pith. sign in

arxiv: 2512.24270 · v2 · pith:YB6XHFBNnew · submitted 2025-12-30 · ⚛️ physics.soc-ph

Strategic Network Abandonment

Sandro Claudio Lera , Andreas Haupt This is my paper

Pith reviewed 2026-05-25 06:56 UTC · model grok-4.3

classification ⚛️ physics.soc-ph
keywords strategic network abandonmentstrategic complementaritiesbootstrap percolationcascade dynamicssocio-economic networksmetastable plateausnetwork fragilityexit cascades
0
0 comments X

The pith

The strength of strategic complementarities determines whether network decay spreads locally like percolation or collapses globally after metastable plateaus.

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

The paper introduces a framework in which agents in socio-economic networks choose activity levels and exit when an outside option improves enough to exceed their network utility. Their decisions feed back through the network, so one agent's exit can change others' incentives. The governing factor is the strength of strategic complementarities, the degree to which each agent's payoff depends on others remaining active. When complementarities are weak, exits accumulate through local neighborhoods in a threshold process similar to bootstrap percolation, so vulnerable clusters can be spotted in advance. When complementarities are strong, exits can stay contained for long periods then suddenly propagate through the entire system, producing abrupt, system-wide collapse.

Core claim

As outside opportunities rise, agents exit endogenously, triggering equilibrium readjustments that may dissipate locally or propagate through the network. The resulting decay dynamics are governed by the strength of strategic complementarities. When complementarities are weak, decay follows a heterogeneous threshold process analogous to bootstrap percolation: failures are driven by local neighborhoods, vulnerable clusters can be identified ex ante, and large cascades emerge only through bottom-up accumulation of fragility. When complementarities are strong, departures propagate globally, producing rupture-like dynamics characterized by metastable plateaus and abrupt system-wide collapse.

What carries the argument

Strategic complementarities, the degree to which each agent's incentives depend on the actions of others, which switches the decay regime between local threshold accumulation and global rupture.

If this is right

  • Supporting central agents is most effective under strong complementarities.
  • Targeting marginal agents is essential when complementarities are weak.
  • Vulnerable clusters can be identified ex ante under weak complementarities.
  • Standard spectral or structural indicators have limited predictive power under strong complementarities.

Where Pith is reading between the lines

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

  • The same distinction between local and global regimes could apply to other systems where participation depends on others, such as platform adoption or group projects.
  • Empirical work could estimate complementarity strength from observed participation data to select the right intervention target.
  • The model implies that gradual policy changes improving outside options can produce qualitatively different outcomes depending on the underlying complementarity level.

Load-bearing premise

Agents remain active only if participation yields higher utility than an improving outside option, and this comparison together with the network structure fully determines equilibrium activity levels and exit cascades.

What would settle it

Measure participation levels in an observed socio-economic network while outside options gradually improve, then check whether exits accumulate through local clusters or instead show long stable periods followed by sudden system-wide drops.

Figures

Figures reproduced from arXiv: 2512.24270 by Andreas Haupt, Sandro Claudio Lera.

