicted quantity being equivalent to its input by construction, so it does not constitute circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central theoretical claims depend on the cavity/mean-field closure, the configuration model assumption, and the specific model mechanisms of perfect synchronization and reinforced recovery. No free parameters are fitted to data; λ, γ, and degree distributions are inputs. The threshold method adds an ad hoc numerical criterion.

assumptions (5)
  • domain assumption Cavity approximation: a node in the cavity state cannot transmit but can receive; neighbor states are treated as statistically independent given degree.
    Used to derive Eqs. (3)-(5) and (12); standard in edge-based compartmental theory but an approximation that fails for clustered networks.
  • domain assumption Configuration model: multiplex networks are generated from a joint degree distribution p(k1,k2) with negligible intra-layer degree correlations.
    Section II.A; the theory assumes the network is tree-like and fully described by the joint degree distribution.
  • domain assumption Perfect inter-layer synchronization: when an individual's replica in one layer becomes S or R, the counterpart in the other layer changes state simultaneously.
    Section II.B; a strong modeling assumption that may not hold for real multi-platform users.
  • domain assumption Reinforced recovery mechanism: a spreader recovers with probability 1-(1-γ)^n, where n counts spreader/recovered neighbors in both layers.
    Section II.B and Eq. (7); defines the model's recovery dynamics.
  • ad hoc to paper The outbreak threshold is the value of β where the leading eigenvalue of the Jacobian deviates from zero.
    Section III.B; not a standard formal stability condition, the threshold is identified numerically from the Jacobian at the disease-free fixed point.

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Pith. "Pith review of Misinformation spreading on correlated multiplex networks." pith.science (2026). https://pith.science/paper/LYRFBZCM

@misc{pith2026190900397,
  author       = {Pith},
  title        = {Pith review of: Misinformation spreading on correlated multiplex networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LYRFBZCM}},
  note         = {Machine review of arXiv:1909.00397}
}
abstract

The numerous expanding online social networks offer fast channels for misinformation spreading, which could have a serious impact on socioeconomic systems. Researchers across multiple areas have paid attention to this issue with a view of addressing it. However, no systematical theoretical study has been performed to date on observing misinformation spreading on correlated multiplex networks. In this study, we propose a multiplex network-based misinformation spreading model, considering the fact that each individual can obtain misinformation from multiple platforms. Subsequently, we develop a heterogeneous edge-base compartmental theory to comprehend the spreading dynamics of our proposed model. In addition, we establish an analytical method based on stability analysis to obtain the misinformation outbreak threshold. On the basis of these theories, we finally analyze the influence of different dynamical and structural parameters on the misinformation spreading dynamics. Results show that the misinformation outbreak size $R(\infty)$ grows continuously with the effective transmission probability $\beta$ once $\beta$ exceeds a certain value, that is, the outbreak threshold $\beta_c$. A large average degrees, strong degree heterogeneity, or positive inter-layer correlation will reduce $\beta_c$, accelerating the outbreak of misinformation. Besides, increasing the degree heterogeneity or a more positive inter-layer correlation will both enlarge (reduce) $R(\infty)$ for small (large) values of $\beta$. Our systematic theoretical analysis results agree well with the numerical simulation results. Our proposed model and accurate theoretical analysis will serve as a useful framework to understand and predict the spreading dynamics of misinformation on multiplex networks, and thereby pave the way to address this serious issue.

Figures

Figures reproduced from arXiv: 1909.00397 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) Illustrations of (a) the generalized [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) Leading eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) Dynamics of spreading of misinformat [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) (a) Simulation results and (b) theore [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Effects of degree heterogeneity on th [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) Spreading dynamics of misinformatio [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (Color online) Influence of inter-layer correlation [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

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Reference graph

Works this paper leans on

55 extracted references · 49 canonical work pages

  1. [1]

    merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

    2010

  2. [2]

    merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

    2010

  3. [3]

    merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  4. [4]

    Vosoughi , author D

    author author S. Vosoughi , author D. Roy , \ and\ author S. Aral ,\ @noop journal journal Science \ volume 359 ,\ pages 1146 ( year 2018 ) NoStop

    work page 2018

  5. [5]

    author author D. M. \ Lazer , author M. A. \ Baum , author Y. Benkler , author A. J. \ Berinsky , author K. M. \ Greenhill , author F. Menczer , author M. J. \ Metzger , author B. Nyhan , author G. Pennycook , author D. Rothschild , et al. ,\ @noop journal journal Science \ volume 359 ,\ pages 1094 ( year 2018 ) NoStop

    work page 2018

  6. [6]

