Competitive Social Mobilization in Threshold Models of Collective Action

Bianca Y. S. Ishikawa; Jos\'e F. Fontanari

arxiv: 2601.18682 · v1 · pith:I5YSCAHLnew · submitted 2026-01-26 · ⚛️ physics.soc-ph · nlin.AO

Competitive Social Mobilization in Threshold Models of Collective Action

Bianca Y. S. Ishikawa , Jos\'e F. Fontanari This is my paper

Pith reviewed 2026-05-16 10:49 UTC · model grok-4.3

classification ⚛️ physics.soc-ph nlin.AO

keywords competitive social mobilizationthreshold modelscollective actionquenched dynamicsannealed dynamicssocial fragmentationconsensusdiscontinuous transitions

0 comments

The pith

The stability of participation thresholds determines whether competing social movements end in one dominant consensus or complete fragmentation.

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

This paper extends threshold models of collective behavior to situations where multiple mutually exclusive movements compete for participants. It finds that the key factor is whether individuals' thresholds for joining stay fixed or can change. When thresholds are fixed, raising the cost of participation first helps one movement dominate but then suddenly causes all movements to fail as too few people are willing to start. When thresholds adjust, higher costs instead drive the population toward a single winning movement. These sharp transitions mean that efforts to control social mobilization by increasing resistance succeed or fail depending on how stable people's opinions are.

Core claim

The outcome of social competition depends critically on the stability of individual dispositions. In quenched environments where participation thresholds are fixed, increasing resistance initially allows a dominant movement to suppress its competitors; however, further resistance triggers a sudden collapse into total fragmentation as low-threshold instigators become too rare to sustain growth. Conversely, in annealed environments where opinions are fluid, higher resistance paradoxically drives a winner-takes-all consensus. In this fluid scenario, massive movements can only be avoided through a deliberate divide-and-conquer strategy. In both cases, the transitions between pulverized and large

What carries the argument

A generalized threshold model of collective action that distinguishes between quenched environments with fixed participation thresholds and annealed environments with fluid opinions.

If this is right

In fixed-threshold settings, moderate resistance strengthens dominant movements while high resistance leads to fragmentation.
In fluid-threshold settings, increased resistance promotes consensus on a single movement.
Preventing large movements in fluid environments requires divide-and-conquer tactics rather than uniform resistance increases.
Both quenched and annealed cases exhibit discontinuous transitions between fragmented and unified states.

Where Pith is reading between the lines

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

Social media algorithms that stabilize user opinions might promote fragmentation when resistance to join is high.
Campaigns to mobilize for causes could benefit from assessing whether target audiences have stable or shifting thresholds before applying pressure tactics.
Extensions incorporating overlapping interests between movements could reveal more nuanced competition dynamics in real societies.

Load-bearing premise

Movements are strictly mutually exclusive with participation decisions following simple threshold rules without overlapping interests or external influences.

What would settle it

Empirical measurement of whether individuals' joining thresholds for social causes remain constant over time, combined with observation of mobilization outcomes when participation costs are raised in populations with stable versus changing thresholds.

Figures

Figures reproduced from arXiv: 2601.18682 by Bianca Y. S. Ishikawa, Jos\'e F. Fontanari.

**Figure 2.** Figure 2: Left panel) Mean fraction of agents in the largest group, [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Mean fraction of isolates ϕ∞ (left panel) and mean density of groups µ∞ (right panel) for the quenched scenario as a function of the attachment exponent γ. The data points represent Monte Carlo simulation results for system sizes ranging from N = 4000 to N = 64000, as indicated. These quantities are measured in the final stationary state, which is reached when the remaining isolates cannot join any of the … view at source ↗

**Figure 4.** Figure 4: Mean density of groups µ∞ (upper panel) and the mean fraction of agents in the largest group ρ∞ (lower panel) for the annealed scenario as a function of the attachment exponent γ. The data points represent Monte Carlo simulation results for system sizes ranging from N = 4000 to N = 1024000 as indicated. These quantities are measured in the final stationary state, characterized by the complete exhaustion of… view at source ↗

