Murphy Charles, Thibeault Vincent, Allard Antoine, Desrosiers Patrick
Département de physique, de génie physique et d'optique, Université Laval, Québec, QC, G1V 0A6, Canada.
Centre interdisciplinaire en modélisation mathématique, Université Laval, Québec, QC, G1V 0A6, Canada.
Nat Commun. 2024 May 25;15(1):4478. doi: 10.1038/s41467-024-48020-x.
Predicting the evolution of a large system of units using its structure of interaction is a fundamental problem in complex system theory. And so is the problem of reconstructing the structure of interaction from temporal observations. Here, we find an intricate relationship between predictability and reconstructability using an information-theoretical point of view. We use the mutual information between a random graph and a stochastic process evolving on this random graph to quantify their codependence. Then, we show how the uncertainty coefficients, which are intimately related to that mutual information, quantify our ability to reconstruct a graph from an observed time series, and our ability to predict the evolution of a process from the structure of its interactions. We provide analytical calculations of the uncertainty coefficients for many different systems, including continuous deterministic systems, and describe a numerical procedure when exact calculations are intractable. Interestingly, we find that predictability and reconstructability, even though closely connected by the mutual information, can behave differently, even in a dual manner. We prove how such duality universally emerges when changing the number of steps in the process. Finally, we provide evidence that predictability-reconstruction dualities may exist in dynamical processes on real networks close to criticality.
利用单元间的相互作用结构预测大型系统的演化是复杂系统理论中的一个基本问题。从时间观测中重构相互作用结构的问题也是如此。在此，我们从信息论的角度发现了可预测性与可重构性之间的复杂关系。我们使用随机图与在该随机图上演化的随机过程之间的互信息来量化它们的相互依赖性。然后，我们展示了与该互信息密切相关的不确定性系数如何量化我们从观测时间序列重构图的能力，以及我们从相互作用结构预测过程演化的能力。我们给出了许多不同系统（包括连续确定性系统）不确定性系数的解析计算，并描述了在精确计算难以处理时的数值方法。有趣的是，我们发现可预测性和可重构性尽管通过互信息紧密相连，但甚至可能以对偶的方式表现出不同的行为。我们证明了在改变过程中的步数时这种对偶性是如何普遍出现的。最后，我们提供证据表明在接近临界状态的真实网络上的动态过程中可能存在可预测性 - 重构对偶性。