    Shao , author G

    author author C. Shao , author G. L. \ Ciampaglia , author O. Varol , author K.-C. \ Yang , author A. Flammini , \ and\ author F. Menczer ,\ @noop journal journal Nature communications \ volume 9 ,\ pages 4787 ( year 2018 ) NoStop

    work page 2018

  7. [7]

    Kumar \ and\ author N

    author author S. Kumar \ and\ author N. Shah ,\ @noop journal journal arXiv preprint arXiv:1804.08559 \ ( year 2018 ) NoStop

    arXiv 2018

  8. [8]

    Bovet \ and\ author H

    author author A. Bovet \ and\ author H. A. \ Makse ,\ @noop journal journal Nature communications \ volume 10 ,\ pages 7 ( year 2019 ) NoStop

    work page 2019

Show all 55 references
  1. [9]

    Castellano , author S

    author author C. Castellano , author S. Fortunato , \ and\ author V. Loreto ,\ @noop journal journal Reviews of Modern Physics \ volume 81 ,\ pages 591 ( year 2009 ) NoStop

    2009

  2. [10]

    Pastor-Satorras , author C

    author author R. Pastor-Satorras , author C. Castellano , author P. Van Mieghem , \ and\ author A. Vespignani ,\ @noop journal journal Reviews of Modern Physics \ volume 87 ,\ pages 925 ( year 2015 ) NoStop

    2015

  3. [11]

    Wang , author M

    author author W. Wang , author M. Tang , author H. E. \ Stanley , \ and\ author L. A. \ Braunstein ,\ @noop journal journal Reports on Progress in Physics \ volume 80 ,\ pages 036603 ( year 2017 ) NoStop

    2017

  4. [12]

    Newman ,\ @noop title Networks: an introduction \ ( publisher Oxford University Press ,\ year 2010 ) NoStop

    author author M. Newman ,\ @noop title Networks: an introduction \ ( publisher Oxford University Press ,\ year 2010 ) NoStop

    2010

  5. [13]

    Jalili , author Y

    author author M. Jalili , author Y. Orouskhani , author M. Asgari , author N. Alipourfard , \ and\ author M. Perc ,\ @noop journal journal Royal Society open science \ volume 4 ,\ pages 160863 ( year 2017 ) NoStop

    2017

  6. [14]

    Karp , author C

    author author R. Karp , author C. Schindelhauer , author S. Shenker , \ and\ author B. Vocking ,\ in\ @noop booktitle Proc. 41st Symp. Foundations of Computer Science (FOCS). \ ( organization IEEE ,\ year 2000 )\ pp.\ pages 565--574 NoStop

    2000

  7. [15]

    Moreno , author M

    author author Y. Moreno , author M. Nekovee , \ and\ author A. F. \ Pacheco ,\ @noop journal journal Physical Review E \ volume 69 ,\ pages 066130 ( year 2004 ) NoStop

    2004

  8. [16]

    Zhao , author J

    author author L. Zhao , author J. Wang , author Y. Chen , author Q. Wang , author J. Cheng , \ and\ author H. Cui ,\ @noop journal journal Physica A: Statistical Mechanics and its Applications \ volume 391 ,\ pages 2444 ( year 2012 ) NoStop

    2012

  9. [17]

    Zhou , author Z

    author author J. Zhou , author Z. Liu , \ and\ author B. Li ,\ @noop journal journal Physics Letters A \ volume 368 ,\ pages 458 ( year 2007 ) NoStop

    2007

  10. [18]

    Moreno , author R

    author author Y. Moreno , author R. Pastor-Satorras , \ and\ author A. Vespignani ,\ @noop journal journal The European Physical Journal B \ volume 26 ,\ pages 521 ( year 2002 ) NoStop

    2002

  11. [19]

    author author R. M. \ May \ and\ author A. L. \ Lloyd ,\ @noop journal journal Physical Review E \ volume 64 ,\ pages 066112 ( year 2001 ) NoStop

    2001

  12. [20]

    Gross , author C

    author author T. Gross , author C. J. D. \ D’Lima , \ and\ author B. Blasius ,\ @noop journal journal Physical Review Letters \ volume 96 ,\ pages 208701 ( year 2006 ) NoStop

    2006

  13. [21]

    Parshani , author S

    author author R. Parshani , author S. Carmi , \ and\ author S. Havlin ,\ @noop journal journal Physical Review Letters \ volume 104 ,\ pages 258701 ( year 2010 ) NoStop

