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16.network_science_models.slides.html
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16.network_science_models.slides.html
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<h3 id="网络科学理论">网络科学理论<a class="anchor-link" href="#网络科学理论">¶</a></h3><hr>
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<h1 id="网络科学模型">网络科学模型<a class="anchor-link" href="#网络科学模型">¶</a></h1><hr>
<hr>
<p>王成军</p>
<p>wangchengjun@nju.edu.cn</p>
<p>计算传播网 <a href="http://computational-communication.com">http://computational-communication.com</a></p>
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<p><img src ='./img/cocktail.png' width = 500>
The random network model</p>
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<h1 id="Erdös-Rényi-model-(1960)">Erdös-Rényi model (1960)<a class="anchor-link" href="#Erdös-Rényi-model-(1960)">¶</a></h1><p>Definition: A random graph is a graph of N nodes where each pair of nodes is connected by probability p.</p>
<h3 id="G(N,-L)-Model">G(N, L) Model<a class="anchor-link" href="#G(N,-L)-Model">¶</a></h3><p>N lableled nodes are connected with L randomly placed links. Erdös-Rényi(1959)</p>
<h3 id="G(N,-p)-Model">G(N, p) Model<a class="anchor-link" href="#G(N,-p)-Model">¶</a></h3><p>Each pair of N labeled nodes is connected with probability p. Gilbert (1959)</p>
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<h2 id="To-construct-a-random-network:">To construct a random network:<a class="anchor-link" href="#To-construct-a-random-network:">¶</a></h2><ul>
<li>1) Start with $N$ isolated nodes.</li>
<li>2) Select a node pair and generate a random number between 0 and 1.
If the number exceeds $p$, connect the selected node pair with a link,
otherwise leave them disconnected.</li>
<li>3) Repeat step (2) for each of the $\frac{N(N-1)}{2}$ node pairs.</li>
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<p>The probability that a random network has exactly $L$ links and $N$ nodes:</p>
<h1 id="$p_L-=-C_{\frac{N(N-1)}{2}}^L-p^L-(1-p)^{\frac{N(N-1)}{2}--L}$">$p_L = C_{\frac{N(N-1)}{2}}^L p^L (1-p)^{\frac{N(N-1)}{2} -L}$<a class="anchor-link" href="#$p_L-=-C_{\frac{N(N-1)}{2}}^L-p^L-(1-p)^{\frac{N(N-1)}{2}--L}$">¶</a></h1>
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<p>$P_L$is a binomial distribution, the <strong>expected</strong> number of links in a random graph is</p>
<h1>$ <L> = \sum_{L = 0}^{\frac{N(N-1)}{2}} L p_L = p \frac{N(N-1)}{2}$</h1>
<p>$L_{max} = \frac{N(N-1)}{2}$</p>
<p>Thus, <strong>the average degree</strong> of a random network is :</p>
<h1>$<k> = \frac{2<L>}{N} = p(N-1)$</h1>
<p>$k_{max} = N-1$</p>
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<h1 id="In-summary:">In summary:<a class="anchor-link" href="#In-summary:">¶</a></h1><ul>
<li>the number of links in a random network varies between realizations. </li>
<li>Its expected value is determined by <strong>N</strong> and <strong>p</strong>. </li>
<li>If we increase p a random network becomes denser: <ul>
<li>The average number of links increase linearly from <L> = 0 to Lmax </li>
<li>and the average degree of a node increases from <k> = 0 to <k> = N-1.</li>
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</li>
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<h1 id="Degree-Distribution">Degree Distribution<a class="anchor-link" href="#Degree-Distribution">¶</a></h1><p>In a given realization of a random network some nodes gain numerous links, while others acquire only a few or no links. These differences are captured by the degree distribution, $p_k$.</p>
