Stochastic modeling of Structure and Dynamics in the Bitcoin Lightning Network

Abstract

The Bitcoin Lightning Network (LN) has emerged as a prominent Layer-2 solution de- signed to address the scalability limitations of the Bitcoin blockchain. However, as the network grows, understanding both its structural evolution and the reliability of its payment-routing mechanisms becomes increasingly important. This thesis investigates these two fundamental aspects through stochastic modeling and empirical analysis. First, we analyze the topological evolution of the Lightning Network. Empirical evi- dence reveals a persistent negative degree assortativity (disassortativity), a feature that classical generative models, such as the Barabási-Albert model, fail to reproduce asymp- totically. We introduce dynamic random graph models that extend preferential attach- ment by allowing edges to disappear over time at rates depending on node degree or channel capacity. We show that edge disappearance alone is sufficient to induce the disassortative mixing observed. Second, we address the reliability of payment routing in capacity-constrained networks inspired by the Lightning Network. We model the balance evolution of payment channels as a stochastic process governed by repeated routing of payments over shortest paths. By analyzing this process on both complete and general graphs, we derive upper and lower bounds for the time until the first payment failure occurs due to liquidity depletion. We establish that this failure time is governed by the ratio between the squared capacity of an edge and its betweenness centrality k2/g(e). Taken together, the results of this thesis provide principled insights into how decentral- ized payment networks evolve structurally in terms of topology and balance distributions