Average homogeneous and non-homogeneous weighted receiving time in recursive weighted Koch networks
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摘要: 本文介绍用递归分割方法得到的实数系统上递归的齐次和非齐次的加权科赫网络模型，其主要是受机场网络和代谢网络的经验观测的启发.其中对于齐次模型，它依赖比例因子t ∈（0，1）；对非齐次的模型，我们通常取不同的比例因子t，s或t，r，s∈（0，1）.作为基本的动力学过程，我们研究递归的齐次与非齐次的加权科赫网络的随机行走，即每一步后都将一致移动到任意一个其位于边界$L_\ell, \ell=0,1,\cdots,m$，上的领域Γ（j）中.为了更方便研究齐次与非齐次模型，我们会再次用到递归分割法以及奇异值分解法来计算所有的节点与目标节点之间最长路径的平均加权（MWLP）的总和，其中目标节点是合并节点{pi：i=1，2，3}中的某个节点.最终，在庞大的网络中，平均的齐次与非齐次加权接收时间将关于网络秩序次线性.Abstract: In this paper, we introduce the recursive homogeneous weighted Koch network model for real systems with a scaling factor t ∈ (0, 1) and the non-homogeneous model with scaling factors t, s∈ (0, 1) or t, r, s ∈ (0, 1). These models were constructed using the recursive division method and motivated by experimental study of aviation networks and metabolic networks. As a process of fundamental dynamics, we study the recursive homogeneous and non-homogeneous weighted Koch networks with a random walk; for all steps, the walker who is starting from an existing node moves uniformly to one of its nearest neighbors Γ(j) lying on the layers $L_\ell, \ell=0,1,\cdots,m$. In order to study homogeneous and non-homogeneous models, the recursive division method and singular value decomposition were used to calculate the sum of the mean weighted longest paths (MWLP) for all nodes absorbed at the target node placed in one of the merging nodes {pi:i=1, 2, 3}. Finally, in a large network, the average weighted receiving time (AWRT) for homogeneous and nonhomogeneous models grows sub-linearly with the network's order.
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Fig. 1 (a) w represents three edges with weight w in a triangle; (b) G0 (i.e., G(0)) with w=1
Fig. 2 (a) The construction of G(1) as layers; (b) G(1) is regarded as the merging of H11 and G11
Fig. 3 (a) The construction of G(2) as layers; (b) G(2) is regarded as the merging of H21 and G21
Fig. 4 (a) The construction of G(n) as layers; (b) G(n) is regarded as the merging of Hni and Gni
Fig. 6 (a) The construction of $G(1)$ as layers; (b) $G(1)$ is regarded as the merging of $H_1^1$ and $G_1^1$
Fig. 7 (a) The construction of $G(2)$ as layers; (b) $G(2)$ is regarded as the merging of $H_2^1$ and $G_2^1$
Fig. 8 (a) The construction of $G(n)$ as layers; (b) $G(n)$ is regarded as the merging of $H_n^i$ and $G_n^i$
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