<h1 id="$p_k$-is-the-probability-that-a-randomly-chosen-node-has-degree-$k$.">$p_k$ is the probability that a randomly chosen node has degree $k$.<a class="anchor-link" href="#$p_k$-is-the-probability-that-a-randomly-chosen-node-has-degree-$k$.">¶</a></h1>
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<p>In a random network the probability that <strong>node i</strong> has exactly <strong>$k$</strong> links is the product of three terms:</p>
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<li>The probability that k of its links are present, or <strong>$p^k$</strong>.</li>
<li>$k_{max} = N-1$, The probability that the remaining (N-1-k) links are missing, or $(1-p)^{N-1-k}$</li>
<li>The number of ways we can select $k$ links from $N- 1$ potential links a node can have, or $C_{N-1}^k$</li>
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<p>Consequently the degree distribution of a random network is:</p>
<h1 id="$p_k-=-C_{N-1}^k-p^k-(1-p)^{N-1-k}$">$p_k = C_{N-1}^k p^k (1-p)^{N-1-k}$<a class="anchor-link" href="#$p_k-=-C_{N-1}^k-p^k-(1-p)^{N-1-k}$">¶</a></h1><p>which follows <strong>the binomial distribution</strong>. The shape of this distribution depends on the system size $N$ and the
probability $p$.</p>
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<h1 id="POISSON-DISTRIBUTION">POISSON DISTRIBUTION<a class="anchor-link" href="#POISSON-DISTRIBUTION">¶</a></h1><p>Most real networks are sparse, meaning that for them <k> ≪ N. In this limit the degree distribution is well approximated by the Poisson distribution</p>
<h1>$p_k = e^{-<k>} \frac{<k>^k}{k!}$</h1>
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<p><img src='./img/er_evolution.png' width = 800px></p>
<p><strong>The evolution of random networks</strong>
The relative size of the giant component in function of the average degree $<k>$ in the Erdős-Rényi model. The figure illustrates the phase tranisition at $<k> = 1$, responsible for the emergence of a giant component with
nonzero $N_G$. <strong>REAL NETWORKS ARE SUPERCRITICAL</strong></p>
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<h1 id="The-small-world-phenomenon">The small world phenomenon<a class="anchor-link" href="#The-small-world-phenomenon">¶</a></h1><p>also known as <strong>six degrees of separation</strong>,
has long fascinated the general public. It states that if you choose any two
individuals anywhere on Earth, you will find a path of at most six acquaintances
between them.</p>
<ul>
<li>What does short (or small) mean, i.e. short compared to what? </li>
<li>How do we explain the existence of these short distances?</li>
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<h1>Consider a random network with average degree $<k>$.</h1>
<p>A node i in this network has on average:</p>
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<li>$<k>$ nodes at distance one (d=1). </li>
<li>$<k>^2$ nodes at distance two (d=2). </li>
<li>$<k>^3$ nodes at distance three (d =3). </li>
<li>...</li>
<li>$<k>^d$ nodes at distance d.</li>
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<h1 id="为什么？">为什么？<a class="anchor-link" href="#为什么？">¶</a></h1><p><img src="./img/homework.jpg" alt=""></p>
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<h1 id="从For循环的角度理解">从For循环的角度理解<a class="anchor-link" href="#从For循环的角度理解">¶</a></h1><ul>
