Employing the Structural Power to Achieve Supply-Demand Balanced Payment Channel Networks

Shuyao Xiao; Shengling Wang; Hongwei Shi; Weicheng Wang; Anlin Chen

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Employing the Structural Power to Achieve Supply-Demand Balanced Payment Channel Networks

Shuyao Xiao, Shengling Wang, Hongwei Shi, Weicheng Wang, Anlin Chen

arXiv:2609.03600v1Sep 3, 2026cs.NI

TL;DR

PCNs improve blockchain transaction efficiency through off-chain transfers, but balance shortages remain a challenge because supply must support transaction demands. The paper links balance deficits to topology, introduces PTE and MaxPTE for topology-level reconfiguration, and reports improved balance efficiency without prior demand prediction.

Problem
PCNs address blockchain scalability through off-chain transactions, yet balance shortages remain a critical challenge as network topology affects balance deficits.
Method
The paper introduces payment topological entropy (PTE) to characterize structural liquidity patterns and uses MaxPTE to reorganize channel topology and existing balances.
Results
MaxPTE reduces balance deficits by 27.25%, improves average maximum flow by 12.23%, and decreases transaction failure probability by 25.96% compared with baseline methods.
Takeaways & Limitations
Topology-level reconfiguration can improve supply-demand alignment and liquidity utilization without increasing total balance or relying on prior demand prediction.

Abstract

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Blockchain technology faces scalability challenges because transactions must be validated and recorded across the network. Payment channel networks (PCNs) improve efficiency by moving transactions off-chain and recording only critical interactions on the mainnet. However, PCNs require pre-deposited channel balances (supply) to match transaction demands (demand), and insufficient balances cause supply shortages. Existing approaches, including transaction path optimization and channel balance allocation, address this problem but incur high costs due to dynamic adaptation to fluctuating demands. We reveal a correlation between balance deficits in PCNs and network topology by uncovering the structural organization of liquidity allocation. This enables supply-demand conflicts to be mitigated from a static topological perspective through channel reconfiguration, without additional balance replenishment. We further introduce payment topological entropy (PTE), an information-theoretic metric that quantifies each node's deviation from the global average connection pattern and captures structural properties of balance supply. Based on PTE, we design MaxPTE, a topology optimization algorithm that reorganizes balance allocation across channels through structural reconfiguration, aligning static balance distribution with dynamic transaction demand. Extensive experiments show that MaxPTE reduces balance deficits by 27.25%, increases average maximum flow by 12.23%, and decreases transaction failure probability by 25.96%, outperforming existing benchmarks. The method also remains robust across diverse balance-demand distributions, improving supply-demand balance without prior demand prediction.

I. INTRODUCTION

Payment channel networks improve blockchain scalability through off-chain transfers, but statically locked channel balances can mismatch dynamic payment demand. The paper investigates topology as a structural lever for reducing deficits and proposes PTE-guided channel reconfiguration.

Motivation: PCNs reduce on-chain load by executing frequent transactions off-chain and submitting only critical information to the mainnet.Channels are established with smart contracts, while mainnet updates occur at creation, settlement, or disputes.
Problem: Static channel locking makes each channel an independent supply pool, so excess demand can cause failures despite liquidity elsewhere.Balances cannot move across channels or be expanded after establishment.
Existing approaches: Existing path optimization and balance planning address shortages through demand adaptation or balance redistribution, which may incur adjustment overhead in dynamic environments.The paper characterizes these approaches as costly and temporary when demands fluctuate.
Empirical motivation: Different PCN topologies produce significantly different balance deficits under the same demand, indicating topology affects balance shortages.The experiments were averaged over 100 trials in the two-network comparison and extended to additional three-node patterns.
Proposed approach: PTE measures node-level deviation from the global connection pattern, while MaxPTE reorganizes channels to align static balance allocation with dynamic demand.The proposed mechanism uses topology reconfiguration rather than additional balance replenishment.

II. PAYMENT TOPOLOGICAL ENTROPY

Payment topological entropy quantifies how PCN nodes’ local balance-allocation patterns differ from a network-wide incoming pattern. It is constructed from normalized outgoing balances and total variation distance.

Definition: PTE is defined as the average structural deviation of all PCN nodes.The deviation is measured using total variation distance.
Node representation: Each node’s normalized outgoing distribution records the proportions of its reserved balance allocated across outgoing channels.For node i, w_ij equals c_ij divided by the node’s total reserved outgoing balance.
Network reference: The network-level incoming distribution is obtained by aggregating the normalized outgoing balance distributions of all nodes.It represents the average proportion of normalized outgoing balance received by each node.
Distance measure: TVD between a node’s outgoing distribution and the network incoming distribution measures that node’s structural deviation from the global balance-circulation pattern.Larger values indicate more distinctive local allocation patterns, while smaller values indicate closer conformity.