**Figure 5.** Figure 5: (Left panel) Mean fraction of agents in the largest group [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: (Left panel) Mean size of the second largest group, [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

read the original abstract

Social mobilization often fails not for a lack of collective interest, but because of fierce competition between rival movements for the same limited pool of participants. We generalize the classic threshold model of collective behavior to analyze this competitive aggregation, exploring how populations with diverse participation thresholds navigate multiple, mutually exclusive causes. Focusing on the conditions necessary for a single consensus movement to encompass an entire population, our analysis reveals that the outcome of social competition depends critically on the stability of individual dispositions. In quenched environments where participation thresholds are fixed, increasing resistance initially allows a dominant movement to suppress its competitors; however, further resistance triggers a sudden collapse into total fragmentation as low-threshold instigators become too rare to sustain growth. Conversely, in annealed environments where opinions are fluid, higher resistance paradoxically drives a winner-takes-all consensus. In this fluid scenario, massive movements can only be avoided through a deliberate divide-and-conquer strategy. In both cases, the transitions between pulverized and massive movements are discontinuous. These findings demonstrate that the effectiveness of social control depends entirely on environmental stability: raising the cost of participation can either forge unity or shatter collective action into insignificance.

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.

Desk Editor's Note private letter to a colleague

The paper's core result is that fixed versus fluid thresholds reverse how resistance affects competitive mobilization outcomes.

read the letter

The main thing to know is that this work extends single-movement threshold models to competition between several movements and finds that the stability of participation thresholds reverses the impact of resistance on whether you get consensus or fragmentation. In the fixed case moderate resistance favors one dominant movement but higher resistance triggers sudden breakup; in the fluid case higher resistance instead favors a single winner. Both shifts appear discontinuous in the reported runs. This is a direct generalization that produces clear qualitative differences from the classic setup without adding many free parameters beyond resistance itself. The derivations track logically from the model rules and the quenched-annealed split is handled cleanly. Credit for spelling out the implications for social control without overclaiming real-world fit. The soft spot is the numerical evidence for discontinuity. The jumps are shown in finite realizations, yet the note on possible finite-size artifacts is fair to raise—if the participation jump shrinks as N grows, the transitions could smooth out and the claim that stability alone dictates the qualitative outcome would need softening. The abstract does not detail scaling checks, so that verification matters. The model keeps the usual assumptions of mutually exclusive movements and pure threshold rules, which is standard but limits how far the predictions travel to settings with overlapping interests. This is for researchers already familiar with threshold models who want to see the competitive layer added. A reader working on opinion dynamics or collective action gets usable new predictions. The extension is straightforward enough and the results are testable in principle, so it deserves a serious referee even if revisions on the scaling analysis are likely. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The paper generalizes the classic threshold model of collective behavior to competitive social mobilization among mutually exclusive movements. It examines how increasing participation resistance affects outcomes in quenched environments (fixed individual thresholds), where moderate resistance enables a dominant movement to suppress rivals but high resistance triggers sudden fragmentation, versus annealed environments (fluid opinions), where higher resistance drives winner-takes-all consensus. Both regimes exhibit discontinuous transitions between fragmented and massive movements, with the conclusion that social control effectiveness depends on environmental stability.

Significance. If the reported discontinuous transitions hold beyond finite-size effects, the work offers a clear mechanistic distinction between fixed and fluid disposition regimes in competitive collective action, with potential implications for understanding when raising participation costs unifies or fragments movements. The parameter-free character of the qualitative predictions (driven solely by the stability of thresholds) is a strength, though the model’s strict mutual exclusivity assumption limits direct applicability to overlapping real-world causes.

major comments (2)

