network_ana

main.tex (5602B)

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 \markright{Popović\hfill 6-th Exercise \hfill}
 
 
 \title{University of Vienna\\ Faculty of Mathematics\\
     \vspace{1.25cm}Seminar: Introduction to complex network analysis \\ 6-th
 Exercise
 }
 \author{Milutin Popović}
 \begin{document}
 \maketitle
 
 \section{Exercise 7}
 \subsection{Bianconi-Barabasi model and network failure}
 Consider a Bianconi-Barabasi model with two fitness values $\eta = 1$ and
 $\eta = a$ where $0 \leq a \leq 1$, with a fitness distribution
 \begin{align}
     \varrho(\eta) = \frac{1}{2} \delta (\eta - 1) + \frac{1}{2} \delta (\eta
     - a),
 \end{align}
 where $\delta(\eta)$ is the Dirac delta distribution.
 
 We know that the dynamic epxonent $\beta(\eta)$ satisfies
 \begin{align}
     \beta(\eta) = \frac{\eta}{C},
 \end{align}
 where
 \begin{align}\label{eq: c}
     C = \in \varrho(\eta) \frac{\eta}{1-\beta(\eta)} d\eta.
 \end{align}
 To calculate $C$ we plug $\varrho(\eta)$ into equation \ref{eq: c}
 \begin{align}
     C &= \frac{1}{2}\left(
         \int \delta(\eta - 1) \frac{\eta}{1-\beta(\eta)}+
         \int \delta(\eta - a) \frac{\eta}{a-\beta(\eta)}
     \right) =\\
       &= \frac{1}{2} \left(\frac{1}{1-\beta(1)} + \frac{1}{1-\beta(a)}\right).
 \end{align}
 Into the calculation we substitute $\beta(\eta)$, thereby getting
 \begin{align}
     C = \frac{1}{2} \left(\frac{C}{C-1} + \frac{Ca}{C-a}\right),
 \end{align}
 this expression has two solutions
 \begin{align}
     C_1 &= \frac{1}{4} \left(\sqrt{9a^2 - 14a + 9} +3a +3 \right), \\
     C_2 &= \frac{1}{4} \left(-\sqrt{9a^2 - 14a + 9} +3a +3 \right).
 \end{align}
 We choose $C = C_1$ because
 \begin{align}
     C = C_1 = \frac{1}{4} \left(\sqrt{9a^2 - 14a + 9} +3a +3 \right) =
     \begin{cases}
         \frac{3}{2} \;\;\;\;\; \text{for} \;\;\; a=0 \\
         2 \;\;\;\;\; \text{for} \;\;\; a=1
     \end{cases}
 \end{align}
 satisfies to the calculation of the reduced model for $a = 0$, which is the
 Barabasi-Albert model. Thus the stationary $p_k$ degree distribution can be
 calculated
 \begin{align}
     p_k &~ C\int d\eta \frac{\varrho(\eta)}{\eta} \left(\frac{m}{k}
         \right)^{\frac{C}{\eta} +1} =\\
         &= \frac{C}{2} \left(\frac{m}{k}\right) \left(\left(\frac{m}{k}\right)^C + \frac{1}{4}
         \left(\frac{m}{k}\right)^{\frac{C}{a}} \right)
 \end{align}
 Thereby the degree exponent $\gamma$ is estimated to be around
 \begin{align}
     \gamma \simeq C + 1
 \end{align}
 Which agrees with the boundaries $a=0$ and $a=1$. The figure below shows the
 relations
 \begin{figure}[H]
     \centering
     \includegraphics[width=\textwidth]{./ex_71.png}
     \caption{Constant $C$ and degree exponent $\gamma$ against $0\leq a \leq$}
 \end{figure}
 \subsection{The Breakdown Threshold}
 The breakdown threshold $f_c$ of networks with a given degree distribution
 $p_k$ can be written as
 \begin{align}
     f_c = 1- \frac{1}{\kappa - 1} \;\;\;\;\; \kappa = \frac{\langle k^2
     \rangle}{\langle k \rangle},
 \end{align}
 For Scale Erdős Rényi networks the distribution is
 \begin{align}
     p_k = \begin{bmatrix}N \\ k\end{bmatrix} p^k (1-p)^{N-k},
 \end{align}
 with $N$ number of nodes for a node with degree $k$ and probability of
 connectivity $p$. Thereby we can calculate the first and second moment of the
 degree $\langle k^m \rangle$ for $m = 1, 2$ just by extending the summation
 to the continuum
 \begin{align}
     \langle k \rangle &= \int_0^N k p_k dk = Np\\
     \langle k^2 \rangle &= \int_0^N k^2 p_k dk = p(1-p)N + p^2N^2.\\
 \end{align}
 Thereby the degree dynamics is
 \begin{align}
     \kappa = 1+(N-1)p,
 \end{align}
 and the for the breakdown threshold
 \begin{align}
     f_c = 1 - \frac{1}{(N-1)p} \underset{N\rightarrow
     \infty}{{\longrightarrow}} 1
 \end{align}
 
 For the scale free network we have a exponential distribution
 \begin{align}
     p(k) = c \cdot k^{-\gamma}.
 \end{align}
 We calculated the degree dynamics $\kappa$ in exercise 5, with
 $k_{\text{max}} = k_{\text{min}} N^{\frac{1}{N-\gamma}}$ we have the
 following
 \begin{align}
     \kappa = k_{\text{min}}\frac{2- \gamma}{3-\gamma}
     \frac{N^{\frac{3-\gamma}{\gamma-1}} - 1}{N^{\frac{2-\gamma}{\gamma-1}} -
     1} \;\;\;\; \underset{N\rightarrow \infty}{{\longrightarrow}} \;\; \infty
 \end{align}
 thereby the breakdown threshold is
 \begin{align}
     f_c = 1 -  \frac{1}{k_{\text{min}}\frac{2- \gamma}{3-\gamma}
     \frac{N^{\frac{3-\gamma}{\gamma-1}} - 1}{N^{\frac{2-\gamma}{\gamma-1}} -
     1} - 1}  \;\;\;\; \underset{N\rightarrow \infty}{{\longrightarrow}} \;\; 1
 \end{align}
 
 \nocite{code}
 \nocite{barabasi2016network}
 \printbibliography
 
 \end{document}