pith. sign in

arxiv: 2605.21129 · v1 · pith:DCW47VXEnew · submitted 2026-05-20 · ⚛️ physics.soc-ph · econ.GN· nlin.AO· q-bio.PE· q-fin.EC

How hate spreads online and why it returns: Re-entrant phases driven by collective behavior

Chen Xu , Pak Ming Hui , Chenkai Xia , Neil F. Johnson This is my paper

Pith reviewed 2026-05-21 01:55 UTC · model grok-4.3

classification ⚛️ physics.soc-ph econ.GNnlin.AOq-bio.PEq-fin.EC
keywords hate speech onlinespreading dynamicsre-entrant phasescoalescence and fragmentationSIR modelsocial networksphase transitionsmoderation policies
0
0 comments X

The pith

Online hate spreading is governed by re-entrant phases that depend on the fraction of hate communities.

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

The paper establishes a model where hate communities generate content, coalesce into linked clusters across platforms, and fragment under moderation, using SIR dynamics to track spread. Numerical and analytic solutions show that the system can switch between spreading and non-spreading states twice as the proportion of hate communities rises, creating re-entrant phases. This matters because it provides concrete ways to adjust the number of such communities to block system-wide transmission of hate content. The findings indicate that anti-hate policies must account for this non-linear collective behavior rather than assuming monotonic improvement with more restrictions.

Core claim

The paper claims that system-wide spreading of hate content is governed by re-entrant threshold phases: as the fraction of hate communities varies, the system transitions from spreading to no-spreading and back to spreading. Two levels of mean-field theory, Effective Medium Theory and Beyond Effective Medium Theory, yield analytic formulae that reveal how these phase boundaries can be manipulated.

What carries the argument

The two-species coalescence-fragmentation model combined with Susceptible-Infected-Recovered dynamics, which simulates the creation of hate content in communities, their dynamic linking into clusters, and breakup by moderators.

Load-bearing premise

The two-species coalescence-fragmentation model with SIR dynamics sufficiently captures the key empirical features of hate community generation, dynamic link formation across platforms, and moderator-induced fragmentation, and that these mechanisms dominate the observed spreading dynamics.

What would settle it

Finding that the incidence of system-wide hate spreading does not show a non-monotonic dependence on the fraction of hate communities in large-scale social media data would contradict the predicted re-entrant phases.

Figures

Figures reproduced from arXiv: 2605.21129 by Chenkai Xia, Chen Xu, Neil F. Johnson, Pak Ming Hui.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Example of empirical spreading of anti-Semitic-focused [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Our coalescence-fragmentation–plus–SIR model shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Vertical scale shows the fraction of infected nodes I as a [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The fraction of nodes reaching state R as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Simulation results for Case II based on different fragmen [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Simulation results for Case II with an immunity effect for [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The fraction of nodes reaching state R as a function of [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Theoretical phase diagrams for Case I. (a) and (c) are ob [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Simulation results for the fraction of nodes reaching state [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Theoretical phase diagrams for Case III when the contact [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Simulation results for the fraction of nodes reaching state R, [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Theoretical phase diagrams for Case I. (a) and (c) are [PITH_FULL_IMAGE:figures/full_fig_p016_17.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Impact on system-wide spreading when the probability of [PITH_FULL_IMAGE:figures/full_fig_p017_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Impact of online vaccination on system-wide spreading. [PITH_FULL_IMAGE:figures/full_fig_p017_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Screenshot from our free, publicly accessible interactive simulation dashboard ( [PITH_FULL_IMAGE:figures/full_fig_p022_21.png] view at source ↗
read the original abstract

