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Strategic Network Abandonment

Sandro Claudio Lera, Andreas Haupt

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.

Strategic Network Abandonment

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.
Paper Structure (7 sections, 2 theorems, 49 equations, 4 figures)

This paper contains 7 sections, 2 theorems, 49 equations, 4 figures.

Table of Contents

  1. Decay Process is Well-Defined
  2. Derivation of Local Threshold Rule
  3. Condition for Global Collapse
  4. Step 1: Poor nodes always drop out
  5. Step 2: Drop vectors
  6. Step 4: Lower bound on $\Delta^R$ and nonnegativity of the inverse
  7. Step 5: Drops on subsets $S\subseteq R$

Key Result

Theorem 1

Let $A \in \mathbb{R}^{N\times N}$ be a (nonnegative) adjacency matrix and let $\beta \geqslant 0$. For any node subset $S \subseteq \{1,\dots,N\}$, write $A[S]$ for the principal submatrix of $A$ induced by $S$, and $\mathbf 1_S$ for the $|S|$-dimensional all-ones vector. Fix $c \geqslant 0$. If fo then their union also satisfies

Figures (4)

  • 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 \ref{['eq:equilibrium_utility']} from game \ref{['eq:linquad_utility']} with $\alpha=1$, $\beta=0.3$ and $\beta \rho(A) = 0.9$. 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 lowest-utility agent to exit without triggering further departures. (c) A further increase in the outside option leads to the exit of one additional agent, again without a cascade. (d) A further increase of the outside option triggers a cascade, where the abandonment of a third agent induces the abandonment of two more agents. We refer to Box \ref{['box:iterative-algorithm']} for details.
  • 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 powerlaw-cluster network with $\rho(A)=3$ under two levels of strategic complementarity. For small $\beta = 0.05$, the decay proceeds in abrupt, punctuated steps consistent with the $k$-core peeling dynamics from Eq. \ref{['eq:integer_threshold']}. For larger $\beta = 0.30$, stronger feedbacks diffuse the effect of each agent’s departure, producing a smoother, more continuous reduction in network size. Right: Gini coefficients of degree and Bonacich centralities during the decay process with $\beta=0.30$. Because Bonacich centrality incorporates higher-order dependencies, it exhibits greater dispersion than degree, reflecting the more heterogeneous influence structure that stabilizes high-$\beta$ networks against synchronized collapse.
  • Figure 3: Stochasticity of collapse in the high-$\beta$ regime. We generate $20$ realizations of a $300$-agent power-law cluster network with $m=2$, $p=0.1$ and $\beta \rho(A) = 0.9$. 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 strong sensitivity to small structural differences. Middle: The resolvent norm $\|(I - \beta A)^{-1}\|_2$, a measure of global susceptibility, declines smoothly with no precursors and does not predict the impending rupture. Right: The inverse participation ratio (IPR) of the leading eigenvector increases only marginally until immediately before collapse, reflecting the lack of structural reorganization prior to failure.
  • Figure 4: Comparison of network support schemes. We consider the same powerlaw-cluster graph as in Figure \ref{['fig:complementary_dependence']} with $\beta \rho(A)=0.5$, subject to varying outside options. Inset: Solution of Eq. \ref{['eq:minimal_support_problem']}, i.e. total support $\|\mathbf{y}^*\|$ required to keep all agents active as the outside option increases. Top: Number of active agents remaining in the network under both support schemes: the bottom-up scheme (BU) directly targets weak agents and maintains full network integrity. The welfare-maximizing top-down policy (TD), with the same budget as BU, fails to sustain participation of low-utility agents. Middle: Evolution of total welfare under the two support schemes. The BU approach raises welfare even as outside options rise, whereas a TD one leads to declining aggregate utility. Bottom: Relative effectiveness of the two support schemes as a function of the spectral radius.

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Definition 1
  • Theorem 2
  • proof