Question 1 (Graph Theory)

In an undirected network, suppose there are exactly 4 triangles and exactly 6 additional connected triplets. According to the definition of the global clustering coefficient, what is the value of \(C_\Delta\)?

  1. \(\frac{1}{2}\)
  2. \(\frac{2}{3}\)
  3. \(\frac{3}{4}\)
  4. \(\frac{6}{7}\)
  5. None of the above
Original idea by: Antonio De Cesare Del Nero

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Question 2 (Random Networks)

In a random network with \(N \gg 1\) and average degree \(\langle k \rangle = 3\), what is the expected fraction of nodes with degree 0 or 1 ? \(2e^{-3}\) \(3e^{-3}\) \(4e^{-3}\) \(1 - e^{-3}\) None of the above Original idea by: Antonio De Cesare Del Nero
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Question 5 (Degree Correlations)

Consider a Barbell graph consisting of two cliques \(K_n\) connected by a path of length \(L\). Assume both \(n\) and \(L\) may grow asymptotically. Evaluate the following statements regarding the assortativity coefficient \(r\): I. If \(L = o(n^2)\), the assortative contribution of the cliques asymptotically dominates the bridge structure. II. If \(L = \Theta(n^3)\), the graph necessarily becomes strongly disassortative. III. When \(L \to \infty\) with fixed \(n\), the relative contribution of hub-hub edges becomes asymptotically negligible. IV. A large number of degree-\(2\) to degree-\(2\) edges does not necessarily imply \(r \to 1\). T, F, T, T T, T, F, T F, F, T, T T, F, F, T None of the above Original idea by: Antonio De Cesare Del Nero
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