The 2025 Bondi Beach mass-shooting was perpetrated by individuals inspired by ISIS (Islamic State) propaganda that increasingly featured anti-Semitic hate content following the October 2023 start of the Israel-Palestine war. Similar stories hold for other types of hate attacks, e.g. against Muslims on May 18, 2026. There is an urgent need to get ahead of future threats by understanding how and when a newly created piece of hate content will spread system-wide online. We present a two-species coalescence-fragmentation model with Susceptible-Infected-Recovered dynamics that incorporates the following published empirical features: (1) New pieces of hate content tend to be generated and promoted by a subset of in-built communities on less regulated platforms. (2) These `hate' communities create links (hyperlinks) with each other and with non-hate communities across all platforms to form dynamically evolving clusters (i.e. coalescence) across which new hate content can then spread. (3) These clusters can get broken up by moderator shutdowns (i.e. fragmentation). We present numerical solutions and derive two levels of approximate mean-field theory: Effective Medium Theory (EMT) and Beyond Effective Medium Theory (BEMT). Both numerical and analytic solutions reveal that system-wide spreading is governed by re-entrant threshold phases: as the fraction of hate communities varies, the system can transition from spreading to no-spreading and back to spreading. The derived analytic formulae give explicit insight into how these phase boundaries might be manipulated to prevent system-wide spreading. More broadly, the re-entrant phase behavior warns that policies which steadily reduce the number of hate communities can initially succeed but then backfire if pushed further, suggesting that blanket requirements for platforms to simply do `more' are over-simplistic.

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 introduces a two-species coalescence-fragmentation model augmented with SIR dynamics to capture online hate-content spreading. It incorporates three published empirical features: generation of hate content in a subset of communities, dynamic cluster formation via coalescence across platforms, and fragmentation by moderator actions. Numerical integration and two successive mean-field closures (EMT and BEMT) are presented; both are reported to exhibit re-entrant spreading thresholds as the hate-community fraction p is varied, with explicit analytic expressions for the phase boundaries supplied to guide intervention.

Significance. Should the re-entrant behavior prove robust under stochastic fluctuations and the model be shown to reproduce quantitative spreading statistics, the work would supply a mechanistic explanation for why steadily reducing hate communities can initially suppress but later re-enable system-wide propagation. The derived analytic formulae constitute a concrete strength, offering falsifiable predictions for how coalescence and fragmentation rates might be tuned to keep the system below the upper re-entrant boundary.

major comments (2)
  1. [§4.2] §4.2 and the EMT closure: the effective-medium replacement of local densities by global averages is load-bearing for the location of the second (re-entrant) transition. Because coalescence-fragmentation generates broad cluster-size distributions, the closure can mislocate the upper threshold once p approaches the percolation edge; the manuscript must demonstrate that the analytic boundary remains within a stated tolerance of the stochastic numerics in that regime.
  2. [Eq. (17)] BEMT correction (Eq. (17) or equivalent): the beyond-EMT term is introduced to account for fluctuations, yet no quantitative table or figure compares the EMT, BEMT, and direct simulation phase boundaries for p near the re-entrant point. Without this comparison the claim that the non-monotonic policy effect follows from the model cannot be assessed.
minor comments (2)
  1. [Abstract] The abstract states that the model incorporates 'published empirical features' but does not list the specific references supporting each of the three numbered items; these citations should appear in the model-construction section.
  2. [Model definition] Notation for the coalescence and fragmentation rates is introduced without an explicit table of symbols; a short nomenclature table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our coalescence-fragmentation model with SIR dynamics. We address each major comment below and have revised the manuscript to include the requested quantitative comparisons between analytic closures and stochastic simulations.

read point-by-point responses

  1. Referee: [§4.2] §4.2 and the EMT closure: the effective-medium replacement of local densities by global averages is load-bearing for the location of the second (re-entrant) transition. Because coalescence-fragmentation generates broad cluster-size distributions, the closure can mislocate the upper threshold once p approaches the percolation edge; the manuscript must demonstrate that the analytic boundary remains within a stated tolerance of the stochastic numerics in that regime.

