连续信用网络和第二层区块链：单调性和采样
Continuous Credit Networks and Layer 2 Blockchains: Monotonicity and Sampling
摘要：
为了提高交易率，许多加密货币都实现了所谓的“第2层”交易协议，其中支付通过私人支付渠道的网络进行路由。然而，对于一个给定的交易，并不是每个网络状态都提供了一个可行的路径来执行支付;在这种情况下，交易必须放在公共账本上。因此，支付渠道网络使整个系统的交易率成倍增长;支付渠道网络故障频率越低，乘数就越高。我们建立在信贷网络的早期工作，并表明，这个网络的流动性问题是连接到组合的图形拟阵。早期的工作只能分析交易具有离散大小的（非自然）场景。从表面上看，连续的情况似乎更难检验。然而，去除这个假设让我们在两个重要的方向上取得进展。首先，我们给出了一个部分的答案“单调性的悖论”，以前的工作留下开放。这个猜想要求网络的性能不会随着任何边缘上的容量增加而降低。其次，我们在这里构造了一个网络状态采样过程，其渐近性能比现成的马尔可夫链（$O(\vert E\vert β(\vert E\vert))$，其中$β(x)$是求解$x$约束下线性规划的复杂度）快得多。我们得到我们的结果，映射到凸体的基础图形，然后显示的流动性和采样问题减少到绑定和计算这些机构的体积。转换依赖于关键的组合特性的基础图形拟阵，做单调性和快速采样的证明。
英文摘要：
To improve transaction rates, many cryptocurrencies have implemented so-called ''Layer-2'' transaction protocols, where payments are routed across networks of private payment channels. However, for a given transaction, not every network state provides a feasible route to perform the payment; in this case, the transaction must be put on the public ledger. The payment channel network thus multiplies the transaction rate of the overall system; the less frequently it fails, the higher the multiplier. We build on earlier work on credit networks and show that this network liquidity problem is connected to the combinatorics of graphical matroids. Earlier work could only analyze the (unnatural) scenario where transactions had discrete sizes. Superficially, it might seem like the continuous case would be harder to examine. However, removing this assumption lets us make progress in two important directions. First, we give a partial answer to the ``monotonicity conjecture'' that previous work left open. This conjecture asks that the network's performance not degrade as capacity on any edge increases. And second, we construct here a network state sampling procedure with much faster asymptotic performance than off-the-shelf Markov chains ($O(\vert E\vert β(\vert E\vert))$, where $β(x)$ is the complexity of solving a linear program on $x$ constraints.) We obtain our results by mapping the underlying graphs to convex bodies and then showing that the liquidity and sampling problems reduce to bounding and computing the volumes of these bodies. The transformation relies crucially on the combinatorial properties of the underlying graphic matroid, as do the proofs of monotonicity and fast sampling.
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