    2010

  14. [22]

    author author D. J. \ Watts ,\ @noop journal journal Proceedings of the National Academy of Sciences of the United States of America \ volume 99 ,\ pages 5766 ( year 2002 ) NoStop

    2002

  15. [23]

    author author P. S. \ Dodds \ and\ author D. J. \ Watts ,\ @noop journal journal Physical review letters \ volume 92 ,\ pages 218701 ( year 2004 ) NoStop

    2004

  16. [24]

    Wang , author H

    author author W. Wang , author H. E. \ Stanley , \ and\ author L. A. \ Braunstein ,\ @noop journal journal New Journal of Physics \ volume 20 ,\ pages 013034 ( year 2018 ) NoStop

    2018

  17. [25]

    Zheng , author L

    author author M. Zheng , author L. L \"u , author M. Zhao , et al. ,\ @noop journal journal Physical Review E \ volume 88 ,\ pages 012818 ( year 2013 ) NoStop

    2013

  18. [26]

    Wang , author L

    author author Z. Wang , author L. Wang , author A. Szolnoki , \ and\ author M. Perc ,\ @noop journal journal The European physical journal B \ volume 88 ,\ pages 124 ( year 2015 a ) NoStop

    2015

  19. [27]

    Boccaletti , author G

    author author S. Boccaletti , author G. Bianconi , author R. Criado , author C. I. \ Del Genio , author J. G \'o mez-Garde \ n es , author M. Romance , author I. Sendi \ n a-Nadal , author Z. Wang , \ and\ author M. Zanin ,\ @noop journal journal Physics Reports \ volume 544 ,...

    2014

  20. [28]

    Kivel \"a , author A

    author author M. Kivel \"a , author A. Arenas , author M. Barthelemy , author J. P. \ Gleeson , author Y. Moreno , \ and\ author M. A. \ Porter ,\ @noop journal journal Journal of Complex Networks \ volume 2 ,\ pages 203 ( year 2014 ) NoStop

    2014

  21. [29]

    Wang , author A

    author author Z. Wang , author A. Szolnoki , \ and\ author M. Perc ,\ @noop journal journal Scientific reports \ volume 3 ( year 2013 ) NoStop

    2013

  22. [30]

    Gao , author S

    author author J. Gao , author S. V. \ Buldyrev , author H. E. \ Stanley , \ and\ author S. Havlin ,\ @noop journal journal Nature Physics \ volume 8 ,\ pages 40 ( year 2012 ) NoStop

    2012

  23. [31]

    Pan , author W

    author author L. Pan , author W. Wang , author S. Cai , \ and\ author T. Zhou ,\ @noop journal journal arXiv preprint arXiv:1905.09176 \ ( year 2019 ) NoStop

    1905 arXiv

  24. [32]

    Wang , author L

    author author Z. Wang , author L. Wang , \ and\ author M. Perc ,\ @noop journal journal Physical Review E \ volume 89 ,\ pages 052813 ( year 2014 ) NoStop

    2014

  25. [33]

    author author G. F. \ de Arruda , author F. A. \ Rodrigues , \ and\ author Y. Moreno ,\ @noop journal journal Physics Reports \ ( year 2018 ) NoStop

    2018

  26. [34]

    De Domenico , author C

    author author M. De Domenico , author C. Granell , author M. A. \ Porter , \ and\ author A. Arenas ,\ @noop journal journal Nature Physics \ volume 12 ,\ pages 901 ( year 2016 ) NoStop

    2016

  27. [35]

    Wang , author Q

    author author H. Wang , author Q. Li , author G. D’Agostino , author S. Havlin , author H. E. \ Stanley , \ and\ author P. Van Mieghem ,\ @noop journal journal Physical Review E \ volume 88 ,\ pages 022801 ( year 2013 ) NoStop

    2013

  28. [36]

    Dickison , author S

    author author M. Dickison , author S. Havlin , \ and\ author H. E. \ Stanley ,\ @noop journal journal Physical Review E \ volume 85 ,\ pages 066109 ( year 2012 ) NoStop

    2012

  29. [37]

    Saumell-Mendiola , author M

    author author A. Saumell-Mendiola , author M. \'A . \ Serrano , \ and\ author M. Bogun \'a ,\ @noop journal journal Physical Review E \ volume 86 ,\ pages 026106 ( year 2012 ) NoStop

    2012

  30. [38]

    Wang , author M

    author author W. Wang , author M. Tang , author H. E. \ Stanley , \ and\ author L. A. \ Braunstein ,\ @noop journal journal Physical Review E \ volume 98 ,\ pages 062320 ( year 2018 ) NoStop

