Closed-form expression for the volume of W_G

Derive a closed-form formula for |W_G|, the number of feasible wealth distributions in a payment channel network G(V,E,cap), where W_G is defined as the set of wealth distributions ω: V → {0,…,C} for which there exists a liquidity function λ ∈ L_G satisfying ∑_{e∈E: v∈e} λ(e,v) = ω(v) for all v ∈ V. The formula should express |W_G| directly in terms of the topology (V,E) and channel capacities cap(e).

Background

The paper introduces W_G as the subset of wealth distributions that are feasible within a given payment channel network G(V,E,cap), constrained by conservation of liquidity in each channel. While the total number of on-chain wealth distributions |𝒲(C,n)| is known via the stars-and-bars formula, feasibility in the off-chain setting shrinks the set to W_G and depends on the network topology and capacities.

The authors note that they estimate |W_G| via Monte Carlo sampling because they lack a closed-form formula. Establishing such a formula would directly quantify feasibility and allow precise evaluation of how topology and capacities restrict wealth distributions.