This investigation, grounded in a mathematical theory, provides insight into how economic rationality and network topology contribute to this issue. The foundational document, titled "A Mathematical Theory of Payment Channel Networks," available at GitHub, lays the groundwork for understanding these dynamics.

Subsequent research has led to the development of two notebooks that further delve into the intricacies of liquidity state estimation and fee optimization within the LN. The first notebook, accessible here, demonstrates a strong correlation between the network's circuit rank and the number of depleted channels, suggesting a predictable pattern driven by efforts to maximize the network's fee potential. The concept of fee potential is crucial here, as it represents the sum of expected earnings from routing payments through a node's channels, hinting at node operators' motivations to optimize this metric.

The second notebook, found here, carries out simulations to investigate if nodes' default behaviors lead to an optimized fee potential state similar to solving the problem globally. The findings suggest that through regular transactions, the network naturally gravitates towards a state where the fee potential is nearly maximized, and the number of depleted channels aligns with the network's circuit rank.

These studies underscore the predictive power of utilizing integer linear programming to forecast the network's liquidity distribution based on publicly available data, such as gossip. This approach highlights potential privacy concerns as it reveals how much can be inferred about network states from external observations.

However, several limitations and open questions remain. These include the influence of unknown wealth distributions, static assumptions about network topology and fees, and the use of simplified payment models in simulations. Despite these challenges, the research concludes that less probing may be needed to estimate liquidity locations, a spanning tree of non-depleted channels likely persists, resembling a hub-and-spoke model, and the direction of liquidity drain in channels is influenced by the broader network context rather than merely the balance of payments.

Looking forward, the discussion opens up avenues for developing strategies for node operators to manage their liquidity more effectively, considering fee adjustments. It also raises the question of whether maintaining more channels than a basic spanning tree, which incurs additional costs, offers sufficient value in terms of redundancy versus the risk of depletion. Additionally, the stability of the spanning tree amidst changing wealth distributions and the potential for leveraging this knowledge to predict liquidity locations for optimized payment routing are identified as areas for further exploration. Finally, the need for more accurate datasets to validate these predictions against actual network behavior is highlighted, signaling a path for future research endeavors.