4.13.2 Contagion & Financial-Network Models

Basic Prerequisites: 01 · From Zero.

1. Intuition & Practical Objective

Contagion is the mechanism by which an idiosyncratic failure becomes a system failure. This page’s objective: build the standard tools - the exposure matrix, the loss-propagation cascade, and the connectivity-dependence result - and learn from the numbers that more connectivity is neither always safe nor always dangerous.

The core object is the exposure matrix $L = (L_{ij})$ , where $L_{ij}$ is the claim bank $i$ holds on bank $j$ . Rows are creditors, columns debtors. Two derived quantities:

Out-degree (bank $i$ ‘s number of counterparties it lends to) - with a fixed total lending $L_{i}$ split evenly, a higher out-degree means smaller individual exposures.
Capital buffer $E_{i}$ - losses absorbed before default.

The cascade rule is the one from §01 generalised to an arbitrary graph: a bank defaults when cumulative losses exceed capital; the default then imposes its full claim on each creditor. Iterate to a fixed point.

The headline empirical pattern (Gai–Kapadia 2010) is robust-yet-fragile: moderate connectivity is the most dangerous - individual exposures are still big, but the channel connecting them now spans the system - whereas very high connectivity diversifies each exposure to the point where the shock is absorbed. The experiment below reproduces a non-monotone default count with a subtle twist you should notice.

2. Mathematical Ground Truth & Derivations

The cascade in matrix form (Eisenberg–Noe 2001 picture). Let $e_{i}$ be bank $i$ ‘s external/net capital, $L_{ij}$ interbank claims, and $\overset{x}{ˉ}_{i} = \sum_{j} L_{ij}$ its total claims. In a clearance round, bank $i$ ‘s value after payments satisfies, for a default-absorbing allocation,

V_{i} = e_{i} + j \sum L_{ij} 1_{{V_{j} \geq \overset{x}{ˉ}_{j}}} + j \sum \frac{L _{ij}}{\sum _{k} L _{kj}} V_{j} 1_{{V_{j} < \overset{x}{ˉ}_{j}}},

where a defaulting $j$ pays creditors proportionally to its remaining value $V_{j}$ (a clearing condition). The default of one bank reduces other banks’ assets through the second term, which can push them negative, which changes their payments, etc. - a fixed-point problem. The simple sequential/cascade version (what we compute) drops the proportional-recoveries subtlety and treats a default as a full write-off, which is conservative and standard for first-pass contagion analysis.

Connectivity and exposure concentration. Fix total lending $L$ and connectivity probability $p$ . Expected out-degree $\approx p (N - 1)$ , so each exposure $\approx L / (p (N - 1))$ . A defaulting counterparty’s hit on a creditor is $L / (p (N - 1))$ . Two regimes:

Sparse, $p (N - 1)$ small: each exposure $> E$ → one default bankrupts its creditor → chain of concentrated hits. Fragile.
Dense, $p (N - 1)$ large: each exposure $≪ E$ → a default hurts but doesn’t kill → “diversification by connectivity” contains it.

Hence the default count is non-monotone in $p$ - which is exactly why “more connected = safer” is a false slogan.

3. Computational Implementation - contagion on a random network

We build a random directed interbank network, knock out a few banks, and let the cascade run. Lower connectivity concentrates losses (big per-edge exposures) into effective contagion; very high connectivity diversifies them away. Stdlib only.

Read the output. Sparse/moderate connectivity ( $p = 0.03$ – $0.12$ ) turns the four seeded failures into near-complete collapse (35–40/40 defaults): each bank has few, large exposures (with $L = 12$ and out-degree $\approx 1$ – $5$ , a single exposure $10 > 4 = E$ wipes a full neighbour). But at $p = 0.25$ (out-degree $\approx 10$ ) each claim is only $12/10 = 1.2 < E = 4$ , so a neighbour survives the loss and the contagion sputters at $5/40$ . This non-monotonicity is the robust-yet-fragile result: diversifying the network past a point converts a single-bank failure from a shock-wave into a contained dent - but only because each edge is then too small to breach a neighbour’s capital. The danger hump sits in the middle, where edges are still big enough to kill and many enough to transmit.

4. Failure Modes & First-Principles Breakdowns

Partial observability of $L$ . The entire method needs the adjacency/claim matrix, but interbank exposures are opaque (bilateral netting, off-balance-sheet, CCP positions). Missing edges are exactly the ones that matter for contagion - the model is most fragile where it is most useful.
Full recovery is not full write-off. Sequential models treat a default as 100% loss to creditors; in reality recovery/fire-sale values are partial and endogenous to how many default simultaneously (the recovery rate itself is a systemic quantity). Eisenberg–Noe gives a fixed-point correction, but with unknown collateral it stays approximate.
Static graph, dynamic system. Networks rewire during stress (counterparties cut lines, flight-to-quality); the pre-crisis $L$ you measured is not the post-crisis $L$ . Connectivity is endogenous to the stress it transmits.

5. References

Eisenberg & Noe, Systemic Risk in Financial Systems, Management Science 47(2):236–249 (2001)
Gai & Kapadia, Contagion in Financial Networks, Proc. R. Soc. A 466 (2010)
Allen & Gale, Financial Contagion, JPE 108 (2000)
Gregory, The xVA Challenge (2025)

6. Connected Graph Bridges

Back: 01 · From Zero · Index Hub
Forward: 03 · Systemic Risk Measures
Sibling mechanisms: Margin & Funding Spirals (the fire-sale channel) · Counterparty Contagion

Kwant Atlas

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