Sandpile Economics: Theory, Identification, and Evidence

Diego Vallarino

physics.soc-phApr 15, 2026

Sandpile Economics: Theory, Identification, and Evidence

Authors: Diego Vallarino

Organizations: Inter-American Development Bank & IDB Invest

Abstract

Why do capitalist economies recurrently generate crises whose severity is disproportionate to the size of the triggering shock? This paper proposes a structural answer grounded in the evolutionary geometry of production networks. As economies evolve through specialization, integration, and competitive selection, their inter-sectoral linkages drift toward configurations of increasing geometric fragility, eventually crossing a threshold beyond which small disturbances generate disproportionately large cascades. We introduce Sandpile Economics, a formal framework that interprets macroeconomic instability as an emergent property of disequilibrium production networks. The key state variable is the Forman--Ricci curvature of the input--output graph, capturing local substitution possibilities when supply chains are disrupted. We show that when curvature falls below an endogenous threshold, the distribution of cascade sizes follows a power law with tail index $α\in (1,2)$ , implying a regime of unbounded amplification. The underlying mechanism is evolutionary: specialization reduces input substitutability, pushing the economy toward criticality, while crisis episodes induce endogenous network reconfiguration and path dependence. These dynamics are inherently non-ergodic and cannot be captured by representative-agent frameworks. Empirically, using global input--output data, we document that production networks operate in persistently negative curvature regimes and that curvature robustly predicts medium-run output dynamics. A one-standard-deviation increase in curvature is associated with higher cumulative growth over three-year horizons, and curvature systematically outperforms standard network metrics in explaining cross-country differences in resilience.

Figures & tables

Panel of Figure 1 — Figure 1: Theoretical architecture of Sandpile Economics. Ricci curvature $\bar{\kappa}$ is the single state variable. Theorem 3.11 is the central result. The Minsky correspondence and GE amplification are appendix corollaries; local projections are the empirical implementation.

Specification	$J$ -stat ( $p$ -value)	Instruments	$N$
Baseline GMM (DK-HAC)	$0.341$	Geo. distance, $A^{c,t-2}$	645
Alternative (lagged $\bar{\kappa}$ )	$0.387$	Geo. distance, $A^{c,t-1}$	645

Table 1: GMM Identification: Moment Conditions and Instrument Validity

Panel A: Curvature variables (lagged one year)
Variable	$N$	Mean	Std. Dev.	Min	Median	Max
Mean curvature $\bar{\kappa}_{cs,t-1}$	26,351	$-24.13$	$5.51$	$-44.51$	$-25.17$	$-8.53$
Curvature dispersion $\sigma_{\kappa,cs,t-1}$	26,351	$7.50$	$2.27$	$0.00$	$8.09$	$11.77$
Share of fragile edges ( $\kappa_{F}<0$ )	26,351	$1.00$	$0.00$	$1.00$	$1.00$	$1.00$
Panel B: Outcome variables
Output growth $\Delta_{1}\log y_{cs,t}$	26,351	$0.052$	$0.190$	$-11.070$	$0.045$	$8.269$

Table 2: Summary Statistics: WIOD Estimation Panel, 2001–2013

Panel of Figure 2 — Figure 2: Log-log CCDF of absolute output contractions by curvature quartile. Dashed blue: most fragile ( $\bar{\kappa}<-29.0$ , $n=1{,}847$ ); solid orange: most resilient ( $\bar{\kappa}>-19.5$ , $n=1{,}203$ ). Thick lines: fitted power-law overlays; dotted lines: fitted exponential overlays. The power law fits the fragile quartile tail visually over more than two decades; the exponential fit collapses well before the data. The difference in slopes— $\hat{\alpha}=1.51$ vs $\hat{\alpha}=2.14$ —corresponds to the structural prediction of Theorem 3.11 : more negative curvature generates heavier tails. The KS test rejects exponential tails at the 1% level in both quartiles; the Vuong test selects the power law over the log-normal ( $\mathrm{LR}=3.42$ , $p=0.003$ ).

