network_ana

Complex Network Anlysis
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     48 \markright{Popović\hfill 1st Exercise \hfill}
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     50 
     51 \title{University of Vienna\\ Faculty of Mathematics\\ \vspace{1.25cm}Seminar: Introduction to complex network analysis \\ 1st
     52 Exercise
     53 }
     54 \author{Milutin Popovic}
     55 \date{03. November, 2021}
     56 
     57 \begin{document}
     58 \maketitle
     59 
     60 \section{Ordering Graphs}
     61 Without computation we can order the following graphs in figure \ref{fig: graphs}
     62 by
     63 \begin{figure}[H]
     64     \centering
     65     \includegraphics[width=\textwidth]{./graphs.png}
     66     \caption{Three graphs, labeled as A, B and C\label{fig: graphs}}
     67 \end{figure}
     68 \begin{enumerate}
     69     \item Diameter $B \rightarrow C \rightarrow A$
     70     \item Density  $A \rightarrow C \rightarrow B$
     71     \item Average clustering coefficient  $A \rightarrow B \rightarrow C$
     72 \end{enumerate}
     73 
     74 \section{Three Graphs}
     75 We have three graphs $X_1, X_2, X_3$. From which two are real world graphs
     76 and one is a ER network, below ist the data of these three networks, where
     77 $n$ is the number of nodes, $L$ the number of edges and $\langle C \rangle$
     78 is the average clustering coefficient
     79 
     80 \begin{center}
     81 \begin{tabular}{ l | c c c }
     82     \hline
     83           & $n$ & $L$ & $\langle C\rangle$\\
     84     \hline
     85     $X_1$ & 4941   & 6594     & 0.08   \\
     86     $X_2$ & 125    & 560      & 0.07      \\
     87     $X_3$ & 256985 & 7778954  & 0.009      \\
     88     \hline
     89 \end{tabular}
     90 \end{center}
     91 According to the distribution law of the ER random network we can calculate
     92 the avarage clustering coefficient with the number of nodes and number of
     93 edges of the network.
     94 \begin{align}
     95     \langle C\rangle = \frac{\langle k \rangle}{N} = \frac{L}{N^2} \simeq
     96     \begin{cases}
     97         0.0026 & \text{for} \;\;\; X_1 \\
     98         0.038  & \text{for} \;\;\; X_2 \\
     99         0.000117 & \text{for} \;\;\; X_3 \\
    100     \end{cases}
    101 \end{align}
    102 For graph $X_2$ the avrage clustering coefficient is $\langle C\rangle =
    103 0.07$, and according to the calculation $0.038$ which is the closest we got
    104 with in comparison to other graphs. Meaning $X_2$ is most likely the graph
    105 that models an ER network .
    106 
    107 \nocite{code}
    108 \printbibliography
    109 
    110 \end{document}