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首页> 外文会议>International workshop on complex networks and their applications >Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime
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Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime

机译：热力学条件下随机几何图的特征值和谱维

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Network geometries are typically characterized by having a finite spectral dimension (SD), d_s that characterizes the return time distribution of a random walk on a graph. The main purpose of this work is to determine the SD of a variety of random graphs called random geometric graphs (RGGs) in the thermodynamic regime, in which the average vertex degree is constant. The spectral dimension depends on the eigenvalue density (ED) of the RGG normalized Laplacian in the neighborhood of the minimum eigenvalues. In fact, the behavior of the ED in such a neighborhood characterizes the random walk. Therefore, we first provide an analytical approximation for the eigenvalues of the regularized normalized Laplacian matrix of RGGs in the thermodynamic regime. Then, we show that the smallest non zero eigenvalue converges to zero in the large graph limit. Based on the analytical expression of the eigenvalues, we show that the eigenvalue distribution in a neighborhood of the minimum value follows a power-law tail. Using this result, we find that the SD of RGGs is approximated by the space dimension d in the thermodynamic regime.
机译：网络几何形状的特征通常是具有有限的频谱尺寸（SD）d_s，该尺寸表征图表上随机游走的返回时间分布。这项工作的主要目的是确定在热力学条件下平均顶点度是恒定的各种随机图的SD，称为随机几何图（RGG）。光谱尺寸取决于最小特征值附近的RGG归一化拉普拉斯算子的特征值密度（ED）。实际上，ED在这种邻域中的行为是随机游走的特征。因此，我们首先为热力学条件下RGGs的正规化归一化Laplacian矩阵的特征值提供解析近似。然后，我们表明最小的非零特征值在大图限制中收敛到零。基于特征值的解析表达式，我们表明最小值附近的特征值分布遵循幂律尾部。利用这一结果，我们发现RGG的SD在热力学条件下近似为空间尺寸d。

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