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# 数据科学代写|复杂网络代写Complex Network代考|COMP5313 Heterogeneous k-core

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## 数据科学代写|复杂网络代写Complex Network代考|Heterogeneous k-core

In the heterogeneous $k$-core problem (Cellai, Lawlor, Dawson, and Gleeson, 2011, 2013a; Baxter, Dorogovtsev, Goltsev, and Mendes, 2011), a threshold degree value $k$ varies from vertex to vertex. Let $k_i, i=1,2, \ldots, N$ be the full set of threshold degrees for all vertices of a graph. The heterogeneous $k$-core of a graph is its largest subgraph with each its vertex $i$ of degree at least $k_i$. In general, the heterogeneous $k$-core consists of a giant component ( giant heterogeneous $k$-core) and numerous finite ones. The heterogeneous $k$-core is the result of the progressive ‘deactivation’ of initially active vertices with
for the generating function $H_{1 k}(x)$ of the probability that an end of a randomly chosen edge in the $k$-core belongs to a finite corona cluster of a given size,
$$H_{1 k}(x)=1-\frac{k P_k(k)}{\langle q\rangle_k}+x \frac{k P_k(k)}{\langle q\rangle_k}\left[H_{1 k}(x)\right]^{k-1} .$$
The additional term $1-k P_k(k) /\langle q\rangle_k$, compared to Eq. (6.69), is the probability that the end of an edge does not belong to the corona. We estimate the generating function $H_k(x)$ for the size distribution of a corona cluster attached to a vertex in the $k$-core,
$$H_k(x) \approx \sum_q P_k(q)\left[H_{1 k}(x)\right]^q,$$
and finally obtain
$$\left\langle s_{\mathrm{crn}}\right\rangle(p)=\left.\frac{d H_k(x)}{d x}\right|_{x=1}=\frac{k P_k(k)}{1-k(k-1) P_k(k) /\langle q\rangle_k} \propto \frac{1}{\sqrt{p-p_c}} .$$

## 数据科学代写|复杂网络代写Complex Network代考|Dynamics of pruning

The pruning process resulting in the $k$-core is a simple paradigm for various cascading failures phenomena, where elements ‘weaker’ than a given threshold are progressively removed from a system, and so it deserves a thorough consideration. Let at each time step, $t=1,2, \ldots$, all vertices with degrees $q<k$ be removed from a network. The criterion for the removal of vertices is local, namely, their degrees, which greatly simplifies the problem. During the pruning, the network progressively shrinks to null or to the $k$-core, and the question is how the structure of the yet unpruned network evolves with time. It is convenient to fix the number of vertices, $N$, so, instead of pruning vertices with their edges, we prune only their edges, progressively increasing the number of isolated vertices. Clearly, this does not change the process. Let the network be infinite, locally tree-like and, furthermore, uncorrelated, where the last assumption is made here only for the sake of compactness. In addition, we assume that during the pruning process, the network remains uncorrelated. Let us introduce $r(t)$, the probability that, following a randomly chosen edge in the network at time $t$, we find that its end vertex has degree less than $k$ :
$$r(t)=\sum_{q<k} \frac{q P(q, t)}{\langle q\rangle(t)}$$
where $\langle q\rangle(t)$ is the average degree of the network at time $t$,
$$\langle q\rangle(t)=\sum_q q P(q, t) .$$

## 数据科学代写|复杂网络代写Complex Network代考|Heterogeneous $\mathbf{k}$-core

$$H_{1 k}(x)=1-\frac{k P_k(k)}{\langle q\rangle_k}+x \frac{k P_k(k)}{\langle q\rangle_k}\left[H_{1 k}(x)\right]^{k-1} .$$

$$H_k(x) \approx \sum_q P_k(q)\left[H_{1 k}(x)\right]^q,$$

$$\left\langle s_{\mathrm{crn}}\right\rangle(p)=\left.\frac{d H_k(x)}{d x}\right|_{x=1}=\frac{k P_k(k)}{1-k(k-1) P_k(k) /\langle q\rangle_k} \propto \frac{1}{\sqrt{p-p_c}} .$$

## 数据科学代写|复杂网络代写Complex Network代考|Dynamics of pruning

$$r(t)=\sum_{q<k} \frac{q P(q, t)}{\langle q\rangle(t)}$$

$$\langle q\rangle(t)=\sum_q q P(q, t) .$$

## MATLAB代写

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