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# 数据科学代写|复杂网络代写Complex Network代考|ANND Distribution in the Barabási-Albert networks

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## 数据科学代写|复杂网络代写Complex Network代考|ANND Distribution in the Barabási-Albert networks

Network in the Barabási-Albert model [1] is iteratively constructed according to the following rules. Let $t$ denote the iteration, then after each new node $t$ is added, it is connected to $m$ existing nodes in the network which are selected with probability dependent on their degree (i.e. preferential attachment mechanism).
Let $d_i(t), s_i(t)=\sum_{j:(i, j) \in E} d_j(t)$ and $\alpha_i(t)=\frac{s_i(t)}{d_i(t)}$ denote the degree of node $i$, the total degree of all neighbors of node $i$, the average degree of all neighbors of node $i$, at given iteration $t$, respectively. Then the empirical values of $\Phi(k)$ can be obtained as follows:
$$\Phi_t(k) \sim \frac{1}{\left|E_k\right|} \sum_{\left{i: d_i=k\right}} \alpha_i,$$
where the sum is taken over all nodes of degree $k$, and $\left|E_k\right|$ denotes the number of such nodes in the network.

The values of $d_i(t)$ or $s_i(t)$ may change at iteration $t$ in the following cases:

• if newborn node $t+1$ links to node $i$, then $s_i(t+1)=s_i(t)+m$, and $d_i(t+1)=$ $d_i(t)+1$. The probability of this case is $\frac{d_i(t)}{2 t}$
• if newborn node $t+1$ links to one of the neighbors of node $i$ by one of its edges $j=1, \ldots, m$, then $s_i(t+1)=s_i(t)+1$, while $d_i(t+1)=d_i(t)$. The probability of this case is $\frac{s_i(t)}{2 t}$.

Let us now find the expected value of $\alpha_i(t)$ using mean field approach. Let us find how the value of $\alpha_i(t)$ transforms after adding new node $t+1$ with $m$ links. We have
\begin{aligned} \Delta \alpha_i(t+1)=\alpha_i(t+1)-\alpha_i(t) & =\frac{d_i(t)}{2 t}\left(\frac{s_i(t)+m}{d_i(t)+1}-\frac{s_i(t)}{d_i(t)}\right)+ \ \frac{s_i(t)}{2 t}\left(\frac{s_i(t)+1}{d_i(t)}-\frac{s_i(t)}{d_i(t)}\right) & =\frac{m}{2 t}-\frac{1}{2 t\left(d_i(t)+1\right)}+\alpha_i(t) \frac{1}{2 t\left(d_i(t)+1\right)} \end{aligned}

## 数据科学代写|复杂网络代写Complex Network代考|Triadic Closure Model Analysis

One of the accountable models utilizing the triadic closure mechanism was examined by Holme and Kim [7]. It is an expansion of the Barabási-Albert model of preferential attachment [3] and uses the idea of triadic closure previously specified in papers $[10,17]$. This model generates networks with heavy-tailed degree distributions (as well as BA model), but with a much higher clustering similar to real-world networks.

The triadic closure model by Holme and Kim [7] generates growth networks as follows. At each iteration $t$ :

1. One newborn node $t$ is merged;
2. $m$ links are attached to the existing nodes of the network which connect the newborn node with them as follows:
(a) the first link connects node $t$ with node $i$ using preferential attachment mechanism (i.e. the probability of being connected to node $i$ is proportional to its degree $\left.d_i(t)\right)$;
(b) the remaining $m-1$ edges link the new node $t$ as follows:
(b1) with a fixed probability $0<p<1$, the link is attached to an arbitrary neighbor of the node $i$ (so-called triad formation);
(b2) with probability $1-p$, the link is attached to one of existing nodes using preferential attachment.

It was shown in [7] that the model may produce networks with various levels of clustering by selecting $p$ and $m$. On the other hand, their degree distributions follow a power law with exponent $\gamma=-3$ for any $p$, i.e. it is the same as in the BA model. Throughout the rest of this section we will assume $p \neq 0$ and $m \geq 2$.
Let nodes $j$ and $i$ be neighbors, i.e. $(j, i) \in E(t)$. It was shown in paper [7] that the clustering coefficient tends to some constant value $\theta=\theta(p, m)$ with an increase in the number of iterations $t$. Then the probability that randomly chosen neighbor of node $j$ is also the neighbor of node $i$ can be approximated by the value of the averaged clustering coefficient $\theta(t)$ which can be approximated by constant $\theta$.

## 数据科学代写|复杂网络代写Complex Network代考|ANND Distribution in the Barabási-Albert networks

Barabási-Albert模型[1]中的网络按照以下规则迭代构建。设$t$表示迭代，则每增加一个新节点$t$后，它就连接到网络中已有的$m$个节点，这些节点以概率依赖于它们的程度来选择(即优先连接机制)。

$$\Phi_t(k) \sim \frac{1}{\left|E_k\right|} \sum_{\left{i: d_i=k\right}} \alpha_i,$$

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