复杂网络上的间接渗流与相变

复杂网络是一种描述自然和社会系统各种结构和功能的模型，现如今它已经成为涵盖数学，物理，生物，计算机，社会科学等多个领域的交叉方向。渗流理论起源于统计物理，几十年来数学家和物理学家对网格上的渗流现象做了大量研究。随着20世纪末网络科学的兴起，渗流理论被广泛运用于复杂网络研究，诸如网络鲁棒性，疾病传播，演化博弈等问题，现已成为研究复杂网络的一种重要手段。长期以来，学者们提出很多种渗流模型来模拟现实世界中可能的各种网络演化和动力学过程，然而绝大多数渗流模型仅仅考虑顶点自身或者顶点邻居的直接作用。事实上，来自自身和邻居以外的间接作用对网络的影响不容忽视，而对网络上间接作用的研究尚且不多。

本文研究了一般的间接渗流模型，考虑了顶点初始时随机活跃的比例，从网络的度分布出发，分别在有向网络和无向网络上，建立了间接渗流的自洽方程，给出了求解网络上最大活跃连通分支的规模的公式，并且在ER随机网络上进行模拟，得到了与理论的数值解相一致的模拟结果，进一步还分析了间接渗流的相变情况，发现随着控制参数的变化，相变会在连续相变，混合相变，跳跃相变三种情况中变化，而且在有向和无向网络中，间接渗流体现出几乎相同的特征。

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