摘要翻译：
我们研究了不相关随机网络的拉普拉斯算子，并作为应用考虑了网络上的跳跃过程（扩散、随机游动、信号传播等）。我们制定了一个严格的方法来解决这些问题。我们导出了一组精确封闭的积分方程组，它提供了拉普拉斯算子预解的平均值。这使我们能够描述一个信号的传播和网络上的随机游动。我们证明了该问题的决定参数是网络中顶点的最小度，而度分布的高次部分并不是必需的。Laplacian谱$\lambda_c$的下沿位置与配位数$q_m$的正则Bethe晶格中的位置相同。即$\lambda_c>0$，如果$q_m>2$，而$\lambda_c=0$，如果$q_m\leq2$。在这两种情况下，特征值$\rho(\lambda)\to0$的密度为$\lambda\to\lambda_c+0$,但在$\lambda_c$附近的极限行为有很大不同。根据与起始顶点的距离，跳跃传播子是一个稳定的高斯运动，随时间展宽。此图与正则Bethe晶格的图定性一致。我们的分析结果包括在$\lambda_c附近的谱密度$\rho(\lambda)$以及自相关子和传播子的长时间渐近性。
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英文标题：
《Laplacian spectra of complex networks and random walks on them: Are
  scale-free architectures really important?》
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作者：
A. N. Samukhin, S. N. Dorogovtsev, J. F. F. Mendes
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最新提交年份：
2007
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分类信息：

一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
  We study the Laplacian operator of an uncorrelated random network and, as an application, consider hopping processes (diffusion, random walks, signal propagation, etc.) on networks. We develop a strict approach to these problems. We derive an exact closed set of integral equations, which provide the averages of the Laplacian operator's resolvent. This enables us to describe the propagation of a signal and random walks on the network. We show that the determining parameter in this problem is the minimum degree $q_m$ of vertices in the network and that the high-degree part of the degree distribution is not that essential. The position of the lower edge of the Laplacian spectrum $\lambda_c$ appears to be the same as in the regular Bethe lattice with the coordination number $q_m$. Namely, $\lambda_c>0$ if $q_m>2$, and $\lambda_c=0$ if $q_m\leq2$. In both these cases the density of eigenvalues $\rho(\lambda)\to0$ as $\lambda\to\lambda_c+0$, but the limiting behaviors near $\lambda_c$ are very different. In terms of a distance from a starting vertex, the hopping propagator is a steady moving Gaussian, broadening with time. This picture qualitatively coincides with that for a regular Bethe lattice. Our analytical results include the spectral density $\rho(\lambda)$ near $\lambda_c$ and the long-time asymptotics of the autocorrelator and the propagator.
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PDF链接：
https://arxiv.org/pdf/706.1176