Chapter 2: The Balance Sheet as a Simplicial Complex
A node is a firm. An edge is a contract. A triangle is a cycle of obligations.
The example
Three banks: A, B, C. A has lent £100 to B. B has lent £80 to C. C has posted £60 of collateral back to A.
Draw this as a graph: three nodes, three directed edges. This is the obligation complex \(\Gamma\) for this three-firm system.
Each edge is a bilateral contract — a promise from one party to another. Taken individually, each is well-defined and enforceable. But the three together form a triangle, and triangles have properties that edges do not.
Simplices
A simplicial complex is built from:
- 0-simplices (nodes): financial institutions, accounts, currencies
- 1-simplices (edges): bilateral contracts, loans, swaps, repos
- 2-simplices (triangles): three mutually connected nodes — a funding cycle
- 3-simplices (tetrahedra): four mutually connected nodes — a four-party cycle
The balance sheet of the financial system is the 0- and 1-skeleton of this complex: nodes and edges only. Risk management asks whether higher simplices — triangles and tetrahedra — are present, and if so, whether they are filled or hollow.
Filled vs hollow
A filled triangle {A, B, C} means there exists a direct three-way settlement mechanism: if A, B, and C all default simultaneously, a single clearing agent can resolve all three claims at once. CCP clearing partially achieves this for standardised products.
A hollow triangle {A, B, C} means the three bilateral contracts exist but no three-way settlement mechanism does. The triangle is present as a cycle in the graph, but there is no 2-simplex filling it in. The obligations are mutually consistent (each bilateral contract is valid) but there is no agent who can see and resolve all three simultaneously.
This distinction — filled vs hollow — is the geometric content of the difference between bilateral risk (\(H^0\)) and triangular risk (\(H^1\)).
The Betti numbers
Given the obligation complex \(\Gamma\), three numbers summarise its topology:
| Number | Name | Financial meaning |
|---|---|---|
| \(\beta_0\) | Connected components | Number of isolated sub-networks |
| \(\beta_1\) | Independent cycles | Number of unfilled funding loops |
| \(\beta_2\) | Hollow voids | Number of irresolvable four-party conflicts |
\(\beta_0 = 1\) means the network is connected — every firm can reach every other through some chain of obligations. \(\beta_1 = 0\) means every funding loop is filled — no unresolved cycles. \(\beta_2 = 0\) means every four-party conflict can be resolved — no hollow tetrahedra.
The 2008 financial crisis had \(\beta_2 > 0\). The rest of this book explains what that means and why it matters.
What standard risk management sees
A standard risk model operates on edges only: it measures the exposure on each bilateral contract, stress-tests each counterparty, and aggregates. It does not ask whether the triangle formed by three bilateral contracts is filled or hollow. It cannot see \(\beta_1\) or \(\beta_2\).
This is not a limitation of computational power. It is a conceptual limitation: the standard framework has no language for the topology of the network. Providing that language is the purpose of this book.