Figure 1
Figure 1. Figure 1: The Strategic Network Abandonment Model. (a) The initial state of a network of 20 agents (nodes), with node sizes proportional to pay-off utility (3) from game (1) with α = 1, β = 0.3 and βρ(A) = 0.9. The outside utility is set equal to 5.05 and hence below the minimal utility of 5.1 that the agent with the lowest utility gets for being in the network. (b) The outside option is increased, causing the lowes… view at source ↗
Figure 2
Figure 2. Figure 2: Empirical and model examples of network decay. Left: Temporal evolution of activity in three socio-economic systems: an abandoned ERC20 cryptocurrency project (Populous coin), participation in an online community (the sourdough subreddit), and the number of registered businesses in a U.S. county (Youngstown, Ohio). All exhibit gradual rather than abrupt decline. Middle: Simulated decay of a 300-agent power… view at source ↗
Figure 3
Figure 3. Figure 3: Stochasticity of collapse in the high-β regime. We generate 20 realizations of a 300-agent power-law cluster network with m = 2, p = 0.1 and βρ(A) = 0.9. All panels plot the evolution of each realization as the outside option increases. Left: The number of remaining agents declines gradually before experiencing an abrupt collapse, but the collapse point varies substantially across realizations, indicating … view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of network support schemes. We con￾sider the same powerlaw-cluster graph as in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Socio-economic networks, from cities and firms to collaborative projects, often appear resilient for long periods before experiencing rapid, cascading decline as participation erodes. We explain such dynamics through a framework of strategic network abandonment, in which interconnected agents choose activity levels in a network game and remain active only if participation yields higher utility than an improving outside option. As outside opportunities rise, agents exit endogenously, triggering equilibrium readjustments that may either dissipate locally or propagate through the network. The resulting decay dynamics are governed by the strength of strategic complementarities, measuring how strongly an agent's incentives depend on the actions of others. When complementarities are weak, decay follows a heterogeneous threshold process analogous to bootstrap percolation: failures are driven by local neighborhoods, vulnerable clusters can be identified ex ante, and large cascades emerge only through bottom-up accumulation of fragility. When complementarities are strong, departures propagate globally, producing rupture-like dynamics characterized by metastable plateaus, abrupt system-wide collapse, and limited predictive power of standard spectral or structural indicators. The comparative effective of intervention depends on the strength of complementarity as well: Supporting central agents is most effective under strong complementarities, whereas targeting marginal agents is essential when complementarities are weak. Together, our results reveal how outside options, network structure, and strategic interdependence jointly determine both the fragility of socio-economic networks and the policies required to sustain them.

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

1 major / 0 minor

Summary. The manuscript introduces a framework of strategic network abandonment in which agents in a socio-economic network game remain active only if participation yields higher utility than an improving outside option. As outside opportunities rise, endogenous exits trigger equilibrium readjustments whose character depends on the strength of strategic complementarities: weak complementarities produce local, heterogeneous-threshold cascades analogous to bootstrap percolation, while strong complementarities produce global, rupture-like dynamics featuring metastable plateaus and abrupt system-wide collapse. The paper further claims that optimal intervention targets (central vs. marginal agents) likewise depend on complementarity strength.

Significance. If the modeling framework is internally consistent and the equilibrium-adjustment process is well-specified, the distinction between local and global decay regimes could provide a useful organizing lens for understanding sudden network collapses and for designing targeted interventions. The abstract presents the framework as a new modeling approach rather than a tautological re-description of existing results.

major comments (1)
  1. [Abstract, paragraph 2] Abstract, paragraph 2: The claimed distinction between local (bootstrap-percolation-like) and global (rupture-like) decay regimes rests on the assertion that, after an agent exits, the resulting equilibrium readjustments are uniquely determined by network structure and utilities. The abstract does not specify the equilibrium-selection rule or adjustment process (simultaneous best response, sequential myopic dynamics, etc.). In network games with strategic complementarities, multiple Nash equilibria are generic; without an explicit selection criterion this assumption is load-bearing for the local-vs-global propagation claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The single major comment identifies a legitimate presentational gap in the abstract regarding equilibrium selection; we address it directly below.

read point-by-point responses

  1. Referee: [Abstract, paragraph 2] Abstract, paragraph 2: The claimed distinction between local (bootstrap-percolation-like) and global (rupture-like) decay regimes rests on the assertion that, after an agent exits, the resulting equilibrium readjustments are uniquely determined by network structure and utilities. The abstract does not specify the equilibrium-selection rule or adjustment process (simultaneous best response, sequential myopic dynamics, etc.). In network games with strategic complementarities, multiple Nash equilibria are generic; without an explicit selection criterion this assumption is load-bearing for the local-vs-global propagation claim.