Curvature quartile	$\hat{\alpha}$	95% CI	$x_{\min}$	$n$
Bottom (most fragile): $\bar{\kappa}<-29.0$	$1.51$	$[1.43,\;1.60]$	$0.031$	$1{,}847$
Middle two quartiles: $-29.0\leq\bar{\kappa}\leq-19.5$	$1.78$	$[1.72,\;1.85]$	$0.028$	$3{,}612$
Top (most resilient): $\bar{\kappa}>-19.5$	$2.14$	$[2.03,\;2.26]$	$0.024$	$1{,}203$
Full sample	$1.83$	$[1.79,\;1.87]$	$0.027$	$6{,}662$

Table 3: Power-Law Tail Estimation by Curvature Quartile

Panel A: Varying $x_{\min}$ (fragile quartile)
Specification	Filter	$\hat{\alpha}$	95% CI	$n$
$x_{\min}=0.010$ (inclusive)	Negative only	$1.49$	$[1.42,\;1.57]$	$2{,}418$
$x_{\min}=0.031$ (KS-optimal)	Negative only	$1.51$	$[1.43,\;1.60]$	$1{,}847$
$x_{\min}=0.050$ (restrictive)	Negative only	$1.54$	$[1.44,\;1.65]$	$1{,}203$
Panel B: Varying filter (full sample)
All episodes $\|\Delta_{1}\log y\|$	Both signs	$1.86$	$[1.82,\;1.90]$	$9{,}347$

Table 4: Robustness of

\hat{\alpha}

to Threshold and Filter Choices

Panel of Figure 3 — Figure 3: Distribution of cascade episode sizes (log-binned frequency vs. log size). A cascade episode is a cluster of co-occurring sectoral contractions within the same country and within a two-year window. Total episodes: 5,253 across 40 countries and 13 years. The power law ( $\hat{\alpha}_{\mathrm{ep}}=1.62$ ) fits the distribution across the full range; the exponential alternative is rejected ( $p<0.001$ , KS test). The slope $\hat{\alpha}_{\mathrm{ep}}=1.62$ is consistent with the WIOD-network theoretical prediction from equation ( 13 ) evaluated at the cross-country median curvature ( $\hat{\alpha}_{\mathrm{theory}}=1.65$ ). The heavy right tail—episodes involving more than 20 sectors— corresponds to the $\alpha<2$ regime identified in Table 3 .

Statistic	Simulated	Theoretical	Empirical
Tail exponent $\hat{\alpha}$	$1.79$	$1.83$	$1.83$
$\mathbb{P}(S>10)$	$0.118$	$0.113$	—
$\mathbb{P}(S>50)$	$0.041$	$0.038$	—
$\mathbb{P}(S>100)$	$0.021$	$0.019$	—
Fraction $\alpha<2$ (by country)	$87\%$	$>\!0$ predicted	$100\%$

Table 5: Simulated vs. Theoretical Cascade Distribution

Model	Adj. $R^{2}$	AIC	BIC	$N$
Ricci curvature (winner)	$\mathbf{0.00534}$	$-12{,}900.6$	$-12{,}876.1$	26,351
Betweenness centrality	$0.00018$	$-12{,}764.2$	$-12{,}739.7$	26,351
PageRank centrality	$0.00002$	$-12{,}760.0$	$-12{,}735.5$	26,351
Herfindahl index (HHI)	$0.00117$	$-12{,}790.4$	$-12{,}765.8$	26,351
Eigenvector centrality	$0.00006$	$-12{,}762.0$	$-12{,}745.6$	26,351
Full model (all metrics jointly)	$0.00567$	$-12{,}906.5$	$-12{,}857.4$	26,351

Table 6: Predictive Horse Race: Ricci Curvature vs. Classical Network Metrics

	Direct effect ( $\hat{\beta}_{h}$ )			Amplification ( $\hat{\gamma}_{h}$ )
$h$	Coeff.	SE	$t$	Coeff.	SE	$t$	$N_{h}$
1	$0.000212$	$0.000301$	$0.71$	$-0.000256$	$0.000520$	$-0.50$	26,351
2	$0.000630$	$0.000430$	$1.47$	$-0.001320$	$0.000730$	$-1.80^{*}$	24,324
3	$0.001540$	$0.000540$	$2.86^{***}$	$-0.003457$	$0.000890$	$-3.89^{***}$	22,297
4	$0.002070$	$0.000680$	$3.03^{***}$	$-0.002890$	$0.001120$	$-2.58^{**}$	20,270
5	$0.002379$	$0.000820$	$2.89^{***}$	$-0.002760$	$0.001370$	$-2.01^{**}$	18,243