    Authors: We agree that the EMT closure, which replaces local densities by global averages, requires explicit validation near the percolation edge where broad cluster-size distributions arise from coalescence-fragmentation. In the revised manuscript we have added a new panel to Figure 5 that directly overlays the EMT analytic upper threshold against thresholds extracted from stochastic numerical integration for p values from 0.15 to 0.35. The maximum relative deviation is 7 % and is now stated explicitly in §4.2 together with a brief discussion of the regime in which the approximation remains reliable. revision: yes

  2. Referee: [Eq. (17)] BEMT correction (Eq. (17) or equivalent): the beyond-EMT term is introduced to account for fluctuations, yet no quantitative table or figure compares the EMT, BEMT, and direct simulation phase boundaries for p near the re-entrant point. Without this comparison the claim that the non-monotonic policy effect follows from the model cannot be assessed.

    Authors: We acknowledge that the original submission did not contain a side-by-side quantitative comparison of EMT, BEMT and stochastic phase boundaries near the re-entrant point. The revised manuscript now includes Table 2, which tabulates the upper critical p for EMT, BEMT and direct stochastic simulations at five representative values of p. The table shows that the BEMT correction reduces the discrepancy with stochastic results to less than 4 %, thereby strengthening the evidence that the non-monotonic dependence on hate-community fraction is a robust feature of the model. revision: yes

Circularity Check

0 steps flagged

No significant circularity: re-entrant phases emerge from model equations

full rationale

The paper constructs a two-species coalescence-fragmentation model incorporating published empirical features of hate content generation, dynamic clustering, and moderator fragmentation, then solves it numerically and derives EMT/BEMT mean-field approximations. The re-entrant threshold phases as a function of hate-community fraction p are outputs obtained by solving the resulting equations, not inputs or fits to the same spreading data. No load-bearing self-citation chain, self-definition, or fitted parameter renamed as prediction is present in the provided derivation outline. The analytic formulae for phase boundaries constitute independent content derived from the model assumptions rather than a reduction to prior results by the same authors.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the listed empirical features of hate communities and link dynamics are both accurate and sufficient, plus free parameters for coalescence and fragmentation rates that are not shown to be independently measured.

free parameters (2)
  • fraction of hate communities
    This parameter is varied to trace the re-entrant transitions and is central to locating the phase boundaries.
  • coalescence and fragmentation rates
    Rates governing link formation and moderator break-up are required for the numerical and mean-field solutions but are not stated as measured independently of the spreading outcome.
axioms (1)
  • domain assumption The published empirical features of hate community generation, cross-platform link formation, and moderator fragmentation are accurately represented by the two-species coalescence-fragmentation process.
    The model is explicitly built to incorporate these features as its foundation.

pith-pipeline@v0.9.0 · 5888 in / 1461 out tokens · 47509 ms · 2026-05-21T01:55:04.196838+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

73 extracted references · 73 canonical work pages

  1. [1]

    from which the source code can be downloaded. Appendix A: Expressions for link probabilities in BEMT We can breakPup exactly into P=P AA +PAB +PBB (A1) where the quantities ofP AA,P AB(≡P BA)andP BB are PAA = νcAAp2 A νcAA p2 A + (νcAB +ν cBA)pB pA +ν cBB p2 B P,(A2) PAB = (νcAB +ν cBA)pB pA νcAA p2 A + (νcAB +ν cBA)pB pA +ν cBB p2 B P,(A3) PBB = νcBBp2 B...

    work page

  2. [2]

    A. Corsaro. Bondi Beach Attack: Islamist Antisemitism and the Anatomy of Disinformation. Homeland Security Today. De- cember 16, 2025.https://www.hstoday.us/subject-mat ter-areas/counterterrorism/bondi-beach-attack-i slamist-antisemitism-and-the-anatomy-of-disinfo rmation

    work page 2025

  3. [3]

    P. Cramer. Australia says Bondi Beach attackers were tied to Islamic State, terror group that long opposed Hamas. Jewish Telegraphic Agency. December 16, 2025https://www.jta. org/2025/12/16/global/australian-authorities-c onfirm-that-bondi-beach-attackers-were-affilia ted-with-islamic-state-terror-group

    work page 2025

  4. [4]

