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# 数据科学代写|复杂网络代写Complex Network代考|TSKS33 Overlapping edges

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## 数据科学代写|复杂网络代写Complex Network代考|Overlapping edges

The locally tree-like interdependent networks allow an easy analytical treatment, however neglecting correlations between layers, typical in real network. One simple type of such correlations, overlapping edges from different layers, still does not spoil the local tree-likeness and so it is treatable analytically (Cellai, López, Zhou, Gleeson, and Bianconi, 2013b; Hu, Zhou, Zhang, Han, Rozenblat, and Havlin, 2013). Let us, for instance, consider a 2-layer multiplex with some of edges in one layer overlapping with the edges in the other layer. In other words, two edges from different layers, can have the same pair of end vertices. We can treat the overlapping edges as a special type of edges in a such a multiplex. Hu, et al. noticed that the clusters of overlapping edges can be treated as supernodes connected with each other by non-overlapping edges. Therefore the presence of any concentration of overlapping edges, smaller than 1, does not change qualitatively the hybrid phase transition associated with the emergence of the giant mutually connected component, as was confirmed by Min, Lee, Lee, and Goh (2015). In a multiplex of this kind, each vertex has three degrees, $q_a, q_b$, and $\tilde{q}$, being, respectively, the number of connections only in layer $a$, only in layer $b$, and the number of overlapping connections. If other correlations are absent, then the multiplex is defined by the joint degree distribution $P\left(q_a, q_b, \tilde{q}\right)$ and it is locally tree-like. Figure $8.10$ shows the phase diagram for a symmetric 2-layer multiplex with the joint degree distribution $P\left(q_a, q_b, \tilde{q}\right)=P\left(q_a\right) P\left(q_b\right) \widetilde{P}(\tilde{q})$, where $P(q)$ and $\widetilde{P}(\tilde{q})$ are Poisson distributions with the first moments $\langle q\rangle$ and $\langle\tilde{q}\rangle$, respectively. In the case of $\langle q\rangle=0$, we get an ordinary Erdős-Rényi graph whose edges are overlapping edges of the multiplex, and a giant mutually connected component emerges at $\langle q\rangle_c=1$ without a discontinuity. For $\langle q\rangle>0$, the phase transition is hybrid. ${ }^6$

In $M$-layer multiplexes, the number of the possible types of overlaps equals $2^M-M-1$ (in particular, in a 3-layer multiplex with layers $a$, $b, c$, there are 4 combinations of overlapping edges: $a b, b c, a c$, and $a b c$ ), and for $M>2$ the situation becomes more complicated than for a 2-layer multiplex. Nonetheless, it was found by the message passing techniques that the qualitative conclusions are similar for all $M \geq 2$ (Cellai, Dorogovtsev, and Bianconi, 2016).

## 数据科学代写|复杂网络代写Complex Network代考|Finite multiplexes

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|Overlapping edges

(Cellai、López、Zhou、Gleeson 和 Bianconi，2013b; Hu、Zhou、Zhang、Han，罗森布拉特和哈夫 林，2013 年) 。例如，让我们考虑一个 2 层多路复用，其中一层中的一些边缘与另一层中的边缘重疍。换句话 说，来自不同层的两条边可以有相同的一对端点。我们可以将重胥边视为此类多路复用中的一种特殊类型的 边。胡等。注意到重疍边的簇可以被视为通过非重合边相互连接的超节点。因此，如 Min、Lee、Lee 和 Goh (2015 年) 所证实的那样，任何小于 1 的重再边缘浓度的存在都不会从质上改变与巨大的相互连接组件的出现 相关的混合相变。在这种多路复用中，每个顶点都有三个度数， $q_a, q_b$ ，和 $\tilde{q}$ ，分别是仅在层中的连接数 $a$ ，仅在 层 $b$ ，以及重胥连接的数量。如果不存在其他相关性，则多重性由联合度分布定义 $P\left(q_a, q_b, \tilde{q}\right)$ 它在局部是树状 的。数字 $8.10$ 显示具有联合度分布的对称 2 层多路复用的相图 $P\left(q_a, q_b, \tilde{q}\right)=P\left(q_a\right) P\left(q_b\right) \widetilde{P}(\tilde{q})$ ，在哪里 $P(q)$ 和 $\widetilde{P}(\tilde{q})$ 是具有一阶矩的泊松分布 $\langle q\rangle$ 和 $\langle\tilde{q}\rangle$ ，分别。如果是 $\langle q\rangle=0$ ，我们得到一个普通的 Erdős-Rényi 图，其边是多路复用的重苩边，并且一个巨大的相互连接的组件出现在 $\langle q\rangle_c=1$ 没有间断。为了 $\langle q\rangle>0 ＼mathrm{~ ， 相 ~}$ 变是混合的。 ${ }^6$

## MATLAB代写

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