[Numerical results] Numerical results section: The claim of discontinuous transitions in both quenched and annealed regimes is based on observed jumps in participation fraction at critical resistance values, but the manuscript provides no finite-size scaling analysis (e.g., how jump size or susceptibility scales with N). If the discontinuity vanishes as N→∞, the central assertion that environmental stability alone dictates the qualitative outcome would require revision.
[Model definition] Model definition (early sections): The assumption that movements are strictly mutually exclusive and that agents choose at most one cause is load-bearing for the fragmentation vs. consensus dichotomy, yet the paper does not explore robustness when agents can hold overlapping interests or when external fields are added; this directly affects the “divide-and-conquer” strategy conclusion.

minor comments (2)

[Abstract/Introduction] The abstract and introduction would benefit from explicit citation of the original Granovetter threshold model and subsequent extensions to competitive settings to clarify the precise generalization.
[Figures] Figure captions should include the system size N used for each curve and whether error bars represent standard deviation over realizations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's thorough review and constructive feedback on our manuscript arXiv:2601.18682. We address each major comment below and outline the changes we will make to strengthen the work.

read point-by-point responses

Referee: [Numerical results] Numerical results section: The claim of discontinuous transitions in both quenched and annealed regimes is based on observed jumps in participation fraction at critical resistance values, but the manuscript provides no finite-size scaling analysis (e.g., how jump size or susceptibility scales with N). If the discontinuity vanishes as N→∞, the central assertion that environmental stability alone dictates the qualitative outcome would require revision.

Authors: We agree that a finite-size scaling analysis is necessary to rigorously establish the discontinuous nature of the transitions. While the jumps are observed consistently across the system sizes simulated in the current work, we will add this analysis in the revision. Specifically, we will examine how the participation jump size and susceptibility scale with N to confirm that the transitions remain discontinuous in the thermodynamic limit, thereby supporting the central claim regarding environmental stability. revision: yes
Referee: [Model definition] Model definition (early sections): The assumption that movements are strictly mutually exclusive and that agents choose at most one cause is load-bearing for the fragmentation vs. consensus dichotomy, yet the paper does not explore robustness when agents can hold overlapping interests or when external fields are added; this directly affects the “divide-and-conquer” strategy conclusion.

Authors: The strict mutual exclusivity is a deliberate modeling choice to capture direct competition for a limited pool of participants, which underpins the fragmentation-consensus distinction. We will include additional discussion in the revised manuscript on the robustness of our findings to overlapping interests and external fields, noting how these extensions might modulate the outcomes while preserving the qualitative role of threshold stability. A full exploration of these variants is beyond the present scope but represents a natural direction for future work. revision: partial

Circularity Check

0 steps flagged

No circularity detected in model derivation

full rationale

The paper generalizes the classic threshold model of collective action to competitive settings and derives outcomes for quenched versus annealed environments directly from the specified participation rules, mutual exclusivity, and stability parameters. All reported transitions and qualitative behaviors (fragmentation, consensus, discontinuities) emerge from numerical realizations of the dynamics rather than from parameter fitting, self-citation chains, or definitional equivalence. No load-bearing claim reduces to its own inputs by construction; the analysis remains self-contained against the model's internal equations and simulation protocol.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard threshold decision rules plus the added distinction between fixed and fluid thresholds; no new entities are postulated.

free parameters (1)

participation resistance
Parameter varied to explore effects on dominance versus fragmentation.

axioms (2)

domain assumption Individuals decide to participate based on whether the number of current participants exceeds their personal threshold
Core assumption inherited from classic threshold models of collective behavior.
domain assumption Movements are mutually exclusive so participants can join only one
Required for modeling direct competition for the same pool.

pith-pipeline@v0.9.0 · 5500 in / 1083 out tokens · 38725 ms · 2026-05-16T10:49:15.701699+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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A. Alsulami, A. Glukhov, M. Shishlenin, S. Petrovskii, Dynamical mod- elling of street protests using the Yellow Vest Movement and Khabarovsk as case studies, Sci Rep. 12 (2022) 20447, https://doi.org/10.1038/s41598- 022-23917-z

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D. Helbing, I. Farkas, T. Vicsek, Simulating dynamical features of escape panic, Nature 407 (2000) 487–490, https://doi.org/10.1038/35035023

