Obligation Compression in Directed Settlement Graphs
We model a settlement network as a directed graph in which vertices represent participants and edges represent unresolved obligations. We describe a class of compression procedures that identify cyclic structure within the graph and cancel mutually offsetting obligations prior to any capital movement, and discuss the conditions under which residual exposure is minimized.
1. Motivation
Bilateral settlement treats each obligation as an independent instruction to move capital from one party to another. When many participants transact within the same interval, this produces significant redundant movement: capital travels from A to B and, separately, from B to A, when only the net difference between the two flows carries economic meaning.
We are interested in the structural conditions under which a network of obligations can be reduced to a smaller set of residual transfers without changing the economic outcome for any participant.
2. Graph Formulation
Let G = (V, E) be a directed multigraph where each vertex v ∈ V is a settlement participant and each edge e(i, j, w) ∈ E denotes an obligation of weight w owed by i to j.
A compression of G is a graph G′ such that, for every vertex v, the net signed flow into v is preserved: Σ w(u,v) − Σ w(v,u) is invariant between G and G′. We are interested in compressions that minimize Σ w over E′.
3. Cycle Cancellation
The simplest compression procedure identifies directed cycles within G and cancels the minimum edge weight along each cycle from every edge in that cycle. This is repeated until no cycle with positive minimum weight remains.
This procedure is greedy and does not, in general, produce a globally minimal residual graph, but it is inexpensive to compute per epoch and produces substantial reductions in observed obligation graphs, where cyclic structure is common among participants with recurring corridors.
4. Multilateral Netting as Flow Minimization
A stronger formulation treats compression as a min-cost flow problem: given the net position of every vertex, find the minimum-weight edge set that realizes those net positions. This produces a provably minimal residual graph, at higher computational cost.
In practice, NIL applies cycle cancellation continuously within an epoch and a bounded flow-minimization pass at epoch close, trading exactness for latency during the open window.
5. Observed Compression
Within simulated epochs of comparable structure to typical remittance and small-corridor payment traffic, cycle cancellation alone routinely reduces gross obligation value by 70–90%. These figures are illustrative of the mechanism, not measurements of live financial activity — see the disclosure notice below.