The discussion advances with a clarification that, indeed, if the wealth distribution is known, one can predict the location of the spanning tree by solving an integer linear program. However, the stability of the spanning tree when subjected to changes in wealth distribution due to payments, without modifying the network's topology or fee rates, remains uncertain. This question highlights an area of open inquiry, suggesting that while the initial wealth distribution significantly influences which channels may deplete, the overall structure of the spanning tree might still retain some level of stability under varying wealth distributions.

Further exploration into how wealth distribution affects channel depletion reveals that the location of the spanning tree is less stable than initially anticipated. This instability is illustrated by channels frequently not depleting in simulations, yet showing depletion in about half of all tested wealth distributions. This observation underscores a deviation from optimal fee potential states in the network following payment transactions, which alter wealth distribution and, consequently, the liquidity state of channels.

An interesting phenomenon is observed when considering networks as spanning trees. If nodes within a network with cycles and varying edge fees engage in reciprocal payments, it effectively contributes to the depletion of those cycles. An illustrative example demonstrates that even without source and sink nodes, all channels tend towards depletion while maintaining an overall equal distribution of coins among nodes. This contradicts the intuitive expectation of creating a "most expensive" spanning tree and highlights how global topology and payment flows dictate channel liquidity.

The discourse further delves into the mechanics of channel depletion through a detailed example where random one satoshi payments lead to uniform channel depletion. This example serves to elucidate the complex dynamics at play, including the influence of fee structures and liquidity shifts over time. It becomes evident that channel depletion occurs independently of individual node balances, showcasing the inherent properties of networks with cycles and variable costs.

In response to a proposed simulation aiming to model real-world financial behaviors and liquidity management, the discussion shifts toward the emergent phenomenon of channel depletion driven by fees and the presence of cycles in the network. Instead of adopting a mixed model simulation, the conversation suggests focusing on how node operators could adjust fees for better flow control and adapt route selection based on this understanding. Drawing parallels to natural phenomena and electrical grids, the conversation encapsulates a broader application of the mathematical models used to comprehend the emerging patterns within the Lightning Network, indicative of a profound intersection between theoretical constructs and practical network behavior.