    2018

  31. [39]

    Wang , author M

    author author W. Wang , author M. Tang , author H. Yang , author Y. Do , author Y.-C. \ Lai , \ and\ author G. Lee ,\ @noop journal journal Scientific Reports \ volume 4 ,\ pages 5097 ( year 2014 ) NoStop

    2014

  32. [40]

    Wang , author M

    author author W. Wang , author M. Cai , \ and\ author M. Zheng ,\ @noop journal journal Physica A: Statistical Mechanics and its Applications \ volume 499 ,\ pages 121 ( year 2018 ) NoStop

    2018

  33. [41]

    Catanzaro , author M

    author author M. Catanzaro , author M. Bogun \'a , \ and\ author R. Pastor-Satorras ,\ @noop journal journal Physical review e \ volume 71 ,\ pages 027103 ( year 2005 ) NoStop

    2005

  34. [42]

    Yang , author L

    author author D. Yang , author L. Pan , \ and\ author T. Zhou ,\ @noop journal journal Chaos: An Interdisciplinary Journal of Nonlinear Science \ volume 27 ,\ pages 033113 ( year 2017 ) NoStop

    2017

  35. [43]

    Guo , author D

    author author F. Guo , author D. Yang , author Z. Yang , author Z.-D. \ Zhao , \ and\ author T. Zhou ,\ @noop journal journal Physical Review E \ volume 95 ,\ pages 052314 ( year 2017 ) NoStop

    2017

  36. [44]

    \ Lee , author J

    author author K.-M. \ Lee , author J. Y. \ Kim , author W.-k. \ Cho , author K.-I. \ Goh , \ and\ author I. Kim ,\ @noop journal journal New Journal of Physics \ volume 14 ,\ pages 033027 ( year 2012 ) NoStop

    2012

  37. [45]

    \ Cho , author K.-I

    author author W.-k. \ Cho , author K.-I. \ Goh , \ and\ author I.-M. \ Kim ,\ @noop journal journal arXiv preprint arXiv:1010.4971 \ ( year 2010 ) NoStop

    2010 arXiv

  38. [46]

    author author J. C. \ Miller , author A. C. \ Slim , \ and\ author E. M. \ Volz ,\ @noop journal journal Journal of the Royal Society Interface \ volume 9 ,\ pages 890 ( year 2011 ) NoStop

    2011

  39. [47]

    Volz ,\ @noop journal journal Journal of Mathematical Biology \ volume 56 ,\ pages 293 ( year 2008 ) NoStop

    author author E. Volz ,\ @noop journal journal Journal of Mathematical Biology \ volume 56 ,\ pages 293 ( year 2008 ) NoStop

    2008

  40. [48]

    author author J. C. \ Miller ,\ @noop journal journal Journal of the Royal Society Interface \ volume 6 ,\ pages 1121 ( year 2009 ) NoStop

    2009

  41. [49]

    author author L. D. \ Valdez , author P. A. \ Macri , \ and\ author L. A. \ Braunstein ,\ @noop journal journal Physica A: Statistical Mechanics and its Applications \ volume 392 ,\ pages 4172 ( year 2013 ) NoStop

    2013

  42. [50]

    author author L. D. \ Valdez , author P. A. \ Macri , \ and\ author L. A. \ Braunstein ,\ @noop journal journal PloS ONE \ volume 7 ,\ pages e44188 ( year 2012 a ) NoStop

    2012

  43. [51]

    Valdez , author P

    author author L. Valdez , author P. A. \ Macri , \ and\ author L. A. \ Braunstein ,\ @noop journal journal Physical Review E \ volume 85 ,\ pages 036108 ( year 2012 b ) NoStop

    2012

  44. [52]

    Wang , author M

    author author W. Wang , author M. Tang , author H.-F. \ Zhang , \ and\ author Y.-C. \ Lai ,\ @noop journal journal Physical Review E \ volume 92 ,\ pages 012820 ( year 2015 b ) NoStop

    2015

  45. [53]

    Shu , author Q.-H

    author author P. Shu , author Q.-H. \ Liu , author S. Wang , \ and\ author W. Wang ,\ @noop journal journal Chaos: An Interdisciplinary Journal of Nonlinear Science \ volume 28 ,\ pages 113114 ( year 2018 ) NoStop

    2018

  46. [54]

    merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number orga...

    2010

  47. [55]

    merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number orga...

    2010

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.