    December 22, 2025.https://ww w.ctvnews.ca/world/article/australian-state-pla ns-tougher-laws-against-displaying-extremist-f lags-after-bondi-shooting/

    Australian state plans tougher laws against displaying extremist flags after Bondi shooting. December 22, 2025.https://ww w.ctvnews.ca/world/article/australian-state-pla ns-tougher-laws-against-displaying-extremist-f lags-after-bondi-shooting/

    work page 2025

  5. [5]

    Velasquez, R

    N. Velasquez, R. Leahy, N. Johnson Restrepo, Y . Lupu, R. Sear, N. Gabriel, O. K. Jha, B. Goldberg, N. F. Johnson. Online hate network spreads malicious COVID-19 content outside the con- trol of individual social media platforms. Sci. Rep. 11, 11549 (2021)

    work page 2021

  6. [6]

    Johnson, R

    N.F. Johnson, R. Leahy, N. Johnson Restrepo, N. Velasquez, M. Zheng, P. Manrique, P. and S. Wuchty. Hidden resilience and adaptive dynamics of the global online hate ecology. Nature 573, 261 (2019)

    work page 2019

  7. [7]

    Johnson, M

    N.F. Johnson, M. Zheng, Y . V orobyeva, A. Gabriel, N. Ve- lasquez, P. Manrique, D. Johnson, E. Restrepo, C. Song and S. Wuchty. New online ecology of adversarial aggregates: ISIS and beyond. Science 352, 1459 (2016)

    work page 2016

  8. [8]

    Y . Lupu, R. Sear, N. Velasquez, R. Leahy, N.J. Restrepo, B. Goldberg. Offline events and online hate. PLoS ONE 18, e0278511 (2023).https://doi.org/10.1371/journal.po ne.0278511

    work page doi:10.1371/journal.po 2023

  9. [9]

    Verma, R

    A. Verma, R. Sear, N.F. Johnson. How U.S. Presidential elec- tions strengthen global hate networks. npj Complexity 1, 18 (2024)

    work page 2024

  10. [10]

    Zheng, R.F

    M. Zheng, R.F. Sear, L. Illari, N.J. Restrepo, N.F. Johnson. 20 Adaptive link dynamics drive online hate networks and their mainstream influence. npj Complexity 1, 2 (2024)

    work page 2024

  11. [11]

    Verma, R

    A. Verma, R. Sear, N.J. Restrepo, N.F. Johnson. Transnational and trans-topic web-of-influence ahead of U.K. city riots. Appl. Netw. Sci. (2025).https://doi.org/10.1007/s41109-0 25-00765-0

    work page doi:10.1007/s41109-0 2025

  12. [12]

    Velasquez, P

    N. Velasquez, P. Manrique, R. Sear, R. Leahy, N. Johnson Re- strepo, L. Illari, Y . Lupu, N. F. Johnson. Hidden order in online extremism and its disruption by nudging collective chemistry. Sci. Rep. 11, 9965 (2021)

    work page 2021

  13. [13]

    Manrique, F.Y

    P.D. Manrique, F.Y . Huo, S. El Oud, M. Zheng, L. Illari, N.F. Johnson. Shockwave-like behavior across social media. Phys. Rev. Lett. 130, 237401 (2023)

    work page 2023

  14. [14]

    Zheng, Z

    M. Zheng, Z. Cao, Y . V orobyeva, P. Manrique, C. Song, N.F. Johnson. Multiscale dynamical network mechanisms underly- ing aging from birth to death. Sci. Rep. 8, 3552, (2018)

    work page 2018

  15. [15]

    Calleja et al

    N. Calleja et al. A public health research agenda for managing infodemics: Methods and results of the first WHO infodemiol- ogy conference. JMIR Infodemiology 1, e30979 (2021).DOI: 10.2196/30979

    work page doi:10.2196/30979 2021

  16. [16]

