How Maximum-Entropy Interbank Reconstruction Algorithms Infer Missing Bilateral Exposures: Row/Column Constraints, RAS Scaling, Structural Zeros, Sparsity Bias and Contagion Bounds

Reader question: A bank’s balance sheet may tell us how much it lends to other banks and how much it borrows from them, but not which bank owes which counterparty. How can an algorithm fill in the missing bilateral matrix without inventing more structure than the data justify?

Maximum-entropy reconstruction treats the unknown interbank network as a constrained inference problem. The observed row and column totals must be respected, known impossible links must stay zero, and among all matrices satisfying those constraints the algorithm chooses one that is as diffuse as possible relative to a prior.

This article owns the missing-bilateral-exposure reconstruction problem: aggregate lending totals + aggregate borrowing totals + structural constraints + an entropy objective → a complete non-negative exposure matrix that exactly or approximately matches the known margins.

It does not own clearing defaults, fire-sale contagion, bank resolution, large-exposure rules or payment-system timing. Those are downstream mechanisms. The purpose here is mathematical: how to infer a network from incomplete balance-sheet information while making the uncertainty visible.

This is public computational education. It is not a statement about the solvency of any real institution and not a substitute for confidential supervisory exposure data.

1. The observed data are usually margins, not links

Suppose there are n banks. Let:

X_ij ≥ 0

represent the exposure of lender bank i to borrower bank j.

From public or supervisory balance-sheet aggregates we may know each bank’s total interbank assets:

A_i = Σ_jX_ij,

and total interbank liabilities:

B_j = Σ_iX_ij.

But the interior cells X_ij are missing.

This is an inverse problem. Many different matrices can share exactly the same row and column sums.

2. A feasibility check comes before entropy

If the network is closed, total lending must equal total borrowing:

Σ_iA_i = Σ_jB_j.

If they do not match, the dataset is incomplete, inconsistent or missing an outside sector.

A common solution is to add a rest-of-world or outside-creditor node so that domestic totals can reconcile without pretending every interbank position is internal to the observed sample.

Falsifier: if the margins cannot be reconciled before reconstruction, no entropy method can make the underlying accounting inconsistency disappear.

3. The simplest unconstrained gravity guess

If self-exposures are allowed and no other structural information is available, a diffuse matrix consistent with row and column totals takes the form:

X_ij = A_iB_j/S,

where:

S = Σ_iA_i = Σ_jB_j.

Each lender distributes its assets across borrowers in proportion to their total borrowing, and each borrower receives funding in proportion to lenders’ total supply.

This is not yet enough for a realistic interbank network because a bank cannot lend to itself in the intended bilateral matrix.

4. Structural zeros change the problem

Set:

X_ii = 0

for all banks.

Other links may also be known to be impossible because of jurisdiction, product, legal or reporting constraints.

Once structural zeros are imposed, the simple outer-product formula no longer preserves all margins. The remaining admissible cells must be rebalanced iteratively.

5. Maximum entropy as constrained optimisation

One formulation chooses X to maximize Shannon-type entropy:

H(X) = −Σ_ijX_ij ln X_ij,

subject to:

• row sums equal observed lending totals;
• column sums equal observed borrowing totals;
• all cells are non-negative;
• structural-zero cells remain zero.

When a prior matrix Q exists, a more general formulation minimizes Kullback–Leibler divergence:

D(X||Q) = Σ X_ij ln(X_ij/Q_ij) − X_ij + Q_ij,

over admissible cells.

This asks for the matrix closest to the prior while exactly satisfying the new constraints.

6. Why “maximum entropy” does not mean “maximum uncertainty everywhere”

The algorithm is not free to spread exposures arbitrarily.

Known row sums, column sums and structural zeros are hard information. Entropy only determines how to allocate what remains unknown.

The principle is:

preserve known constraints; avoid adding unsupported concentration.

That makes maximum entropy a disciplined baseline, not a claim that real banks actually diversify exposures as evenly as possible.

7. RAS / iterative proportional fitting

A practical way to solve the constrained problem is iterative proportional fitting, often called the RAS algorithm.

Start with a positive prior matrix Q on admissible links and zeros on forbidden links.

Row step: multiply each row by a factor that makes its sum equal the target A_i.

Column step: multiply each column by a factor that makes its sum equal the target B_j.

Alternating row and column scaling continues until both sets of margins are within tolerance.

The resulting matrix often has multiplicative form:

X_ij = r_i Q_ij s_j

for admissible cells, where r_i and s_j are the final scaling factors.

8. A small worked example

Suppose three banks have interbank lending totals:

A = (60, 30, 10)

and borrowing totals:

B = (20, 50, 30).