<li>从这个节点i走一步，到达他/她的$<k>$个朋友</li>
<li>从节点i走两步，先到他/她的$<k>$个朋友，再到每个朋友的$<k>$个朋友。</li>
<li>。。。</li>
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<p>for $<k> ≈ 1,000$, which is the estimated number of acquaintences an individual has, we expect $10^6$ individuals at distance two and about a billion, i.e. almost the whole earth’s population, at distance three from us.</p>
<p>从一个开始的节点出发走d步，共有节点数量：</p>
<p>$N(d) = 1 + <k> + <k>^2 + <k>^3 + ...+ <k>^d = \frac{<k>^{d+1}-1}{<k>-1}$</p>
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<h1 id="网络直径">网络直径<a class="anchor-link" href="#网络直径">¶</a></h1><p>$N(d_{max}) = N$可以计算最大的距离，或者说网络直径。</p>
<p>若$<k>$ 远大于1，那么$N = <k>^{d_{max}}$，
因此，$d_{max} = \frac{ln N}{ln <k>}$</p>
<h3 id="The-diameter-depends-logarithmically-on-the-system-size.">The diameter depends logarithmically on the system size.<a class="anchor-link" href="#The-diameter-depends-logarithmically-on-the-system-size.">¶</a></h3><h3>The $1/ln <k>$ term implies that the denser the network, the smaller is the distance between the nodes.</h3>
<p>Let us illustrate the implications of (3.19) for social networks. Using $N ≈ 7 ×10^9$ and $<k> ≈ 10^3$, we obtain</p>
<h2 id="$d_{max}-=-\frac{ln-7-×10^9}{ln-10^3}-=-3.28$">$d_{max} = \frac{ln 7 ×10^9}{ln 10^3} = 3.28$<a class="anchor-link" href="#$d_{max}-=-\frac{ln-7-×10^9}{ln-10^3}-=-3.28$">¶</a></h2><p>I. de Sola Pool and M. Kochen. Contacts and Influence. Social Networks, 1: 5-51, 1978.</p>
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<li>1D: For a one-dimensional lattice (a line of length N) the diameter and the average path length scale linearly with $N: dmax\sim<d> \sim N$.</li>
<li>2D: For a square lattice $dmax \sim <d> \sim N^{1/2}$.</li>
<li>3D: For a cubic lattice $dmax \sim <d> \sim N^{1/3}$.</li>
<li>dD: In general, for a d-dimensional lattice $dmax \sim <d> \sim N^{1/d}$.</li>
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<p>In lattices the path lengths are significantly longer than in a random network.</p>
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<h1 id="SIX-DEGREES:-EXPERIMENTAL-CONFIRMATION">SIX DEGREES: EXPERIMENTAL CONFIRMATION<a class="anchor-link" href="#SIX-DEGREES:-EXPERIMENTAL-CONFIRMATION">¶</a></h1><p>The first empirical study of the small world phenomena took place in 1967</p>
<ul>
<li>Stanley Milgram, building on the work of Pool and Kochen, designed an experiment to measure the distances in social networks. </li>
<li>Milgram chose a stock broker in Boston and a divinity student in Sharon, Massachusetts as targets. </li>
<li>He then randomly selected residents of Wichita and Omaha, sending them a letter containing a short summary of the study’s purpose, a photograph, the name, address and information about the target person. </li>
<li>They were asked to forward the letter to a friend, relative or acquantance who is most likely to know the target person.</li>
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<h1 id="He-found-that-the-median-number-of-intermediates-was-5.2,">He found that the median number of intermediates was 5.2,<a class="anchor-link" href="#He-found-that-the-median-number-of-intermediates-was-5.2,">¶</a></h1>
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<p>Using Facebook’s social graph of May 2011, consisting of 721 million active users and 68 billion symmetric friendship links, researchers found:</p>
