双分形网络上的随机游走
Random walks on bifractal networks
AI总结：
本文研究双分形无标度网络上的随机游走，发现游走维数不受局部结构差异影响，而谱维数因起始节点不同取两个值，并给出解析表达式与数值验证。
摘要：
最近的研究表明，具有无标度和分形性质的网络可能表现出双分形特性，其中局部结构由两个不同的分形维数描述。在本研究中，我们通过考察游走维数 $d_{\ ext{w}}$ 和谱维数 $d_{\ ext{s}}$，研究此类分形无标度网络（FSFN）上的随机游走，以理解双分形性如何影响其动力学性质。研究发现，游走维数不受 FSFN 局部分形性差异的影响，无论随机游走从哪个节点出发都保持恒定；而谱维数则取两个值，$d_{\ ext{s}}^{\ ext{min}}$ 和 $d_{\ ext{s}}^{\ ext{max}}(> d_{\ ext{s}}^{\ ext{min}})$，具体取决于起始节点。维数 $d_{\ ext{s}}^{\ ext{min}}$ 刻画了热力学极限下从无限度中心节点出发的随机游走者的返回概率，而 $d_{\ ext{s}}^{\ ext{max}}$ 描述的是从有限度非中心节点出发、且该节点与中心节点无限远时的随机游走者返回概率，并等于全局谱维数 $D_{\ ext{s}}$。两个局部谱维数的存在是 FSFN 双分形性的直接结果。此外，针对由生成器模型形成的 FSFN 以及临界无标度随机图的巨分量，本文给出了 $d_{\ ext{w}}$、$d_{\ ext{s}}^{\ ext{min}}$ 和 $d_{\ ext{s}}^{\ ext{max}}$ 的解析表达式，并通过数值计算加以验证。
英文摘要：
It has recently been shown that networks possessing scale-free and fractal properties may exhibit a bifractal nature, in which local structures are described by two different fractal dimensions. In this study, we investigate random walks on such fractal scale-free networks (FSFNs) by examining the walk dimension $d_{\text{w}}$ and the spectral dimension $d_{\text{s}}$, to understand how the bifractality affects their dynamical properties. The walk dimension is found to be unaffected by the difference in local fractality of an FSFN and remains constant regardless of the starting node of a random walk, whereas the spectral dimension takes two values, $d_{\text{s}}^{\text{min}}$ and $d_{\text{s}}^{\text{max}}(> d_{\text{s}}^{\text{min}})$, depending on the starting node. The dimension $d_{\text{s}}^{\text{min}}$ characterizes the return probability of a random walker starting from an infinite-degree hub node in the thermodynamic limit, while $d_{\text{s}}^{\text{max}}$ describes that of a random walker starting from a finite-degree non-hub node infinitely distant from hub nodes and is equal to the global spectral dimension $D_{\text{s}}$. The existence of two local spectral dimensions is a direct consequence of the bifractality of the FSFN. Furthermore, analytical expressions of $d_{\text{w}}$, $d_{\text{s}}^{\text{min}}$, and $d_{\text{s}}^{\text{max}}$ are presented for FSFNs formed by the generator model and the giant components of critical scale-free random graphs, and are numerically confirmed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
