Simulation and Evaluation¶
We evaluate BPE using a Python agent-based simulation with 50 sinks, 10 sources, and 3 task types over 1000 timesteps.
Setup¶
Each sink has heterogeneous true capacity drawn uniformly from \([10, 100]\) and stake from \([S_{\min}, 10 \cdot S_{\min}]\) with \(S_{\min} = 100\). Sources emit flow rates from \([50, 200]\). Capacity signals include Gaussian noise (\(\sigma = 0.05 \cdot C_{\text{true}}\)). The EWMA parameter is \(\alpha = 0.3\) unless otherwise noted. We compare three strategies: BPE (capacity-proportional), round-robin (equal split), and random (Dirichlet-weighted).
Allocation efficiency is defined as the fraction of total payment that is productive (i.e., absorbable by the receiving sink): \(\eta(t) = \sum_k \min(F_k(t), C_k) \;/\; \sum_k F_k(t)\).
E1: Convergence¶
Allocation efficiency and max queue backlog over 1000 timesteps. BPE achieves 95.7% average efficiency (last 100 steps) vs. 93.5% for round-robin and 79.7% for random.
BPE converges to near-optimal allocation within \(\sim\)50 timesteps as the EWMA smooths noisy capacity signals. Round-robin approximates efficiency only when sink capacities are similar; its fixed-split strategy cannot adapt to heterogeneous capacity. Random allocation shows high variance and consistently suboptimal routing.
E2: Shock Response¶
BPE response to 20% sink failure at \(t = 500\). Efficiency dips briefly then recovers as EWMA smoothing adapts to the new capacity landscape.
At \(t=500\), 20% of sinks are killed (capacity set to zero). BPE’s efficiency drops momentarily as the EWMA-smoothed signals lag the true capacity change, then recovers within \(\sim\!1/\alpha \approx 3\) epochs to a new equilibrium.
E3: Sybil Resistance¶
Sybil attack profitability. Net gain decreases as the number of fragments \(n\) grows, due to $\(\sqrt{\cdot}\)$ cap concavity and per-identity minimum stake cost.
We fix total attacker stake \(S\) and split into \(n \in \{1, 2, 5, 10, 20\}\) identities. Each identity pays \(S_{\min}\) overhead. Total capacity scales as \(\sqrt{n}\) while cost scales linearly. The attack becomes unprofitable at moderate \(n\), validating the theoretical analysis in Security Analysis.
E4: EWMA Parameter Sweep¶
Allocation efficiency for \(α ∈ {0.1, 0.2, 0.3, 0.5, 0.8}\). \(α = 0.3\) balances responsiveness and stability.
Low \(\alpha\) values (0.1, 0.2) yield stable but slowly-adapting signals. High \(\alpha\) (0.8) tracks capacity changes rapidly but amplifies noise, reducing average efficiency. \(\alpha = 0.3\) achieves the best efficiency–stability tradeoff.
E5: Buffer Dynamics¶
Source stall rate and allocation efficiency under different buffer sizes \(Bmax ∈ {0, 0.1, 0.5, 1.0} × Λ\).
With no buffer (\(B_{\max} = 0\)), sources stall frequently whenever instantaneous flow exceeds capacity. Moderate buffers (\(B_{\max} = 0.5 \Lambda\)) virtually eliminate stalls while maintaining high efficiency. Large buffers incur capital inefficiency with minimal marginal benefit, confirming the bounded buffer result of Theorem (Overflow Buffer Bound).
E6: On-Chain Gas Benchmarks¶
We deploy the full contract suite to Base Sepolia and measure gas costs for critical operations. All measurements are single-transaction costs at zero network congestion.