<h1 id="an-average-distance-4.74-between-the-users.">an average distance 4.74 between the users.<a class="anchor-link" href="#an-average-distance-4.74-between-the-users.">¶</a></h1><ul>
<li>Therefore, the study detected only ‘four degrees of separation’, closer to the prediction of than to Milgram’s six degrees.</li>
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<p>L. Backstrom, P. Boldi, M. Rosa, J. Ugander, and S. Vigna. Four degrees of separation. In ACM Web Science 2012: Conference Proceedings, pages 45−54. ACM Press, 2012.</p>
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<h1 id="邻居彼此认识吗？">邻居彼此认识吗？<a class="anchor-link" href="#邻居彼此认识吗？">¶</a></h1><p>Local clustering coefficient</p>
<p>We need to estimate the expected number of links $L_i$ between the node’s $k_i$ neighbors.</p>
<p>In a random network the probability that two of i’s neighbors link to each other is $p$.</p>
<p>As there are $\frac{k_i(k_i - 1)}{2}$ possible links between the $k_i$ neighbors of node i, the expected value of $L_i$ is</p>
<h1>$<L> = p\frac{k_i(k_i - 1)}{2}$</h1>
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<h1 id="局部聚集系数">局部聚集系数<a class="anchor-link" href="#局部聚集系数">¶</a></h1><h2>$<C_i> = \frac{<L>}{\frac{1}{2} k_i(k_i - 1)} = p = \frac{<k>}{N}$</h2>
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<li>For fixed $<k>$, the larger the network, the smaller is a node’s clustering coefficient.</li>
<li>The local clustering coefficient of a node is independent of the node’s degree.</li>
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<h1 id="In-summary">In summary<a class="anchor-link" href="#In-summary">¶</a></h1><ul>
<li>we find that the random network model does not capture the clustering of real networks. </li>
<li>Instead real networks have a much higher clustering coefficient than expected for a random network of similar N and L. </li>
<li>An extension of the random network model proposed by Watts and Strogatz addresses the coexistence of high $<C>$ and the small world property. <ul>
<li>It fails to explain, however, why high-degree nodes have a smaller clustering coefficient than low-degree nodes. </li>
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<h1 id="WS模型-（1998）">WS模型 （1998）<a class="anchor-link" href="#WS模型-（1998）">¶</a></h1><p>Duncan Watts and Steven Strogatz proposed an extension of the random network model motivated by two observations：</p>
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<li>(a) Small World Property： In real networks the average distance between two nodes depends logarithmically on N</li>
<li><p>(b) High Clustering： The average clustering coefficient of real networks is much high- er than expected for a random network of similar N and L</p>
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<li><p>a regular lattice has high clustering but lacks the small-world phenomenon.</p>
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<li>and a random network has low clustering, but displays the small-world property.</li>
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<blockquote><h2 id="Numerical-simulations-indicate-that-for-a-range-of-rewiring-parameters-the-model's-average-path-length-is-low-but-the-clustering-coefficient-is-high">Numerical simulations indicate that for a range of rewiring parameters the model's average path length is low but the clustering coefficient is high<a class="anchor-link" href="#Numerical-simulations-indicate-that-for-a-range-of-rewiring-parameters-the-model's-average-path-length-is-low-but-the-clustering-coefficient-is-high">¶</a></h2><p>hence reproducing the coexistence of high clustering and small-world phenomena</p>