    Authors: We agree that the abstract should make the equilibrium-selection rule explicit. The body of the manuscript (Section 2 and Proposition 1) establishes that, for any fixed set of active agents, the network game admits a unique Nash equilibrium under the maintained assumptions of continuous action spaces and strictly supermodular payoffs with linear best responses; after each exit the remaining agents play this unique equilibrium. The local-versus-global distinction is derived from the comparative statics of this unique equilibrium with respect to the outside option. To eliminate any ambiguity we will revise the abstract to include a concise clause specifying the adjustment process. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework is self-contained theoretical model

full rationale

The paper introduces a modeling framework for network abandonment in games with strategic complementarities and rising outside options. The claimed distinction between bootstrap-percolation-like local decay (weak complementarities) and global rupture-like cascades (strong complementarities) follows directly from the stated assumptions about utility comparisons and equilibrium readjustments, without any reduction of predictions to fitted parameters, self-definitional loops, or load-bearing self-citations. No equations or sections in the abstract or description exhibit the enumerated circularity patterns; the derivation chain remains independent of its own outputs. Equilibrium selection is left implicit, but this is a modeling choice rather than a definitional tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review reveals one core behavioral assumption but no explicit free parameters or invented entities.

axioms (1)
  • domain assumption Agents remain active only if network participation yields higher utility than an improving outside option.
    This rule is invoked as the driver of endogenous exit and subsequent equilibrium readjustments.

pith-pipeline@v0.9.0 · 5761 in / 1223 out tokens · 40755 ms · 2026-05-25T06:56:43.132489+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Penguin, 2011

    Jared Diamond.Collapse: how societies choose to fail or succeed: revised edition. Penguin, 2011

    work page 2011

  2. [2]

    Urban de- cline and durable housing.Journal of political econ- omy, 113(2):345–375, 2005

    Edward L Glaeser and Joseph Gyourko. Urban de- cline and durable housing.Journal of political econ- omy, 113(2):345–375, 2005

    work page 2005

  3. [3]

    Threshold models of collective be- havior.American journal of sociology, 83(6):1420– 1443, 1978

    Mark Granovetter. Threshold models of collective be- havior.American journal of sociology, 83(6):1420– 1443, 1978

    work page 1978

  4. [4]

    A simple model of global cascades on random networks.Proceedings of the National Academy of Sciences, 99(9):5766–5771, 2002

    Duncan J Watts. A simple model of global cascades on random networks.Proceedings of the National Academy of Sciences, 99(9):5766–5771, 2002

    work page 2002

  5. [5]

    Contagion.The Review of Economic Studies, 67(1):57–78, 2000

    Stephen Morris. Contagion.The Review of Economic Studies, 67(1):57–78, 2000

    work page 2000

  6. [6]

    Error and attack tolerance of complex net- works.nature, 406(6794):378–382, 2000

    Réka Albert, Hawoong Jeong, and Albert-László Barabási. Error and attack tolerance of complex net- works.nature, 406(6794):378–382, 2000. 7

    work page 2000

  7. [7]

    Resilience of the internet to ran- dombreakdowns.Physical review letters, 85(21):4626, 2000

    Reuven Cohen, Keren Erez, Daniel Ben-Avraham, and Shlomo Havlin. Resilience of the internet to ran- dombreakdowns.Physical review letters, 85(21):4626, 2000

    work page 2000

  8. [8]

    Breakdown of the internet under intentional attack.Physical review letters, 86(16):3682, 2001

    Reuven Cohen, Keren Erez, Daniel Ben-Avraham, and Shlomo Havlin. Breakdown of the internet under intentional attack.Physical review letters, 86(16):3682, 2001

    work page 2001

  9. [9]

    The evolution of social and economic networks.Journal of economic theory, 106(2):265–295, 2002

    Matthew O Jackson and Alison Watts. The evolution of social and economic networks.Journal of economic theory, 106(2):265–295, 2002

    work page 2002

  10. [10]

    Identification of peer effects through social networks.Journal of econometrics, 150(1):41–55, 2009

    Yann Bramoullé, Habiba Djebbari, and Bernard Fortin. Identification of peer effects through social networks.Journal of econometrics, 150(1):41–55, 2009

    work page 2009

  11. [11]

    Peer effects and social networks in edu- cation.The Review of Economic Studies, 76(4):1239– 1267, 2009

