发表机构
Dipartimento di Fisica dell’Università di Roma ‘Tor Vergata’ e sezione INFN di Roma ‘Tor Vergata’; Università di Roma ‘Tor Vergata’; Dipartimento di Ingegneria Civile, Informatica e delle Tecnologie Aeronautiche, Università degli Studi ‘Roma Tre’; Università degli Studi ‘Roma Tre’; Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada; Universidad de Granada（罗马托尔维加塔大学物理系及罗马托尔维加塔INFN分部; 罗马托尔维加塔大学; 罗马第三大学土木、计算机与航空技术工程系; 罗马第三大学; 格拉纳达大学卡洛斯一世理论与计算物理研究所; 格拉纳达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探究网络的网络的Fiedler维数，解析计算不同基底与纤维组合的维数，发现小世界核心主导Fiedler维数，有限模块化系统的有效维数存在系统性偏差。
AI 中文摘要
模块化网络的弛豫是由其模块决定的，还是由连接这些模块的网络决定的？该答案确定了扩散、共识和同步的时间尺度，所有这些都由Fiedler特征值设定。我们表明，对于捆绑网络，两个层级依次起作用：随机游走者必须先从其模块中逃脱，然后再在连接它们的网络上扩散。平衡时间是这两个时间的总和，即从纤维内部到达基底所需的时间，以及基底的弛豫时间，该时间因纤维的质量而减慢。其结果出乎意料：无论模块多大，小世界核心始终会施加其自身的Fiedler维数；模块仅作为对数修正存在，该修正会减慢平衡速度，并在远超任何真实网络的规模下隐藏真实指数。因此，在有限模块化系统上测量的有效维数可能会出现系统性偏差。我们解析计算了有限维基底与小世界纤维、小世界基底与有限维纤维、有限维基底与有限维纤维、小世界基底与小世界纤维的所有组合，并将其与各类别代表的数值精确谱进行了比较。
英文摘要
Is the relaxation of a modular network dictated by its modules or by the network that connects them? The answer fixes the time scales of diffusion, consensus, and synchronization, all set by the Fiedler eigenvalue. We show that, for bundled networks, the two levels act one after the other: a random walker must first escape from its module and then spread over the network that connects them. The equilibration time is the sum of these two times, the time needed to reach the base from inside a fiber and the relaxation time of the base, slowed down by the mass of the fibers. The consequences are unexpected. However large the modules are, a small-world core always imposes its own Fiedler dimension; the modules survive only as a logarithmic correction, which slows down equilibration and hides the true exponent up to sizes far beyond any real network. Effective dimensions measured on finite modular systems can therefore be systematically biased. We analytically work out all combinations of finite-dimensional and small-world bases and fibers, and compare them with numerically exact spectra of representatives of each category.
Comments10 pages, 3 figures