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Wiedermann, E.K

M. Wiedermann, E.K. Smith, J. Heitzig, J.F. Donges, A network-based microfoundation of Granovetter’s threshold model for social tipping, Sci Rep. 10 (2020) 11202, https://doi.org/10.1038/s41598-020-67102-6. 23

work page doi:10.1038/s41598-020-67102-6 2020

[1] [1]

Coleman, Introduction to Mathematical Sociology, Free Press Glen- coe, London, 1964

J.S. Coleman, Introduction to Mathematical Sociology, Free Press Glen- coe, London, 1964

work page 1964

[2] [2]

Gavrilets, Collective action problem in heterogeneous groups, Philos

S. Gavrilets, Collective action problem in heterogeneous groups, Philos. Trans. R. Soc. Lond. B 370 (2015) 20150016, https://doi.org/10.1098/rstb.2015.0016

work page doi:10.1098/rstb.2015.0016 2015

[3] [3]

Coleman, J

J.S. Coleman, J. James, The Equilibrium Size Distribu- tion of Freely-forming Groups, Sociometry 24 (1961) 36–45, https://doi.org/10.2307/2785927

work page doi:10.2307/2785927 1961

[4] [4]

White, Chance Models of Systems of Casual Groups, Sociometry 25 (1962) 153–172, https://doi.org/10.2307/2785947

H. White, Chance Models of Systems of Casual Groups, Sociometry 25 (1962) 153–172, https://doi.org/10.2307/2785947

work page doi:10.2307/2785947 1962

[5] [5]

Fontanari, Stochastic Simulations of Casual Groups, Mathematics 11 (2023) 2152, https://doi.org/10.3390/math11092152

J.F. Fontanari, Stochastic Simulations of Casual Groups, Mathematics 11 (2023) 2152, https://doi.org/10.3390/math11092152

work page doi:10.3390/math11092152 2023

[6] [6]

M.L. Katz, C. Shapiro, Network Externalities, Competi- tion, and Compatibility, Am.Econ.Rev. 75 (1985) 424–440, https://www.jstor.org/stable/1814809

work page arXiv 1985

[7] [7]

Competing Technologies, Increasing Returns, and Lock- In by Historical Events

W.B. Arthur, Competing technologies, increasing returns, and lock-in by historical events, Econ. J. 99 (1989) 116–131, https://doi.org/10.2307/2234208. 21

work page doi:10.2307/2234208 1989

[8] [8]

McCarthy, M.N

J.D. McCarthy, M.N. Zald, Resource mobilization and social move- ments: A partial theory, Am. J. Sociol. 82 (1977) 1212–1241, https://www.jstor.org/stable/2777934

work page arXiv 1977

[9] [9]

McAdam, S

D. McAdam, S. Tarrow, C. Tilly, Dynamics of Contention, Cambridge University Press, Cambridge, 2001

work page 2001

[10] [10]

Bakke, K.G

K.M. Bakke, K.G. Cunningham, L.J.M. Seymour, A plague of initials: Fragmentation, cohesion, and infighting in civil wars, Perspect. Polit. 10 (2012) 265–283, https://doi.org/10.1017/S1537592712000667

work page doi:10.1017/s1537592712000667 2012

[11] [11]

Schelling

T.C. Schelling, Dynamic models of segregation, J. Math. Sociol. 1 (1971) 143–186, https://doi.org/10.1080/0022250X.1971.9989794

work page doi:10.1080/0022250x.1971.9989794 1971

[12] [12]

Granovetter, Threshold models of collective behavior, Am

M. Granovetter, Threshold models of collective behavior, Am. J. Sociol. 83 (1978) 1420–1443, https://www.jstor.org/stable/2778111

work page arXiv 1978

[13] [13]

Hannan, J

M.T. Hannan, J. Freeman, The population ecology of organizations, Am. J. Soc. 82 (1977) 929–964, https://doi.org/10.1086/226424

work page doi:10.1086/226424 1977

[14] [14]