    Antoni Calvó-Armengol, Eleonora Patacchini, and Yves Zenou. Peer effects and social networks in edu- cation.The Review of Economic Studies, 76(4):1239– 1267, 2009

    work page 2009

  12. [12]

    Identifying influential and susceptible members of social networks.Science, 337(6092):337–341, 2012

    Sinan Aral and Dylan Walker. Identifying influential and susceptible members of social networks.Science, 337(6092):337–341, 2012

    work page 2012

  13. [13]

    Who’s who in networks

    Coralio Ballester, Antoni Calvó-Armengol, and Yves Zenou. Who’s who in networks. wanted: The key player.Econometrica, 74(5):1403–1417, 2006

    work page 2006

  14. [14]

    Public goods in networks.Journal of Economic theory, 135(1):478– 494, 2007

    Yann Bramoullé and Rachel Kranton. Public goods in networks.Journal of Economic theory, 135(1):478– 494, 2007

    work page 2007

  15. [15]

    Strategic interaction and networks.Amer- ican Economic Review, 104(3):898–930, 2014

    Yann Bramoullé, Rachel Kranton, and Martin D’amours. Strategic interaction and networks.Amer- ican Economic Review, 104(3):898–930, 2014

    work page 2014

  16. [16]

    Network games.The review of economic studies, 77(1):218– 244, 2010

    Andrea Galeotti, Sanjeev Goyal, Matthew O Jackson, Fernando Vega-Redondo, and Leeat Yariv. Network games.The review of economic studies, 77(1):218– 244, 2010

    work page 2010

  17. [17]

    On the abandon- ment and survival of open source projects: An empir- ical investigation

    Guilherme Avelino, Eleni Constantinou, Marco Tulio Valente, and Alexander Serebrenik. On the abandon- ment and survival of open source projects: An empir- ical investigation. In2019 ACM/IEEE International Symposium on Empirical Software Engineering and Measurement (ESEM), pages 1–12. IEEE, 2019

    work page 2019

  18. [18]

    Seed size strongly affects cascades on random networks.Phys- ical Review E, 75(5):056103, 2007

    James P Gleeson and Diarmuid J Cahalane. Seed size strongly affects cascades on random networks.Phys- ical Review E, 75(5):056103, 2007

    work page 2007

  19. [19]

    Sharp metastability threshold for two-dimensional bootstrap percolation.Probabil- ity theory and related fields, 125(2):195–224, 2003

    Alexander E Holroyd. Sharp metastability threshold for two-dimensional bootstrap percolation.Probabil- ity theory and related fields, 125(2):195–224, 2003

    work page 2003

  20. [20]

    Sethna, Karin Dahmen, and Christopher R

    James P. Sethna, Karin Dahmen, and Christopher R. Myers. Crackling noise.Nature, 410:242–250, 2001

    work page 2001

  21. [21]

    Critical rup- tures.The European Physical Journal B-Condensed Matter and Complex Systems, 18(1):163–181, 2000

    Anders Johansen and Didier Sornette. Critical rup- tures.The European Physical Journal B-Condensed Matter and Complex Systems, 18(1):163–181, 2000

    work page 2000

  22. [22]

    Alava, Phani K

    Mikko J. Alava, Phani K. V. V. Nukala, and Stefano Zapperi. Statistical models of fracture.Advances in Physics, 55(3-4):349–476, 2006

    work page 2006

  23. [23]

    Chakrabarti

    Srutarshi Pradhan, Alex Hansen, and Bikas K. Chakrabarti. Failure processes in elastic fiber bun- dles.Rev. Mod. Phys., 82:499–555, Mar 2010

    work page 2010

  24. [24]

    Targeting interventions in networks.Econo- metrica, 88(6):2445–2471, 2020

    Andrea Galeotti, Benjamin Golub, and Sanjeev Goyal. Targeting interventions in networks.Econo- metrica, 88(6):2445–2471, 2020. Appendix A. Decay Process is Well-Defined In this appendix, we show that the union of stable sub- sets is stable, which guarantees a unique maximal stable survivor set at each stage of the decay process. In other words, the cascad...

    work page 2020