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<p>D. J. Watts and S. H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393: 409–10, 1998.</p>
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<li><p>The dependence of the average path length $d(p)$ and clustering coefficient $<C(p)>$ on the <strong>rewiring parameter $p$</strong>.</p>
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<li><p>Note that $d(p)$ and $<C(p)>$ have been normalized by $d(0)$ and $<C(0)>$ obtained for a regular lattice.</p>
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<li>The rapid drop in $d(p)$ signals the onset of the small-world phenomenon. <ul>
<li>During this drop, $<C(p)>$ remains high. </li>
<li>Hence in the range $0.001<p<0.1$ short path lengths and high clustering co-exist in the network.</li>
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<p>D. J. Watts and S. H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393: 409–10, 1998.</p>
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<h1 id="In-summary">In summary<a class="anchor-link" href="#In-summary">¶</a></h1><ul>
<li>we find that the random network model does not capture the clustering of real networks. </li>
<li>Instead real networks have a <strong>much higher clustering coefficient</strong> than expected for a random network of similar N and L. </li>
<li>An extension of the random network model proposed by Watts and Strogatz [1998] addresses the coexistence of high <C> and the small world property. </li>
<li>It fails to explain, however, <strong>why high-degree nodes have a smaller clustering coefficient than low-degree nodes</strong>.</li>
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<p><img src = './img/barabasi99.png' width = 700px></p>
<h1 id="REAL-NETWORKS-ARE-NOT-POISSON">REAL NETWORKS ARE NOT POISSON<a class="anchor-link" href="#REAL-NETWORKS-ARE-NOT-POISSON">¶</a></h1>
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<h1 id="BA模型-（1999）">BA模型 （1999）<a class="anchor-link" href="#BA模型-（1999）">¶</a></h1><p>Barabasi (1999) Emergence of scaling in random networks.Science-509-12</p>
<h3 id="Scaling-Free">Scaling-Free<a class="anchor-link" href="#Scaling-Free">¶</a></h3><p><img src ='./img/ba.png' width = 500px></p>
<h3 id="The-Preferential-Attachment-Model">The Preferential Attachment Model<a class="anchor-link" href="#The-Preferential-Attachment-Model">¶</a></h3>
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<h1 id="无标度的意义">无标度的意义<a class="anchor-link" href="#无标度的意义">¶</a></h1>
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<p>度分布的n阶矩被定义为：</p>
<p>$ <k^n> = \sum_{k_{min}}^{k_{max}}k^np_k = \int_{k_{min}}^{k_{max}}k^np_kdk $ （1）</p>
<p>低阶矩具有明确的统计意义：</p>
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<li>n = 1的时候，一阶矩是$<k^{}>$，即平均度。</li>
<li>n = 2的时候，二阶矩是$<k^2>$，可以帮助计算方差 $ \delta^2 = <k^2> - <k^{}>^{2} $，测量了度的离散程度（the spread in the degrees）。</li>
<li>n = 3的时候，三阶矩是$<k^3>$, 决定了度分布的偏度（skewness），测量了$p_k$围绕着$<k>$的对称性。</li>
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<p>对于无标度网络而言，满足幂律分布：</p>
<p>$p(k) = Ck^{-\gamma}$ （2）</p>
<p>由公式（1）和（2）可以得到：</p>
<p>$ <k^n> = \int_{k_{min}}^{k_{max}}k^np_kdk = C \frac{k_{max}^{n- \gamma +1} - k_{min}^{n - \gamma +1}}{n - \gamma + 1} $ （3）</p>