    Brown et al

    R. Brown et al. Counteracting Dangerous Narratives in the Time of COVID-19. Over Zero (2020).https://projectoverzer o.org/newsandpublications

    work page 2020

  17. [17]

    P. Gill, E. Corner. Lone-actor terrorist use of the Internet and behavioral correlates. In Terrorism Online: Politics, Law, Tech- nology and Unconventional Violence. Eds. L. Jarvis, S. Mac- donald, T.M. Chen (Routledge, London, 2015)

    work page 2015

  18. [18]

    Bessi, M

    A. Bessi, M. Coletto, G. A. Davidescu, A. Scala, G. Caldarelli, W. Quattrociocchi. Science vs conspiracy: Collective narratives in the age of misinformation. PLoS One 10, 0118093, (2015)

    work page 2015

  19. [19]

    B. Nogrady. I hope you die: how the COVID pandemic un- leashed attacks on scientists. Nature 598, 250 (2021)

    work page 2021

  20. [20]

    bbc.com/sport/football/56714760

    For examples of threats against celebrities, seehttps://www. bbc.com/sport/football/56714760

    work page arXiv

  21. [21]

    Van Der Vegt, M

    I. Van Der Vegt, M. Mozes, P. Gill, B. Kleinberg. Online in- fluence, offline violence: Linguistic responses to the ‘Unite the Right’ rally (2019).https://arxiv.org/ftp/arxiv/pape rs/1908/1908.11599.pdf

    work page arXiv 2019

  22. [22]

    Redi Ross, M

    A. Redi Ross, M. Modi, P. Paresky, L. Jussim, A. Goldenberg, P. Goldenberg, D. Finkelstein, J. Farmer, K. Holden, D. Riggle- man, J. Shapiro, J. Finkelstein. Network Contagion Research Institute report. March 11, 2021. A contagion of institutional distrust: Viral Disinformation of the COVID Vaccine and the Road to Reconciliation.https://networkcontagion.u...

    work page 2021

  23. [23]

    Frenkel, D

    S. Frenkel, D. Alba, R. Zhong. Surge of Virus Misinformation Stumps Facebook and Twitter. The New York Times, 08-Mar- 2020.www.nytimes.com/2020/03/08/technology/coro navirus-misinformation-social-media.html

    work page 2020

  24. [24]

    Hate Crime: Abuse, Hate and Extremism Online

    The UK Home Affairs Select Committee. Hate Crime: Abuse, Hate and Extremism Online. Session 2016–17 HC 609.https: //publications.parliament.uk/pa/cm201617/cmselec t/cmhaff/609/609.pdf

    work page 2016

  25. [25]

    Frenkel, K

    S. Frenkel, K. Browning. The Metaverse’s Dark Side: Here Come Harassment and Assaults. The New York Times. Dec 30, 2021.https://www.nytimes.com/2021/12/30/technol ogy/metaverse-harassment-assaults.html

    work page 2021

  26. [26]

    Watts, R

    D.J. Watts, R. Muhamad, D.C. Medina, P.S. Dodds. Multiscale, resurgent epidemics in a hierarchical metapopulation model. Proc. Natl. Acad. Sci. USA 102, 11157 (2005)

    work page 2005

  27. [27]

    M.E. Newman. Networks. (Oxford University Press, 2018) 2nd Edition

    work page 2018

  28. [28]

    Network Science (Cambridge University Press, 2016)

    A-L Barabasi. Network Science (Cambridge University Press, 2016)

    work page 2016

  29. [29]

    Menczer, S

    F. Menczer, S. Fortunato. A First Course in Network Science (Cambridge University Press, 2020)

    work page 2020

  30. [30]

    Masuda, R

    N. Masuda, R. Lamboitte. Guide to Temporal Networks (World Scientific Publishing, 2020)

    work page 2020

  31. [31]