Starnini, A

M. Starnini, A. Baronchelli, R. Pastor-Satorras, Modeling Human Dy- namics of Face-to-Face Interaction Networks, Phys. Rev. Lett. 110 (2013) 168701, https://doi.org/10.1103/PhysRevLett.110.168701

work page doi:10.1103/physrevlett.110.168701 2013

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Starnini, A

M. Starnini, A. Baronchelli, R. Pastor-Satorras, Model reproduces indi- vidual, group and collective dynamics of human contact networks, Soc. Netw. 47 (2016) 130–137, https://doi.org/10.1016/j.socnet.2016.06.002

work page doi:10.1016/j.socnet.2016.06.002 2016

[16] [16]

Thermal eﬀects and their compen- sation in advanced virgo.Journal of Physics: Confer- ence Series, 363:012016, jun 2012

M.S. Mariano, J.F. Fontanari, The Impact of Social Attractiveness on Casual Group Formation: Power-Law Group Sizes and Suppressed Perco- lation, J. Stat. Mech. 2025 (2025) 113405, https://doi.org/10.1088/1742- 5468/ae1af9

work page doi:10.1088/1742- 2025

[17] [17]

Petrovskii, W

S. Petrovskii, W. Alharbi, A. Alhomairi, A. Morozov, Modelling Population Dynamics of Social Protests in Time and Space: The Reaction-Diffusion Approach, Mathematics 8 (2020) 78, https://doi.org/10.3390/math8010078

work page doi:10.3390/math8010078 2020

[18] [18]

Yamagishi, N

A. Alsulami, A. Glukhov, M. Shishlenin, S. Petrovskii, Dynamical mod- elling of street protests using the Yellow Vest Movement and Khabarovsk as case studies, Sci Rep. 12 (2022) 20447, https://doi.org/10.1038/s41598- 022-23917-z

work page doi:10.1038/s41598- 2022

[19] [19]

Volpert, Mathematical model of repressive response to col- lective action and protest, J

V. Volpert, Mathematical model of repressive response to col- lective action and protest, J. Theor. Biol. 595 (2024) 111970, https://doi.org/10.1016/j.jtbi.2024.111970

work page doi:10.1016/j.jtbi.2024.111970 2024

[20] [20]

Volpert, S

V. Volpert, S. Petrovskii, Reaction-diffusion waves in biology: new trends, recent developments, Phys. Life Rev. 52 (2025) 1–20, https://doi.org/10.1016/j.plrev.2024.11.007. 22

work page doi:10.1016/j.plrev.2024.11.007 2025

[21] [21]

Lorenz-Spreen, B.M Mølgaard, P

P. Lorenz-Spreen, B.M Mølgaard, P. Hövel, S. Lehmann, Accelerat- ing dynamics of collective attention, Nat. Commun. 10 (2019) 1759, https://doi.org/10.1038/s41467-019-09311-w

work page doi:10.1038/s41467-019-09311-w 2019

[22] [22]

L. Weng, A. Flammini, A. Vespignani, F. Menczer, Competition among memes in a world with limited attention, Sci. Rep. 2 (2012) 335, https://doi.org/10.1038/srep00335

work page doi:10.1038/srep00335 2012

[23] [23]

APACrefauthors \ 2001

R. Cont, Empirical properties of asset returns: stylized facts and statistical issues, Quant. Finance 1 (2001) 223–236, https://doi.org/10.1080/713665670

work page doi:10.1080/713665670 2001

[24] [24]

Helbing, I

D. Helbing, I. Farkas, T. Vicsek, Simulating dynamical features of escape panic, Nature 407 (2000) 487–490, https://doi.org/10.1038/35035023

work page doi:10.1038/35035023 2000

[25] [25]

Wiedermann, E.K

M. Wiedermann, E.K. Smith, J. Heitzig, J.F. Donges, A network-based microfoundation of Granovetter’s threshold model for social tipping, Sci Rep. 10 (2020) 11202, https://doi.org/10.1038/s41598-020-67102-6. 23

work page doi:10.1038/s41598-020-67102-6 2020