<p>可以使用wolframalpha的积分计算器积分来进行简单验证，例如x^(n-r)dx从10到100积分 网页链接：<a href="http://www.wolframalpha.com/input/?i=integrate+x%5E%28n-r%29+dx+from+10+to+100">http://www.wolframalpha.com/input/?i=integrate+x%5E%28n-r%29+dx+from+10+to+100</a></p>
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<p>显然：</p>
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<li>当$n - \gamma +1 <= 0$时，随着$k_{max}增加，$$k_{max}^{n- \gamma +1} \rightarrow 0$。所有满足$n <= \gamma -1$的n阶矩都是有限的。</li>
<li>当$n - \gamma +1 >0 $时，随着$k_{max}增加，$$k_{max}^{n- \gamma +1} \rightarrow \infty$。所有满足$n > \gamma -1$的n阶矩都是无极限的。</li>
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<p>对于无标度网络而言，一般幂参数$2 < \gamma < 3$，所以：</p>
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<li><p>对于n = 1的情况，即一阶矩平均度$<k^{}>$是有限的。</p>
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<li><p>但对于n >= 2的情况，即$k^2$或$k^3$是无极限的。二阶和高阶矩无穷大是“无标度”的来源</p>
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<h1 id="无标度的意义">无标度的意义<a class="anchor-link" href="#无标度的意义">¶</a></h1><blockquote><h3>在网络中随机抽取一个节点的度可以显著的不同于平均度$<k>$</h3>
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<p>上图最为直接的描绘出了这种特点，即与正态分布等相比，<code>无标度网络下降的慢</code>。</p>
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<li><p>对于任何指数类型的分布，如泊松分布或高斯分布，随机选取一个节点的度在平均度附近，因此平均度就是这些网络的<code>尺度</code>。</p>
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<li><p>对于一个幂律分布而言，因为二阶矩发散，在网络中随机抽取一个节点的度可以显著的不同于平均度$<k^{}>$， 因此平均度不再是网络的尺度，称之为无标度。</p>
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<p>对于不同的网络，平均度与标准差的取值</p>
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<p><img src='./img/newton.png' width = 700px></p>
<p>英国的天才 第一集 Genius of Britain: The Scientists Who Changed the World (2010) <a href="https://movie.douban.com/subject/4887415/">https://movie.douban.com/subject/4887415/</a></p>
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<h1 id="1.-连续平均场"Continuum-Mean-Field"">1. 连续平均场"Continuum Mean-Field"<a class="anchor-link" href="#1.-连续平均场"Continuum-Mean-Field"">¶</a></h1><h3 id=""Mean-Field":-many-->-one">"Mean-Field": many -> one<a class="anchor-link" href="#"Mean-Field":-many-->-one">¶</a></h3><p>The complex interaction between a large number of components can be simplified into a single averaged effect of all the other individuals on any given one.</p>
<h3 id=""Continuum":-discrete-->-Continuous">"Continuum": discrete -> Continuous<a class="anchor-link" href="#"Continuum":-discrete-->-Continuous">¶</a></h3><p>A discrete variables that is "smooth enough", i.e. increase by one unit in each period, such as time steps [1], spatial jumps [2-3], and population, can be viewed as a continuous geometric structure. Any measure on this structure, like degree [1], number of unique locations visited [2], and distance [3], can be described by differential equations.</p>
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<h2 id="模型设定">模型设定<a class="anchor-link" href="#模型设定">¶</a></h2><ul>
<li>初始状态有$m_0$个节点</li>
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<li>增长原则：每次加入一个节点i （加入时间记为$t_i$）, 每个节点的加入带来m条边，2m个度的增加
<strong> 其中老节点分到的度数是m，新加入的那一个节点分到的度数为m</strong> 那么到时间t的时候，网络的总节点数是$m_0 + t$，网络的总度数为$2mt$。</li>
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<li>优先链接原则：每一次从m条边中占有一条边的概率正比于节点的度$k_i$
<strong> 那么显然，加入的越早（$t_i$越小）越容易获得更多的链接数。</strong> 从时间0开始，每一个时间步系统中的节点度$k_i$是不断增加的。</li>