B. Structural Interpretation of PTE.

PCN liquidity efficiency depends on how balances are distributed across independently locked channels, not only on total balance. PTE interprets differentiated node roles as a more organized balance-supply structure.

Structural liquidity: PCNs with equal total balance can have different deficits because channel-level locking prevents liquidity from being freely reallocated.Balances assigned to different outgoing channels form independent liquidity pools after establishment.
Interpretation: W_out_i represents node i’s local outgoing allocation, while W_in represents the network-level balance-reception pattern.Their comparison characterizes how an individual node’s liquidity role differs from overall circulation.
PTE meaning: Low PTE indicates redundant, similar allocation patterns, whereas high PTE indicates differentiated and complementary liquidity roles.Differentiated roles can reduce redundant allocation and form more effective balance-flow corridors.
PTE computation: In the toy example, node A’s two 60-coin outgoing balances normalize to weights of 0.5 each.The normalized transition matrix records outgoing-balance proportions, followed by column-wise aggregation into the network incoming distribution.

C. Connection between PTE and Topology Reconfiguration

PTE evaluates how liquidity is structurally differentiated across outgoing channels, while topology reconfiguration improves this organization by consolidating redundant channels without changing total liquidity. MaxPTE selects feasible consolidations that increase PTE while preserving network accessibility.

Consolidating similar outgoing channels preserves total balance while reorganizing liquidity toward more distinctive payment directions.The operation changes the normalized outgoing distribution while preserving each node’s total liquidity supply.
PTE-guided consolidation identifies redundant channel configurations without requiring prior transaction-demand knowledge.Among feasible channel pairs, MaxPTE favors consolidations that increase PTE.
MaxPTE constrains consolidation by strong connectivity and bounded channel modifications to preserve payment-network reachability.These constraints balance structural differentiation against the loss of available payment paths.

III. PCN TOPOLOGY OPTIMIZATION ALGORITHM

The paper formulates MaxPTE as a topology optimization procedure that maximizes PTE while preserving connectivity and reorganizing existing liquidity. It iteratively tests channel merges, accepts PTE-improving feasible changes, and operates with polynomial complexity as a periodic maintenance mechanism.

III. PCN TOPOLOGY OPTIMIZATION ALGORITHM: MaxPTE consolidates structurally redundant outgoing channels and reallocates their pre-deposited balances without increasing each node’s total balance.PTE evaluates the resulting structural differentiation and selects the consolidation that most improves existing balance organization.
III. PCN TOPOLOGY OPTIMIZATION ALGORITHM: Larger PTE values tend to correspond to smaller balance deficits under identical balance and demand settings.This motivates PTE as a structural criterion for topology reconfiguration.
III. PCN TOPOLOGY OPTIMIZATION ALGORITHM: MaxPTE maximizes PTE subject to strong connectivity and bounded channel reduction.The iteration parameter k controls the trade-off between structural liquidity differentiation and transaction-path availability.
III. PCN TOPOLOGY OPTIMIZATION ALGORITHM: For each node, the algorithm evaluates outgoing-channel pairs, retains the best PTE-increasing strongly connected merge, and updates the graph incrementally.It iterates for k rounds and can remove at most one selected channel per node in each round.
III. PCN TOPOLOGY OPTIMIZATION ALGORITHM: In the one-iteration example, MaxPTE raises PTE from 0.50 to 0.67 while consolidating node B’s channel balance and preserving total liquidity.The channel toward node A is consolidated into the channel toward node C.
III. PCN TOPOLOGY OPTIMIZATION ALGORITHM: Theorem 1 gives MaxPTE time complexity as O(kn∆2|E|).Here, k is the maximum number of channel reductions per node, ∆ is maximum outgoing degree, and |E| is the number of channels.
III. PCN TOPOLOGY OPTIMIZATION ALGORITHM: MaxPTE is intended for periodic topology maintenance rather than per-transaction routing, and sparse PCNs make local neighborhood structure its main practical influence.The paper reports an average node degree of approximately 7.43 and describes execution during channel establishment or topology adjustment.

IV. EXPERIMENTAL EVALUATION

The evaluation assesses PCN liquidity efficiency and network performance using balance deficit, average maximum flow, and transaction failure probability.

