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CS6917课程简介
Complex problems emerging from various disciplines and interaction of large amount of data can be modelled with networks. Networks are powerful tools for making sense of the world in the big-data era. The scientific study of networks, including computer networks, social networks, and biological networks, has received an enormous amount of interest recently. The rise of the Internet, the wide availability of inexpensive computers, and penetration of smart mobile devices have made it possible to gather and analyze network data on a large scale, and the development of a variety of new theoretical tools has allowed us to extract new knowledge from many different kinds of networks.
Prerequisites
The study of large, complex networks is broadly interdisciplinary and important developments have occurred in many fields, including mathematics, physics, computer and information sciences, biology, and the social sciences. Subjects covered in this course include the measurement and structure of networks, the fundamentals of network theory, computer algorithms, and spectral methods and mathematical models of networks, all at a much greater scale.
CS6917 Complex Networks HELP（EXAM HELP， ONLINE TUTOR）
The cluster size distribution in Erdos Renyi is described by expression
$$
c_k(t)=\frac{k^{k-2}}{k !} t^{k-1} e^{-k t},
$$
where $t=p N$ is the average degree in a network of $\mathrm{N}$ nodes.
A) (3 points) Study this distribution asymptotically at $k \rightarrow \infty$ and $t=1$.
Show that the infinite cluster is formed at this point: distribution moments diverge starting from the 2 nd.
To study the asymptotic behavior of $c_k(t)$ at $k \rightarrow \infty$ and $t=1$, we can use Stirling’s approximation for factorials:
$$k! \approx \sqrt{2 \pi k} \left(\frac{k}{e}\right)^k$$
Using this approximation, we have:
$$c_k(1) \approx \frac{1}{\sqrt{2 \pi k}} k^{k-2} e^{k-1} e^{-k} = \frac{1}{\sqrt{2 \pi k}} k^{k-2} e^{-1}$$
To study the behavior at $k \rightarrow \infty$, we can take the logarithm of $c_k(1)$ and use the asymptotic expansion $\log(1+x) \approx x$ for small $x$:
$$\log c_k(1) \approx \frac{1}{2} \log k – \log e – 1$$
As $k \rightarrow \infty$, the dominant term is $\frac{1}{2} \log k$, so we have $\log c_k(1) \sim \log k$. This means that $c_k(1)$ grows faster than any power of $k$ as $k \rightarrow \infty$.
To study the moment behavior of $c_k(1)$, we can compute the moments $\langle k^n \rangle$:
$$\langle k^n \rangle = \sum_{k=1}^{\infty} k^n c_k(1) \approx \sum_{k=1}^{\infty} k^n \frac{1}{\sqrt{2 \pi k}} k^{k-2} e^{-1}$$
We can use Stirling’s approximation again to simplify the expression:
$$\langle k^n \rangle \approx \frac{1}{\sqrt{2 \pi}} e^{-1} \sum_{k=1}^{\infty} \frac{1}{k^{1/2-n}}$$
This sum converges for $n>3/2$, which means that the moments $\langle k^n \rangle$ diverge starting from the 2nd moment. This divergence indicates the formation of an infinite cluster in the network.
B) (2 points) Study this distribution asymptotically at $k \rightarrow \infty$ and $t=1-\epsilon$ $(\epsilon<1)$. Show that the distribution looks like $c_k(t)=k^{-\alpha} \exp (-k / f(\epsilon))$. Find $\alpha$ and $f(\epsilon)$.
To study the asymptotic behavior of $c_k(t)$ at $k \rightarrow \infty$ and $t=1-\epsilon$, we again use Stirling’s approximation for factorials:
$$k! \approx \sqrt{2 \pi k} \left(\frac{k}{e}\right)^k$$
Using this approximation, we have:
$$c_k(1-\epsilon) \approx \frac{1}{\sqrt{2 \pi k}} k^{k-2} (1-\epsilon)^{k-1} e^{-k(1-\epsilon)}$$
We can rewrite this expression as:
$$c_k(1-\epsilon) \approx \frac{1}{\sqrt{2 \pi k}} \exp\left[(k-2)\log k + (k-1)\log(1-\epsilon) – k(1-\epsilon)\right]$$
To study the asymptotic behavior at $k \rightarrow \infty$, we take the logarithm of $c_k(1-\epsilon)$:
$$\log c_k(1-\epsilon) \approx -\frac{1}{2} \log k + (1-\epsilon) \left[\log(1-\epsilon)-\frac{1}{1-\epsilon}\right]$$
As $k \rightarrow \infty$, the dominant term is $(1-\epsilon) \left[\log(1-\epsilon)-\frac{1}{1-\epsilon}\right]$, so we have $\log c_k(1-\epsilon) \sim (1-\epsilon) \log(1-\epsilon)$. This means that $c_k(1-\epsilon)$ decays exponentially fast in $k$ as $k \rightarrow \infty$.
To find the functional form of this exponential decay, we can rewrite the expression for $c_k(1-\epsilon)$ as:
$$c_k(1-\epsilon) \approx \frac{1}{\sqrt{2 \pi k}} \exp\left[-k(1-\epsilon)\left(1-\frac{(k-2)\log k + (k-1)\log(1-\epsilon)}{k(1-\epsilon)}\right)\right]$$
We can simplify the exponent by using the fact that $(1-\epsilon)\log(1-\epsilon) \sim -\epsilon$ for small $\epsilon$:
$$c_k(1-\epsilon) \approx \frac{1}{\sqrt{2 \pi k}} \exp\left[-k(1-\epsilon)\left(1-\frac{(k-2)\log k}{k(1-\epsilon)}-\frac{(k-1)\epsilon}{k(1-\epsilon)}\right)\right]$$
Taking the limit $k \rightarrow \infty$, we have:
$$c_k(1-\epsilon) \sim \frac{1}{\sqrt{2 \pi k}} \exp\left[-k\left(\frac{\epsilon}{1-\epsilon}+\frac{\log k}{1-\epsilon}\right)\right]$$
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