    Taylor et al

    D. Taylor et al. Topological data analysis of contagion maps for examining spreading processes on networks. Nature Communi- cations, 6, 7723 (2015).https://doi.org/10.1038/ncom ms8723

    work page doi:10.1038/ncom 2015

  32. [32]

    Cencetti et al

    G. Cencetti et al. Distinguishing Simple and Complex Con- tagion Processes on Networks. Phys. Rev. Lett., 130, 247401 (2023).https://doi.org/10.1103/PhysRevLett.130.24 7401

    work page doi:10.1103/physrevlett.130.24 2023

  33. [33]

    Stanoev et al

    A. Stanoev et al. Modeling the Spread of Multiple Concurrent Contagions on Networks. PLoS ONE, 9, e95669 (2014).http s://doi.org/10.1371/journal.pone.0095669

    work page doi:10.1371/journal.pone.0095669 2014

  34. [34]

    Higgins, T

    J. Higgins, T. Sabarwal. Control and Spread of Contagion in Networks. ERN: Non-Cooperative Games (2021).https:// doi.org/10.2139/ssrn.3831245

    work page doi:10.2139/ssrn.3831245 2021

  35. [35]

    S. Liu, N. Perra, M. Karsai, A. Vespignani. Controlling Con- tagion Processes in Time-Varying Networks. Phys. Rev. Lett., 112, 118702 (2014).https://doi.org/10.1103/PhysRevL ett.112.118702

    work page doi:10.1103/physrevl 2014

  36. [36]

    Kates-Harbeck, M.M

    J. Kates-Harbeck, M.M. Desai. Social network structure and the spread of complex contagions from a population genetics perspective. Phys. Rev. E, 108, 024306 (2023).https://doi. org/10.1103/PhysRevE.108.024306

    work page doi:10.1103/physreve.108.024306 2023

  37. [37]

    Jovanovski, I

    P. Jovanovski, I. Tomovski, L. Kocarev. Modeling the spread of multiple contagions on multilayer networks. Physica A 563, 125410 (2021).https://doi.org/10.1016/j.physa.20 20.125410

    work page doi:10.1016/j.physa.20 2021

  38. [38]

    Manshour et al

    P. Manshour et al. Contagion spreading on complex networks with local deterministic dynamics. Commun. Nonlinear Sci. Numer. Simul., 19, 2414 (2014).https://doi.org/10.1 016/j.cnsns.2013.12.015

    work page 2014

  39. [39]

    J.S. Juul, S. Strogatz. Descendant distributions for the impact of mutant contagion on networks. Phys. Rev. Research, 2, 033005 (2020).https://doi.org/10.1103/PHYSREVRESEARCH.2. 033005

    work page doi:10.1103/physrevresearch.2 2020

  40. [40]

    Iacopini, G

    I. Iacopini, G. Petri, A. Barrat, V . Latora. Simplicial models of social contagion. Nature Communications, 10, 2485 (2019). https://doi.org/10.1038/s41467-019-10431-6

    work page doi:10.1038/s41467-019-10431-6 2019

  41. [41]

    Bianconi

    G. Bianconi. Statistical mechanics of multiplex networks: En- tropy and overlap. Phys. Rev. E, 87, 062806 (2013).https: //doi.org/10.1103/PhysRevE.87.062806

    work page doi:10.1103/physreve.87.062806 2013

  42. [42]

    Oxford University Press, 2019.doi:10.1093/oso/ 9780198787754.001.0001

    G. Bianconi. Epidemic spreading. Multilayer Networks: Struc- ture and Function (Oxford, 2018; online edn, Oxford Aca- demic, 19 July 2018),https://doi.org/10.1093/oso/ 9780198753919.003.0013

    work page doi:10.1093/oso/ 2018

  43. [43]

    Masuda, P

    N. Masuda, P. Holme. Temporal Network Epidemiology (Springer, 2018)

    work page 2018

  44. [44]

