Catastrophic disruption cascades driven by the nonlinearity of systemic risk

When one company fails, others can usually find a new supplier. However, if a specific group of companies fails at the same time, they might leave no alternatives. This can cause a massive chain reaction that shuts down much of the economy.

Modern supply chains are highly specialized, interconnected webs of buyer-supplier dependencies. While the failure of a single firm typically causes localized ripples, recent global events like the COVID-19 pandemic have demonstrated that these disruptions can escalate into system-wide collapses. Researchers have long tried to quantify this "systemic risk"—the danger of a local failure cascading through a network. Modeling these shifts has been difficult due to a lack of granular, firm-level data.

A new study reveals that our current understanding of risk may be incomplete. The authors report that the systemic risk caused by the simultaneous failure of a few firms can be vastly larger than the sum of their individual impacts. In the national supply chain of Ecuador, they found instances where combined failures produced risk amplifications of up to a factor of 257.

The limit of sublinear risk models

Traditional methods for quantifying systemic risk often assume a "sublinear" relationship. In finance, the authors highlight the DebtRank model. This model tracks how a bank's distress devalues the assets of its creditors. As shown in, DebtRank is inherently sublinear. This means the total loss from two banks failing simultaneously is always less than or equal to the sum of their individual losses. This happens because the model includes built-in constraints. For example, a bank cannot lose more equity than it actually possesses.

Supply chains, however, operate under different logic. The authors argue that existing models fail to account for a critical compensatory mechanism: supplier substitutability (the ability of a buyer to switch to a different vendor). In a healthy network, if a supplier fails, a customer can often pivot to an alternative provider. This acts like a shock absorber in a vehicle. It dampens the impact of a bump. But the authors demonstrate that when multiple suppliers fail at once, these "shock absorbers" vanish. This turns a manageable hiccup into a catastrophic cascade.

Breaking the substitution mechanism

To capture this nonlinearity, the authors utilize the Economic Systemic Risk Index (ESRI). This metric estimates the fraction of total production lost due to a firm's failure. Their model moves through the network in two directions. First, it calculates downstream propagation (shocks moving from suppliers to customers). This occurs via input shortages. Second, it calculates upstream propagation (shocks moving from customers to suppliers). This occurs via lost demand.

The mathematical core of their approach is the "effective market share" ($\sigma_i(t)$). This is not a static number. It is a dynamic value that changes as the network is shocked. If a firm's competitors in the same industry also fail, that firm's effective market share rises. This makes it much harder for customers to replace them. The authors illustrate this in .

A single supplier failure can be partially compensated for. However, a dual failure leaves the central firm with no alternatives. This forces a total halt in production.

To find these rare, high-risk combinations, the researchers used a "random chemistry" inspired extraction procedure. They do not check every possible combination. Instead, they first shock large, random sets of firms. They then systematically prune them. They do this until they find the smallest subset that still triggers a massive, nonlinear cascade.

Rare but catastrophic amplifications

The study finds that while massive amplification is rare, it is mathematically extreme. Analyzing the Ecuadorian national supply chain, the authors report that only 0.14% of randomly sampled firm pairs exhibit an amplification factor ($\alpha$) greater than 4. This means most pairs behave predictably, but a tiny fraction causes disproportionate damage. Yet, the tail of this distribution is heavy. As shown in, the frequency of these high-amplification events follows a power law.

This implies that extreme "black swan" events occur more often than standard statistics would suggest.

The most striking result involves a specific pair of firms in the Ecuadorian network. Their individual impacts were negligible. However, their joint failure resulted in an ESRI of 0.94. This means 94% of the total network production was lost. Compared to the sum of their individual risks (0.0036), this represents an amplification factor of 257. The authors categorize these events into three structural "modes" in .

One mode involves a small supplier whose failure becomes catastrophic only when a large player in its same industry also fails. This strips the network of its ability to substitute the small player.

Hidden vulnerabilities and model gaps

The authors note that these findings represent an upper bound on potential damage. The model uses a simplified heuristic for supplier replacement. It does not simulate active "rewiring" (the process of a firm actively seeking and signing new contracts). Therefore, the actual speed of a network's recovery might differ. The model also ignores price adjustments. In reality, wealthy firms might outbid others for scarce supplies to keep their lines running.

There are also data-driven limitations. The reconstruction relies on 2015 VAT data. This provides a snapshot of transactions. It lacks real-time information on inventory levels or production lead times. Without knowing how much stock a firm holds, it is hard to predict exactly when a supply delay becomes a production halt. Additionally, the extraction method is stochastic (randomized). While it is effective at finding destructive sets, it cannot mathematically guarantee that every possible combination has been identified.

Identifying the plateau

The research suggests that systemic risk monitoring must evolve. Instead of simply ranking firms by individual importance, managers should look for "plateau-firms." These are groups of firms whose combined failure disables the substitution mechanism for an entire industry.

The verdict depends on the application. For economic planning, this work warns that "safe" looking industries can harbor hidden, nonlinear vulnerabilities. For practitioners building resilient logistics networks, the tool is ready for prototyping. Users should supplement it with real-world data on contract flexibility and inventory buffers. Code for the extraction procedure is reportedly available; see the paper for the canonical link.

Figures from the paper