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<h2 id="度的增长/时间依赖性">度的增长/时间依赖性<a class="anchor-link" href="#度的增长/时间依赖性">¶</a></h2><p>$k_i$在一个时间步获得一个度的概率表示为$\prod (k_i) $， 那么有：</p>
<p>$\prod (k_i) = \frac{k_i}{\sum k_i} = \frac{k_i}{2mt}$</p>
<p>一个时间步，$k_i$随t的变化率可以表达为：</p>
<h1 id="$\frac{\partial-k_i}{\partial-t}-=-\Delta-k-\prod-(k_i)-=-m-\frac{k_i}{2mt}-=-\frac{k_i}{2t}$">$\frac{\partial k_i}{\partial t} = \Delta k \prod (k_i) = m \frac{k_i}{2mt} = \frac{k_i}{2t}$<a class="anchor-link" href="#$\frac{\partial-k_i}{\partial-t}-=-\Delta-k-\prod-(k_i)-=-m-\frac{k_i}{2mt}-=-\frac{k_i}{2t}$">¶</a></h1><h1 id="$\frac{\partial-k_i}{k_i}-=-\frac{\partial-t}{2t}$">$\frac{\partial k_i}{k_i} = \frac{\partial t}{2t}$<a class="anchor-link" href="#$\frac{\partial-k_i}{k_i}-=-\frac{\partial-t}{2t}$">¶</a></h1><h1 id="$\int-\frac{1}{k_i}-d-k_i-=-\int-\frac{1}{2t}-dt$">$\int \frac{1}{k_i} d k_i = \int \frac{1}{2t} dt$<a class="anchor-link" href="#$\int-\frac{1}{k_i}-d-k_i-=-\int-\frac{1}{2t}-dt$">¶</a></h1>
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<h2 id="积分结果">积分结果<a class="anchor-link" href="#积分结果">¶</a></h2><h1 id="$k_i-=-C-(2t)-^-{-0.5}$----（1）">$k_i = C (2t) ^ {-0.5}$ （1）<a class="anchor-link" href="#$k_i-=-C-(2t)-^-{-0.5}$----（1）">¶</a></h1><p>此时，根据模型的初始条件，每个新加入节点获得的度是m：</p>
<p>$k_i(t_i) = m $ 代入公式（1）</p>
<p>可以得到$C = m(2t_i)^{-0.5}$ 公式（2）</p>
<p>代入公式（1），得到：</p>
<p>$k_i = m (\frac{t}{t_i})^{0.5}$ 公式（3）</p>
<p>对于一个节点i，其加入网络的时间$t_i$是固定的，我们可以观察其度$k_i$随着时间的幂律关系。</p>
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<h2 id="累积概率分布">累积概率分布<a class="anchor-link" href="#累积概率分布">¶</a></h2><p>当我们思考一个累积概率分布的时候，我们想要的是$k_i(t) < k$的概率：$P(k_i(t) < k) $</p>
<p>由公式（3），可以知道：</p>
<p>$P(k_i(t) < k) = P( m (\frac{t}{t_i})^{0.5} < k ) = P( t_i > \frac{m^2 t}{k^2} ) = 1 - P(t_i \leqslant \frac{m^2 t}{k^2} )$（4）</p>
<p>在初始状态$ t = 0$, 有$m_0$个节点，那么$t_{m_0} = 0$</p>
<p>假设我们将节点加入的时间步是均匀的，那么$t_i$的概率是一个常数：</p>
<p>$P(t_i) = \frac{1}{m_0 + t}$ 公式（5）</p>
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<h3 id="均匀分布的性质">均匀分布的性质<a class="anchor-link" href="#均匀分布的性质">¶</a></h3><ul>
<li>设连续型随机变量X的概率密度函数为 $f(x)=1/(b-a)，a≤x≤b $, 则称随机变量X服从[a,b]上的均匀分布，记为X~U[a,b]。若[x1,x2]是[a,b]的任一子区间，则 $P{x_1≤x≤x_2}=(x_2-x_1) \frac{1}{b-a}$</li>
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<p>根据均匀分布的性质，将公式（5）代入公式（4）得到：</p>
<p>$P(k_i(t) < k) = 1- \frac{m^2 t}{k^2} P(t_i) = 1 - \frac{m^2 t}{k^2 (m_0 + t)} $ 公式（6）</p>
<p>对累积概率函数求微分，就可以到达概率密度函数:</p>
<p>$P( k ) = \frac{\partial P(k_i(t) < k)}{\partial k} = \frac{2m^2 t}{m_0 + t} \frac{1}{k^3}$ 公式（7）</p>
<p>也就是说：$\gamma = 3$, 与m无关。</p>
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<h1 id="参考文献">参考文献<a class="anchor-link" href="#参考文献">¶</a></h1><ul>
<li>Barabasi 2016 Network Science. Cambridge</li>
<li>Barabasi (1999) Emergence of scaling in random networks.Science-509-12.pdf</li>
<li>Barabasi (1999) Mean-field theory for scale-free random networks. PA.pdf</li>
<li>Albert & Barabasi (2002) Statistical mechanics of complex networks. RMP.pdf</li>
<li>BA Model 计算传播百科 <a href="http://computational-communication.com/wiki/index.php?title=Mean_field_method#BA_Model">http://computational-communication.com/wiki/index.php?title=Mean_field_method#BA_Model</a></li>
<li>Principle of Locality I: Hacking the Continuum Mean-Field Technique in Network Modeling <a href="http://www.jianshu.com/p/97f674267d3e">http://www.jianshu.com/p/97f674267d3e</a></li>
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