Both total 100.

We require a zero diagonal. Start with prior weights of one on every off-diagonal cell:

Q = [[0,1,1],[1,0,1],[1,1,0]].

The first row scaling gives row 1 a total of 60, row 2 a total of 30 and row 3 a total of 10. Column totals will generally then be wrong. Column scaling repairs them, which disturbs rows slightly, so another row pass is needed.

The procedure repeats until both:

max_i|Σ_jX_ij−A_i|

and:

max_j|Σ_iX_ij−B_j|

fall below the numerical tolerance.

The exact cell values are less important than the invariant: the final matrix must reconcile both sets of margins while keeping forbidden cells at zero.

9. Convergence diagnostics

A production implementation should track:

• maximum row residual;
• maximum column residual;
• total-mass difference;
• number of iterations;
• largest scaling factor;
• smallest positive cell;
• whether any admissible row or column has no feasible counterparty.

Huge scaling factors can indicate near-infeasibility even when the algorithm eventually converges.

10. Structural zeros can make the margins impossible

Imagine a borrower needs 90 of funding, but the only lenders allowed to connect to it have combined lending capacity of 50.

No non-negative matrix can satisfy both the margins and the zero pattern.

This is not a numerical failure. It is a feasibility failure.

Falsifier: test whether subsets of constrained rows have enough admissible column capacity and vice versa. If the support pattern cannot carry the required mass, the model needs a revised node set or constraints.

11. Why maximum entropy produces dense networks

If every off-diagonal link is permitted, the entropy objective tends to place positive exposure on many links.

Real interbank markets are often sparser, with core–periphery structure, relationship lending and concentrated counterparties.

Therefore the maximum-entropy matrix can look too diversified even though it matches every balance-sheet total perfectly.

12. The contagion consequence of a dense reconstruction

Spreading one bank’s borrowing over many creditors dilutes each creditor’s direct loss if that bank defaults.

That can reduce apparent default cascades relative to a sparse network where a few creditors carry large exposures.

BIS research by Anand, Craig and von Peter found that, in their stress-testing setting, a minimum-density reconstruction could overestimate contagion while maximum entropy could underestimate it; using the two as benchmarks created a useful range when the true network was unknown.

The key lesson is not “maximum entropy always understates contagion.” It is:

network uncertainty must be propagated into stress-test uncertainty.

13. Mistrulli’s counterexample matters

Paolo Mistrulli compared maximum-entropy estimates with observed Italian interbank bilateral exposures.

The results supported the common finding that maximum entropy often underestimates contagion, but also showed that this is not universal. Depending on actual network structure, recovery assumptions and capitalization, maximum entropy can sometimes overestimate contagion.

This is an important falsifier of a simplistic rule.

14. Inputs and outputs

Inputs can include:

• bank identifiers;
• total interbank assets by bank;
• total interbank liabilities by bank;
• outside-sector totals;
• known bilateral exposures;
• structural-zero mask;
• prior matrix;
• row and column tolerances;
• maximum iterations;
• regularisation or sparsity assumptions.

Outputs can include:

• reconstructed bilateral exposure matrix;
• row and column reconciliation residuals;
• network density;
• largest exposures;
• concentration measures;
• core–periphery diagnostics;
• reconstruction uncertainty scenarios;
• downstream contagion ranges.

15. Evidence polarity

Evidence for confidence includes exact margin reconciliation, structural zeros preserved, stable results under reasonable priors, consistency with any known bilateral links, network statistics resembling observed markets, and downstream conclusions that remain similar across multiple plausible reconstructions.

Evidence against confidence includes poor margin fit, extreme scaling factors, results dominated by arbitrary prior weights, implausibly dense connectivity, known large exposures diluted across many small inferred links, contagion results that change dramatically under an equally plausible sparse reconstruction, or contradictions with confidential validation samples.

16. Counterexample: correct margins, wrong topology

Two exposure matrices can share identical row and column totals while one is fully connected and the other is sparse.

If systemic-risk conclusions differ, margin fit alone is not enough to validate the network.

Falsifier: compare graph density, degree distribution, concentration and contagion outcomes under alternative reconstructions.

17. Counterexample: the zero diagonal is forgotten

If the unconstrained outer-product matrix is used directly, each bank may appear to lend to itself.

That self-exposure absorbs part of both a row and a column total, reducing exposures available to real counterparties.

Falsifier: verify the diagonal is exactly zero before any downstream network metric is calculated.

18. Counterexample: public sample excludes foreign counterparties

Suppose domestic banks’ reported interbank assets exceed liabilities because some assets are claims on foreign banks not included in the node set.