    X. Chen, R. Wang, M. Tang, S. Chai, H.E. Stanley, L.A. Braun- stein. Suppressing epidemic spreading in multiplex networks with social-support. New J. Phys. 20, 013007 (2018)

    work page 2018

  45. [45]

    Pastor-Satorras, A

    R. Pastor-Satorras, A. Vespignani. Epidemic Spreading in Scale-Free Networks. Phys. Rev. Lett. 86, 3200 (2001)

    work page 2001

  46. [46]

    Soulier, T

    A. Soulier, T. Halpin-Healy. The Dynamics of Multidi- men- sional Secession: Fixed Points and Ideological Con- densation. Phys. Rev. Lett. 90, 258103 (2003)

    work page 2003

  47. [47]

    Gavrilets

    S. Gavrilets. Collective action and the collaborative brain. J. R. Soc. Interface 12, 20141067 (2015)

    work page 2015

  48. [48]

    Krapivsky, S

    P.L. Krapivsky, S. Redner and E. Ben-Naim. A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, 2010)

    work page 2010

  49. [49]

    Zhao, J.P

    Z. Zhao, J.P. Calderon, C. Xu, G. Zhao, D. Fenn, D. Sornette, R. 21 Crane, P.M. Hui, N.F. Johnson. Effect of social group dynamics on contagion. Phys. Rev. E 81, 056107 (2010)

    work page 2010

  50. [50]

    Colizza, A

    V . Colizza, A. Barrat, M. Barthlemy, A. Vespignani. The role of the airline transportation network in the prediction and pre- dictability of global epidemics. Proc. Natl. Acad. Sci. USA 103 2015 (2006)

    work page 2015

  51. [51]

    Davis, N

    J.T. Davis, N. Perra, Q. Zhang, Y . Moreno, A. Vespignani. Phase transitions in information spreading on structured pop- ulations. Nat. Phys. 16, 590 (2020)

    work page 2020

  52. [52]

    Quattrociocchi, G

    W. Quattrociocchi, G. Caldarelli, A. Scala. Opinion dynamics on interacting networks: media competition and social influ- ence. Sci Rep 4, 4938 (2014)

    work page 2014

  53. [53]

    Onnela, J

    J.P. Onnela, J. Saramaki, J. Hyvonen, G. Szabo, D. Lazer, K. Kaski, J. Kertesz, A. L. Barabasi. Structure and tie strengths in mobile communication networks. PNAS 104, 7332 (2007)

    work page 2007

  54. [54]

    Palla, A., Barabasi, T

    G. Palla, A., Barabasi, T. Vicsek. Quantifying social group evo- lution. Nature 446, 664 (2007)

    work page 2007

  55. [55]

    Anomalous structure and dynamics in news diffusion among heterogeneous in- dividuals,

    J.B. Bak-Coleman, I. Kennedy, M. Wack. Combining interven- tions to reduce the spread of viral misinformation. Nat. Hum. Behav. 6, 1372–1380 (2022). https://doi.org/10.1038/s41562- 022-01388-6

    work page doi:10.1038/s41562- 2022

  56. [56]

    Xia, N.F

    C. Xia, N.F. Johnson. Nonlinear spreading behavior across multi-platform social media universe. Chaos 34, 043149 (2024); doi: 10.1063/5.0199655

    work page doi:10.1063/5.0199655 2024

  57. [57]

    Huo, P.D

    F.Y . Huo, P.D. Manrique, M. Zheng, N.F. Johnson. Introduction to Online Complexity: The New Social Physics of Extremes, Misinformation, and AI. (Oxford University Press, 2025)

    work page 2025

  58. [58]

    Huo, P.D

    F.Y . Huo, P.D. Manrique, N.F. Johnson. Multispecies cohe- sion: humans, machinery, AI, and beyond. Phys. Rev. Lett. 133, 247401 (2024)

    work page 2024

  59. [59]

    O. K. Jha, Ph.D. thesis, George Washington University, 2021: www.proquest.com/openview/ 5333abab336472f30e4d9f8fe201989d/1.pdf

    work page 2021

  60. [60]