IV. EXPERIMENTAL EVALUATION: The experiments use balance deficit, average maximum flow, and transaction failure probability as evaluation metrics.These metrics are used to assess liquidity efficiency and network performance in PCNs.

A. Necessity Analysis

Fragmented channel balances form independent pools that cannot be directly shared, so aggregate liquidity may still produce local shortages. Consolidation aggregates these balances without changing total supply and makes liquidity available across a wider range of demands.

A. Necessity Analysis: Independent channel pools cannot directly share balances, allowing local shortages despite sufficient aggregate liquidity.This follows from the channel-level locking mechanism.
A. Necessity Analysis: Consolidating n channels with balance µ produces one channel with capacity n × µ while preserving total liquidity supply.The aggregated channel is less fragmented and can serve a wider range of transaction demands.
A. Necessity Analysis: Figure 5 contrasts n independent channels before consolidation with the aggregated channel after consolidation.The diagram illustrates the change from fragmented to combined liquidity allocation.

B. Balance deficit

The evaluation tests MaxPTE-based channel allocation under heterogeneous demands and finds lower balance deficits than original and baseline networks across demand models.

Evaluation setup: The evaluation constructs 10 Lightning Network topologies with 200 nodes and about 750 edges, assigning each node 100 satoshis across payment channels.Baseline comparisons include Random, MaxOut, MinOut, MaxBetweenness, and MaxClustering channel-merging strategies.
Metric and demand models: Balance deficit averages demand minus deposited balance, with overall deficit averaged across all node pairs.The evaluation scales generated demand matrices using a global Demand factor to simulate different traffic intensities.
Demand distributions: Four representative demand distributions—Poisson, Uniform, Power-law, and Gaussian—test robustness across spontaneous, balanced, hub-dominated, and locality-aware traffic patterns.Power-law demands combine connectivity-rank weighting with uniform components, while Gaussian demands concentrate around three highly connected nodes.
Results: 27.25%: MaxPTE reduces average balance deficits compared with the original network across all demand models.The improvement is especially significant under power-law and Gaussian distributions, where balance allocation and connectivity are uneven.
Results: MaxPTE consistently produces lower average balance deficits than both the original and other baseline networks as transaction demand increases.All compared networks preserve the same total number of channels, with k=1 fixed for MaxPTE in this evaluation.

C. Average maximum flow

Average maximum flow measures the mean maximum transferable balance across directed node pairs. MaxPTE increases this network-level flow relative to the original topology and outperforms other channel-merging strategies.

Metric: Average maximum flow is the average of maximum flows between all directed node pairs in the PCN.The directed pair count is n(n −1), where n is the total number of nodes.
Results: 12.23%: MaxPTE improves average maximum flow compared with the original network.Box plots report values under different channel-merging algorithms, with the red center line showing the median across experiments.
Results: MaxPTE significantly outperforms other channel-merging strategies on average maximum flow.The reported increase in maximum flow is associated in the evaluation with greater balance mobility and circulation.

D. Proportion of failure transactions

Transaction failure probability measures the share of requested transfers whose required balance exceeds available maximum flow. MaxPTE achieves the lowest average failure probability among evaluated topologies.

Metric definition: Pfail(Ω) is the fraction of transaction requests that cannot succeed at balance demand Ω.A transaction fails when maximum available flow from source s to destination d is below Ω.
Metric interpretation: Higher Pfail(Ω) indicates that fewer node pairs can forward transactions requiring at least Ω satoshis.This corresponds to a stronger supply–demand imbalance in the network.
Results: 25.96%: MaxPTE reduces the mean transaction failure rate compared with the original topology.Across 10 network topologies, MaxPTE has the lowest average failure probability and remains strongest under high balance demands.

E. Discussion

PCN solutions include routing optimization and channel balance management, but their effectiveness depends on demand prediction and frequent adaptation. MaxPTE instead reorganizes liquidity structurally through topology reconfiguration, improving supply-demand alignment while complementing existing approaches.

PCN solutions are generally categorized into transaction path optimization and channel balance management.
Transaction path optimization: Routing methods optimize transaction paths by selecting or constructing routes with sufficient capacity, performance, or reliability.
Channel balance management: Balance planning methods rebalance or allocate channel balances to reduce shortages across the network.
Limitations of existing methods: Existing methods depend substantially on demand prediction and may require frequent updates as transaction demands and channel conditions change.
MaxPTE: MaxPTE optimizes structural liquidity organization through topology reconfiguration rather than adjusting routing decisions or balance allocations, and remains complementary to those approaches.