Forcing the domestic matrix to balance reallocates cross-border exposures incorrectly.

Repair: introduce an outside or rest-of-world node, or use available country-level cross-border constraints.

19. Counterexample: one known large link should not be “smoothed away”

If supervisory data reveal a specific 40-unit exposure from A to B, that cell is no longer unknown.

The reconstruction should fix the known amount first, subtract it from the relevant row and column margins, and distribute only the remaining unknown mass.

Falsifier: re-sum known bilateral positions after reconstruction. If they changed, the algorithm violated hard information.

20. Counterexample: zero prior creates an accidental structural zero

In KL-based fitting, a prior cell with exactly zero weight may remain zero because assigning positive mass there creates infinite relative-entropy cost.

A careless prior can therefore ban a link unintentionally.

Diagnostic: distinguish deliberate structural zeros from merely small prior probabilities.

21. Sparse alternatives

Maximum entropy is a baseline, not the only reconstruction.

Minimum-density methods try to concentrate exposures on fewer links while respecting margins.

Fitness models estimate link probabilities from node attributes such as size.

Bayesian network reconstruction places distributions over unknown links rather than one point estimate.

Entropy with degree constraints adds information about expected numbers of counterparties.

Ensemble methods generate many matrices consistent with the constraints and propagate reconstruction uncertainty into stress tests.

22. The 2026 multi-country extension

Recent work has extended maximum-entropy reconstruction to multi-country systems by combining bank-level domestic interbank totals with country-level aggregate cross-border exposures.

The problem becomes hierarchical:

1. country-level external positions constrain cross-border blocks;
2. bank-level margins constrain rows and columns within those blocks;
3. domestic blocks still require zero diagonals and local balancing;
4. a global rest-of-world node may be needed to close the system.

This extension is useful because a bank network reconstructed country by country can be internally consistent yet globally inconsistent once cross-border claims are considered.

23. Maximum entropy is upstream of contagion models

The reconstructed matrix can feed an Eisenberg–Noe clearing model, DebtRank-style distress propagation, default cascades or stress testing.

But the downstream model should never forget that the matrix may itself be inferred.

There are therefore two uncertainty layers:

network reconstruction uncertainty + contagion-model uncertainty.

Reporting one deterministic cascade from one maximum-entropy matrix hides the first layer entirely.

24. Connections to the Bukit Timah Tutor knowledge estate

The downstream fixed-point clearing problem is developed in Eisenberg–Noe clearing algorithms.

The wider mechanisms of direct exposures, overlapping portfolios and fire sales are surveyed in interbank network stress algorithms.

Concentration controls connect to Basel large-exposure algorithms.

The lane hub is Finance & Banking Algorithms | Applied Mathematics in Real Financial Systems.

25. What would falsify confidence?

Confidence should be withdrawn if the margin totals are inconsistent; if structural zeros make the target margins infeasible; if the RAS/IPF residuals do not converge; if known bilateral exposures are not preserved; if the reconstructed topology is inconsistent with available network evidence; if systemic-risk conclusions reverse across equally plausible reconstructions; or if a single maximum-entropy matrix is presented as though it were observed fact.

26. Verification and update triggers

Preserve the original margin data, node universe, outside-sector treatment, structural-zero mask, prior matrix, fixed known links, scaling tolerance and iteration history.

Trigger revalidation when bank balance sheets change materially, the sample of institutions changes, new cross-border data become available, large-exposure disclosures reveal previously hidden links, a bank merger changes the node map, or a downstream contagion analysis becomes sensitive to topology that the current reconstruction does not identify robustly.

27. Primary and high-quality references

• Kartik Anand, Ben Craig and Goetz von Peter, Filling in the Blanks: Network Structure and Interbank Contagion, BIS Working Papers No. 455.
• Federal Reserve Bank of Cleveland, Filling in the Blanks: Network Structure and Interbank Contagion, working-paper version and publication record.
• Paolo Emilio Mistrulli, Assessing Financial Contagion in the Interbank Market: Maximum Entropy versus Observed Interbank Lending Patterns, Bank of Italy.
• Simone Sbaraglia, Pin Guo and Stefano Zedda, Extending Maximum-Entropy Interbank Reconstruction to a Multi-Country Framework with Cross-Border Exposures, 2026.
• Federal Reserve Board and OCC, SR 11-7 — Guidance on Model Risk Management, for model uncertainty, benchmarking and validation principles.

Educational boundary: Maximum entropy tells us how to allocate missing information under explicit constraints. It does not turn missing bilateral data into observed truth.