Antonio DCDN - MO412

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

In the optimization model, if \( \delta \) is extremely large, what dominates the connection decision? Node degree Path length \(h_{ij}\) Distance \(d_{ij}\) Random choice None of the above Original idea by: Antonio De Cesare Del Nero

Using the continuum formalism, a scale-free network has \[ p(k)=Ck^{-\gamma}, \qquad k \geq 1, \] with \(\gamma = 2.5\). Which expression gives the second moment \(\langle k^2 \rangle\) for a finite network with cutoff \(k_{\max}\)? \(\langle k^2 \rangle = \int_1^{k_{\max}} k^{-0.5}\,dk\) \(\langle k^2 \rangle = C\int_1^{k_{\max}} k^{-0.5}\,dk\) \(\langle k^2 \rangle = C\int_1^{k_{\max}} k^{-2.5}\,dk\) \(\langle k^2 \rangle = \int_1^{k_{\max}} k^{2.5}\,dk\) None of the above Original idea by: Antonio De Cesare Del Nero

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

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