Potential diffusion processes of real-world systems are relevant to the underlying network structure and dynamical mechanisms. The vast majority of the existing work on spreading dynamics, in response to a large-scale network, is built on the condition of the infinite initial state, i.e., the extremely small seed size. The impact of an increasing seed size on the persistent diffusion has been less investigated. Based on classical epidemic models, this paper offers a framework for studying such impact through observing a crossover phenomenon in a two-diffusion-process dynamical system. The two diffusion processes are triggered by nodes with a high and low centrality, respectively. Specifically, given a finite initial state in networks with scale-free degree distributions, we demonstrate analytically and numerically that the diffusion process triggered by low centrality nodes pervades faster than that triggered by high centrality nodes from a certain point. The presence of the crossover phenomenon reveals that the dynamical process under the finite initial state is far more than the vertical scaling of the spreading curve under an infinite initial state. Further discussion emphasizes the persistent infection of individuals in epidemic dynamics as the essential reason rooted in the crossover, while the finite initial state is the catalyst directly leading to the emergence of this phenomenon. Our results provide valuable implications for studying the diversity of hidden dynamics on heterogeneous networks.

中文翻译：

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异构网络中流行病动力学下非线性交叉的出现

实际系统的潜在扩散过程与基础网络结构和动力学机制有关。响应于大规模网络，目前有关传播动力学的绝大多数工作都建立在无限初始状态（即极小的种子大小）的条件下。种子尺寸的增加对持续扩散的影响已被研究较少。基于经典的流行病模型，本文提供了一个框架，通过观察两扩散过程动力系统中的交叉现象来研究这种影响。这两个扩散过程分别由具有高和低中心性的节点触发。具体来说，假设网络中具有无标度分布的有限初始状态，我们从分析和数值上证明，从某个点来看，低中心性节点触发的扩散过程比高中心性节点触发的扩散过程更快。交叉现象的存在表明，有限初始状态下的动力学过程远大于无限初始状态下扩展曲线的垂直缩放。进一步的讨论强调了流行病学中个体的持续感染是根源于交叉的根本原因，而有限的初始状态是直接导致这种现象出现的催化剂。我们的结果为研究异构网络上隐藏动态的多样性提供了宝贵的启示。交叉现象的存在表明，有限初始状态下的动力学过程远大于无限初始状态下扩展曲线的垂直缩放。进一步的讨论强调了流行病学中个体的持续感染是根源于交叉的根本原因，而有限的初始状态是直接导致这种现象出现的催化剂。我们的结果为研究异构网络上隐藏动态的多样性提供了宝贵的启示。交叉现象的存在表明，有限初始状态下的动力学过程远大于无限初始状态下扩展曲线的垂直缩放。进一步的讨论强调了流行病学中个体的持续感染是根源于交叉的根本原因，而有限的初始状态是直接导致这种现象出现的催化剂。我们的结果为研究异构网络上隐藏动态的多样性提供了宝贵的启示。进一步的讨论强调了流行病学中个体的持续感染是根源于交叉的根本原因，而有限的初始状态是直接导致这种现象出现的催化剂。我们的结果为研究异构网络上隐藏动态的多样性提供了宝贵的启示。进一步的讨论强调了流行病学中个体的持续感染是根源于交叉的根本原因，而有限的初始状态是直接导致这种现象出现的催化剂。我们的结果为研究异构网络上隐藏动态的多样性提供了宝贵的启示。