Antonio DCDN - MO412

Consider the following adjacency matrix of an undirected network: \[ A= \begin{bmatrix} 0&1&1&0&0&0&0\\ 1&0&1&1&0&0&0\\ 1&1&0&1&1&0&0\\ 0&1&1&0&0&1&0\\ 0&0&1&0&0&1&1\\ 0&0&0&1&1&0&1\\ 0&0&0&0&1&1&0 \end{bmatrix} \] Consider the candidate communities \[ C_1=\{1,2,3,4\} \qquad C_2=\{5,6,7\} \] Which statement is correct? \(C_1\) is a clique and \(C_2\) is a strong community \(C_1\) is a strong community but not clique, while \(C_2\) is a clique Both \(C_1\) and \(C_2\) are strong communities but not cliques \(C_1\) is weak only and \(C_2\) is a clique None of the above Original idea by: Antonio De Cesare Del Nero

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