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# 数据科学代写|复杂网络代写Complex Network代考|CS7280 Spectral gap and diameter

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## 数据科学代写|复杂网络代写Complex Network代考|Spectral gap and diameter

In addition to the mixing time, the eigenvalue $\lambda_2$ of the normalized Laplacian matrix of a graph of $N$ vertices enables one to estimate the diameter $d$ of the earlier graph,
$$d \leq \frac{\ln (N-1)}{\ln \left[\left(\lambda_N+\lambda_2\right) /\left(\lambda_N-\lambda_2\right)\right]}$$

(Chung, 1997). ${ }^{10}$ Equations (9.18), (9.23), and (9.25) assume the existence of the spectral gap (whose width is substantially determined by the relative size of a bottleneck in the graph, Eq. (2.84) in Section 2.9). If this is not the case, then often $\rho^{(L)}(\lambda) \sim \rho^{(\mathcal{L})}(\lambda) \sim \lambda^{D_s / 2-1}$ for small $\lambda$, where $D_s$ is the spectral dimension. In this situation, occurring, in particular, in finite-dimensional lattices, long-time relaxation in diffusion and random walk processes become power-law. In this case, for example, the probability to find a walker on its initial vertex decays as $\int d \lambda \rho^{(\widetilde{\mathcal{L}})}(\lambda) e^{-\lambda t} \sim t^{-D_s / 2}$. A finite spectral dimension and a power-law relaxation can happen even in small worlds as is allowed by the inequality in Eq. (2.81) relating $D_s$ and the Hausdorff dimension. For a few examples of this combination, namely $D_s<\infty$ while $D_H=\infty$, in simplicial complexes, see Bianconi and Dorogovstev (2020) and da Silva, Bianconi, da Costa, Dorogovtsev, and Mendes (2018).

## 数据科学代写|复杂网络代写Complex Network代考|Spectral densities

In finite random multiplex networks, the size of the largest mutually connected component fluctuates in different members of a statistical ensemble of multiplexes. Coghi, Radicchi, and Bianconi (2018) observed that in finite multiplexes the size of this component in a single realization can strongly deviate from the average over the entire ensemble. These deviations are particularly strong near $p_c$, the critical point of the corresponding infinite multiplex, and below it. Notably, the distribution of the relative size of the largest mutually connected component, $\mathcal{P}(S)$, has two peaks near the critical point and in some subcritical region. One of the peaks is at small $S$ and the second one is near the value of $S$ immediately after the hybrid phase transition in the corresponding infinite multiplex. This bimodal distribution indicates the mixture of two phases: normal phase and the phase with a ‘giant’ component typical for finite systems with a hybrid phase transition. Compare this with a one-peak distribution $\mathcal{P}(S)$ for a continuous transition (recall Figure 6.33). We refer the reader to Lee, Choi, Stippinger, Kertész, and Kahng (2016a) for theory and measurements of finite-size scaling of a ‘giant’ component in multiplex networks and in other problems with a hybrid phase transition.

Relation between multiplexes and general interdependent networks

Up to now we mostly focused on multiplex networks and that was not only for demonstration purposes. Already Gao, Buldyrev, Stanley, and Havlin (2012b) noticed that, with respect to a giant mutually connected component, a large class of networks of networks is equivalent to multiplexes, and hence the case of multiplexes is particularly important among interdependent networks. Moreover, even a wider class of interdependent networks was found to have the same sets of mutually connected components as in corresponding multiplexes (Bianconi, Dorogovtsev, and Mendes, 2015). Imagine a multiplex in the multilayer representation. Each vertex in the multiplex together with the replicas of this vertex in other layers form a fully connected subgraph- $M$-clique – linked together by interlinks between layers, and no interlinks connect vertices from different $M$-cliques. Let us remove some of the interlinks in such a way that each of these $M$-graphs remain connected. This removal does not change any of paths within layers and keeps intact the connectivity of each of the $M$-graphs. This guarantees that any network generated in this way has the same set of mutually connected components as the original multiplex. On the other hand, interdependent networks with vertices non-separable into sets of replicas can show some peculiarities unseen in usual multiplexes, for example, multiple hybrid transitions (Bianconi and Dorogovtsev, 2014).

## 数据科学代写|复杂网络代写Complex Network代考|Spectral gap and diameter

$$d \leq \frac{\ln (N-1)}{\ln \left[\left(\lambda_N+\lambda_2\right) /\left(\lambda_N-\lambda_2\right)\right]}$$
（钟，1997）。10等式 (9.18)、(9.23) 和 (9.25) 假设存在缯间隙（其宽度基本上由图中形颈的相对大小决定，见第 $2.9$ 节中的等 别是在有限维晶格中，扩散和随机游动过程中的长时间松她弯成昌律。在这种情况下，例如，在其初始页点上找到步行者的概率衰 (2.81) 有关 $D_s$ 和豪斯多夫维度。对于这种组合的几个例子，即 $D_s<\infty$ 尽管 $D_H=\infty$ ，在单纯复形中，参见 Bianconi 和 Dorogovstev (2020) 以及 da Silva、Bianconi、da Costa、Dorogovtsev 和 Mendes (2018)。

## 数据科学代写|复杂网络代写Complex Network代考|Spectral densities

6.33）。我们建议读者参考 Lee、Choi、Stippinger、Kertész 和 Kahng (2016a)，了解多重网絡中”巨型”组件的有限尺寸缩放 的理论和测量以及混合相变的其他问题。

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