The “invisible hand” of the free market is remarkably effective at producing near-equilibrium prices. This is difficult to quantify, however, in the absence of an agreed model for out-of-equilibrium trade. Short of a fully reductionist model, a useful substitute would be a scaling law relating equilibration time and other market parameters. Even this, however, is missing in the literature.

We make progress in this direction. We examine a class of Arrow–Debreu markets with price signaling driven by continuous-time proportional-tâtonnement. We show that the connectivity among the participants in the market determines quite accurately a scaling law for convergence time of the market to equilibrium, and thus determines the effectiveness of the price signaling. To our knowledge this is the first characterization of price stability in terms of market connectivity. At a technical level, we show how convergence in our class of markets is determined by a market-dependent Laplacian matrix.

If a market is not isolated but, rather, subject to external noise, equilibrium theory has predictive value only to the extent to which that noise is counterbalanced by the price equilibration process. Our model quantifies this predictive value by providing a scaling law that relates the connectivity of the market with the variance of its prices.

自由市场的“看不见的手”在产生接近均衡的价格方面非常有效。然而，在没有一个商定的失衡贸易模型的情况下，这很难量化。如果没有完全还原的模型，一个有用的替代方法是与平衡时间和其他市场参数相关的比例法则。然而，即使是这一点，在文献中也是缺失的。

我们在这个方向上取得进展。我们研究了一类具有由连续时间比例调整驱动的价格信号的 Arrow-Debreu 市场。我们表明，市场参与者之间的连通性非常准确地确定了市场收敛时间的比例法则，从而确定了价格信号的有效性。据我们所知，这是市场连通性方面价格稳定性的第一个特征。在技​​术层面，我们展示了我们的市场类别中的收敛是如何由依赖于市场的拉普拉斯矩阵决定的。

如果市场不是孤立的，而是受到外部噪音的影响，那么均衡理论只有在噪音被价格均衡过程抵消的程度上才具有预测价值。我们的模型通过提供将市场的连通性与其价格的方差相关联的比例定律来量化该预测值。