Considering an undirected network with no degree correlation. Let \(e_{jk}\) be the probability to find a node with degree \(j\) and degree \(k\) at the two ends of a randomly selected link and let \(q_k\) be the probability to have a degree \(k\) node at the end of a randomly selected link. You are given the average degree:
\(\langle k\rangle = 10\)
and the following degree-distribution probability masses (all other \(p_k\) can be anything consistent and are not needed):
\(p_{20} = 0.04\)
\(p_{30} = 0.03\)
\(p_{50} = 0.02\)
Using only the information above, compute the 3×3 submatrix of \(e_{jk}\) for degrees 20, 30, and 50. Which option matches this block? Consider the theoretical e_jk for the network with no degree correlation and that the matrix rows and columns correspond to degrees 20, 30, and 50 in that order.
A) \(\left[ \begin{array}{ccc} 0.0064 & 0.0072 & 0.0080\\ 0.0072 & 0.0081 & 0.0090\\ 0.0080 & 0.0090 & 0.0100\end{array}\right]\)
B) \(\left[ \begin{array}{ccc} 0.0032 & 0.0045 & 0.0060\\ 0.0045 & 0.0075 & 0.0080\\ 0.0060 & 0.0080 & 0.0090\end{array}\right]\)
C) \(\left[ \begin{array}{ccc} 0.0080 & 0.0090 & 0.0100\\ 0.0090 & 0.0100 & 0.0110\\ 0.0100 & 0.0110 & 0.0120\end{array}\right]\)
D) \(\left[ \begin{array}{ccc} 0.0064 & 0.0080 & 0.0096\\ 0.0080 & 0.0100 & 0.0120\\ 0.0096 & 0.0120 & 0.0144\end{array}\right]\)
E) None of the above.
Original idea by: Thiago Soares Laitz
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