Table 7: Local Projection IRF: Ricci Curvature and Cumulative Output Growth

May 2, 2026

Hall-Like Transversal Stress and Sandpile Criticality on Real Production Networks

This paper develops a Hall-Sandpile model of economic instability that combines a Hall-like transversal stress mechanism with sandpile threshold dynamics on a real production-network substrate. In analogy with the physical Hall effect, where exposed flows under an external field generate stress in a transversal direction, we model economic shocks as fields that act on flow-intensive, low-redundancy, low-capacity nodes and produce systemic stress through a multiplicative conversion function. The accumulated stress drives a discrete toppling rule and an avalanche dynamics whose effective activation threshold declines with transversal exposure. The model is calibrated on annual World Input--Output Database (WIOD) production networks for 2000--2014 and simulated on the 2014 substrate (2{,}283 country--sector nodes) under three alternative propagation normalisations to avoid mechanical near-criticality from row-stochastic operators. Controlled Monte Carlo experiments over external field intensity and redundancy stress generate four ordered regimes: stable absorption, latent fragility, critical transition, and avalanche regime. Mean avalanche size and the probabilities of finite-size systemic events

\Pr(S\!\geq\!5)

\Pr(S\!\geq\!10)

and

\Pr(S\!\geq\!20)

rise jointly with field intensity and redundancy stress. Tail diagnostics show regime-dependent thickening of the avalanche distribution, but the estimated tail indices remain too high to interpret as evidence of universal power-law criticality. The contribution is therefore a finite-size, real-network description of how transversal stress activates structural fragility, not a claim of self-organised criticality in the global economy.

Diego Vallarino

Inter-American Development Bank, Washington, D.C., United States

Topic pending

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Apr 3, 2026

Identification and Inference in Nonlinear Dynamic Network Models

We study identification and inference in nonlinear dynamic systems defined on unknown interaction networks. The system evolves through an unobserved dependence matrix governing cross-sectional shock propagation via a nonlinear operator. We show that the network structure is not generically identified, and that identification requires sufficient spectral heterogeneity. In particular, identification arises when the network induces non-exchangeable covariance patterns through heterogeneous amplification of eigenmodes. When the spectrum is concentrated, dependence becomes observationally equivalent to common shocks or scalar heterogeneity, leading to non-identification. We provide necessary and sufficient conditions for identification, characterize observational equivalence classes, and propose a semiparametric estimator with asymptotic theory. We also develop tests for network dependence whose power depends on spectral properties of the interaction matrix. The results apply to a broad class of economic models, including production networks, contagion models, and dynamic interaction systems.

Diego Vallarino

Inter-American Development Bank (IDB), Washington, DC, United States

Parameter Identifiability Nonlinear System Identification

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Apr 2, 2026

Network Structure in UK Payment Flows: Evidence on Economic Interdependencies and Implications for Real-Time Measurement

Network analysis of inter-industry payment flows reveals structural economic relationships invisible to traditional bilateral measurement approaches, with significant implications for real-time economic monitoring. Analysing 532,346 UK payment records (2017--2024) across 89 industry sectors, we demonstrate that graph-theoretic features which include centrality measures and clustering coefficients improve payment flow forecasting by 8.8 percentage points beyond traditional time-series methods. Critically, network features prove most valuable during economic disruptions: during the COVID-19 pandemic, when traditional forecasting accuracy collapsed (R2} falling from 0.38 to 0.19), network-enhanced models maintained substantially better performance, with network contributions reaching +13.8 percentage points. The analysis identifies Financial Services, Wholesale Trade, and Professional Services as structurally central industries whose network positions indicate systemic importance beyond their transaction volumes. Network density increased 12.5% over the sample period, with visible disruption during 2020 followed by recovery exceeding pre-pandemic integration levels. These findings suggest payment network monitoring could enhance official statistics production by providing leading indicators of structural economic change and improving nowcasting accuracy during periods when traditional temporal patterns prove unreliable.

Aditya Humnabadkar

Office for National Statistics, UK

Financial Forecasting Time Series Forecasting

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