Znaidi Mohamed Ridha, Sia Jayson, Ronquist Scott, Rajapakse Indika, Jonckheere Edmond, Bogdan Paul
Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA, 90089, USA.
Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, 48109, USA.
Sci Rep. 2023 Oct 20;13(1):17948. doi: 10.1038/s41598-023-44791-3.
Deciphering the non-trivial interactions and mechanisms driving the evolution of time-varying complex networks (TVCNs) plays a crucial role in designing optimal control strategies for such networks or enhancing their causal predictive capabilities. In this paper, we advance the science of TVCNs by providing a mathematical framework through which we can gauge how local changes within a complex weighted network affect its global properties. More precisely, we focus on unraveling unknown geometric properties of a network and determine its implications on detecting phase transitions within the dynamics of a TVCN. In this vein, we aim at elaborating a novel and unified approach that can be used to depict the relationship between local interactions in a complex network and its global kinetics. We propose a geometric-inspired framework to characterize the network's state and detect a phase transition between different states, to infer the TVCN's dynamics. A phase of a TVCN is determined by its Forman-Ricci curvature property. Numerical experiments show the usefulness of the proposed curvature formalism to detect the transition between phases within artificially generated networks. Furthermore, we demonstrate the effectiveness of the proposed framework in identifying the phase transition phenomena governing the training and learning processes of artificial neural networks. Moreover, we exploit this approach to investigate the phase transition phenomena in cellular re-programming by interpreting the dynamics of Hi-C matrices as TVCNs and observing singularity trends in the curvature network entropy. Finally, we demonstrate that this curvature formalism can detect a political change. Specifically, our framework can be applied to the US Senate data to detect a political change in the United States of America after the 1994 election, as discussed by political scientists.
破译驱动时变复杂网络（TVCNs）演化的非平凡相互作用和机制，对于为此类网络设计最优控制策略或增强其因果预测能力起着至关重要的作用。在本文中，我们通过提供一个数学框架来推进TVCNs科学，通过该框架我们可以衡量复杂加权网络内的局部变化如何影响其全局属性。更确切地说，我们专注于揭示网络未知的几何属性，并确定其对检测TVCNs动态过程中相变的影响。在此背景下，我们旨在阐述一种新颖且统一的方法，可用于描述复杂网络中局部相互作用与其全局动力学之间的关系。我们提出一个受几何启发的框架来表征网络状态并检测不同状态之间的相变，以推断TVCNs的动力学。TVCNs的一个相由其福尔曼 - 里奇曲率属性决定。数值实验表明，所提出的曲率形式主义对于检测人工生成网络内的相转变是有用的。此外，我们证明了所提出的框架在识别支配人工神经网络训练和学习过程的相变现象方面的有效性。此外，我们利用这种方法来研究细胞重编程中的相变现象，将Hi-C矩阵的动力学解释为TVCNs，并观察曲率网络熵中的奇点趋势。最后，我们证明这种曲率形式主义可以检测政治变化。具体而言，我们的框架可以应用于美国参议院数据，以检测1994年选举后美国的政治变化，正如政治科学家所讨论的那样。
