Topology maximizing the relative feasible wealth volume r(G)

Determine, for given numbers of users n and total coins C, the payment channel network topology G(V,E,cap) that maximizes r(G) = |W_G| / |đť’˛(C,n)|, where W_G is the set of wealth distributions feasible under G and đť’˛(C,n) is the set of all on-chain wealth distributions. Characterize the optimal topology or develop an efficient method to find it.

Background

The paper defines r(G) as the ratio between the number of feasible off-chain wealth distributions and the total number of on-chain wealth distributions, capturing how network topology restricts feasibility. Larger r(G) implies more payments can be executed off-chain without on-chain intervention.

The authors state they lack a method to find the topology that maximizes r(G). Solving this problem would inform how to design payment channel networks to minimize infeasible payments and improve off-chain scalability.

References

In particular we are not aware of a smart way to find the topology G that maximizes r(G).

— A Mathematical Theory of Payment Channel Networks  (2601.04835 - Pickhardt, 8 Jan 2026) in Polytope of feasible Wealth Distributions, subsection discussing r(G) (around equation for r(G))