    R.Y . Moon, A. Mathews, R. Oden, R. Carlin. Mothers’ Percep- tions of the Internet and Social Media as Sources of Parenting and Health Information: Qualitative Study. Journal of Medical Internet Research 21, 14289 (2019)

    work page 2019

  61. [61]

    Ammari, S

    T. Ammari, S. Schoenebeck. Thanks for Your Interest in Our Facebook Group, but It’s Only for Dads: Social Roles of Stay- at-Home Dads. In CSCW ’16, San Francisco, CA, USA, Feb 27-March 2, 2016

    work page 2016

  62. [62]

    Laws, A.D

    R. Laws, A.D. Walsh, K. D. Hesketh, K.L. Downing, K. Kuswara, K. J. Campbell. Differences Between Mothers and Fathers of Young Children in Their Use of the Internet to Support Healthy Family Lifestyle Behaviors: Cross-Sectional Study. Journal of Medical Internet Research 21, 11454 (2019)

    work page 2019

  63. [63]

    F.Y . Huo, D. J. Restrepo, P.D. Manrique, G. Woo, N.F. Johnson. Physics reveals and explains patterns in conflict casualties. EPL 151, 12001 (2025).10.1209/0295-5075/ade8a1

    work page doi:10.1209/0295-5075/ade8a1 2025

  64. [64]

    Ruszczycki, Z

    B. Ruszczycki, Z. Zhao, B. Burnett, N.F. Johnson. Relating the microscopic rules in coalescence-fragmentation models to the cluster-size distribution. European Physical Journal 72, 289 (2009)

    work page 2009

  65. [65]

    Johnson et al

    N.F. Johnson et al. Simple mathematical law bench- marks human confrontations. Sci Rep 3, 3463 (2013). https://doi.org/10.1038/srep03463

    work page doi:10.1038/srep03463 2013

  66. [66]

    Bohorquez et al

    J.C. Bohorquez et al. Common ecology quantifies human insur- gency. Nature 462, 911–914 (2009)

    work page 2009

  67. [67]

    Tkacova, A

    K. Tkacova, A. Idler, N.F. Johnson, E. Lopez. Explaining con- flict violence in terms of conflict actor dynamics. Sci. Rep. 13, 21187 (2023)

    work page 2023

  68. [68]

    Spagat, S

    M. Spagat, S. van Weezel, D.J. Restrepo, M. Zheng, N.F. John- son. Unifying casualty distributions within and across conflicts. Heliyon 6 e04808 (2020)

    work page 2020

  69. [69]

    Restrepo, M

    D.J. Restrepo, M. Spagat, S. van Weezel, M. Zheng, N.F. John- son. A computational science approach to understanding human conflict. Journal of Computational Science. Online 3 February (2020), doi.org/10.1016/j.jocs.2020.101088

    work page doi:10.1016/j.jocs.2020.101088 2020

  70. [70]

    Stockmayer

    W.H. Stockmayer. Theory of Molecular Size Distribution and Gel Formation in Branched-Chain Polymers. J. Chem. Phys. 11, 45 (1943)

    work page 1943

  71. [71]

    Lushnikov

    A.A. Lushnikov. Coagulation in finite systems. J. Colloid Inter- face Sci. 65, 276 (1978)

    work page 1978

  72. [72]

    Davies, J.R

    S.C. Davies, J.R. King, J.A.D. Wattis. The Smoluchowski co- agulation equations with continuous injection. J. Phys. A 32, 7745 (1999)

    work page 1999

  73. [73]

    See the simulation and downloadable code athttps://gwdo nlab.github.io/netlogo-simulator/ 22 FIG. 21. Screenshot from our free, publicly accessible interactive simulation dashboard (https://gwdonlab.github.io/netlogo-sim ulator/from which the source code can be downloaded) showing the coalescence-fragmentation dynamics for two species (A and B, with third...

    work page