Network Analysis Questions PDF

**1.** Domanda 1 Consider the undirected network represented in the picture. Immagine che contiene cerchio, schermata Descrizione generata automaticamente Consider a symmetric version of the Connections Model with the following utility function ui(g)=∑j≠iδl(i,j)−∑j∈Ni(g)c*ui*​(*g*)=∑*j*​=*i*​*...

**1.** Domanda 1 Consider the undirected network represented in the picture. Immagine che contiene cerchio, schermata Descrizione generata automaticamente Consider a symmetric version of the Connections Model with the following utility function ui(g)=∑j≠iδl(i,j)−∑j∈Ni(g)c*ui*​(*g*)=∑*j*​=*i*​*δl*(*i*,*j*)−∑*j*∈*Ni*​(*g*)​*c*, and suppose δ=0.5*δ*=0.5 and c=0.4*c*=0.4. What is the utility of node 2, u2(g)*u*2​(*g*), in the pictured network? 3\*0.5 + 0.25 - 3\*0.4 4\*0.5 - 3\*0.4 3\*0.5 + 0.25 - 4\*0.4 3\*0.5 + 0.25 - 3\*0.4 - 0.16 **2.** Domanda 2 Consider a symmetric version of Connections Model with δ=0.999*δ*=0.999 and c=1.4*c*=1.4 and a society of n=3*n*=3 people. \[Therefore the values are depicted in the picture.\] In which row(s) are the network(s) **Pairwise stable**?  b d a c **3.** Domanda 3 Consider a symmetric version of Connections Model with δ=0.999*δ*=0.999 and c=1.4*c*=1.4 and a society of n=3*n*=3 people. \[Therefore the values are depicted in the picture.\] In which row(s) are the network(s) **Pareto Efficient**? Immagine che contiene schermata, cerchio Descrizione generata automaticamente b d a c **4.** Domanda 4 Consider a symmetric version of the Connections Model with δ=0.999*δ*=0.999 and c=1.4*c*=1.4 and a society of n=3*n*=3 people. \[Therefore the values are depicted in the picture.\] In which row(s) are the network(s) **Efficient** (maximizing the total sum of utilities)?  b d a c **5.** Domanda 5 Consider a symmetric version of the Connections Model in a society with n=5*n*=5 people and δ=0.5*δ*=0.5. Which are the lower and upper bounds of the cost c*c*, such that star networks are efficient? 0.25, 0.5 0.5, 1 0.25, 0.875 0.25, 1 1 punto **6.** Domanda 6 (Lecture 4.4) Consider a symmetric version of the Connections Model in a society with n=7*n*=7 people and with δ=0.5*δ*=0.5 and c=1*c*=1. Which of the following describes the efficient network architectures? The complete network Star networks involving all nodes The empty network None of the above **7.** Domanda 7 Consider a symmetric version of the Connections Model in a society with n=7*n*=7 people and δ=0.5*δ*=0.5. For which of the following costs c*c* is (are) star networks pairwise stable? 0.8 0.4 0.6 0.2 **8.** Domanda 8 In the picture, consider adding a link 12 to the network g = {13, 24}. So the left network is g, and the right network is g + 12. The number next to each node is the node\'s utility from that network. Regarding the externality to node 3 brought by adding link 12, which of the following statements is correct? Immagine che contiene cerchio, schermata Descrizione generata automaticamente Adding link 12 has a positive externality on node 3. Adding link 12 has a negative externality on node 3. Neither of the above. 1 punto **9.** Domanda 9 Consider the undirected network in the picture using \"Coauthor\" model (as described in this video). What is the utility of agent 1?  2.75 3 3.5 4 1 punto **10.** Domanda 10 Consider undirected networks on 3 nodes, with the utilities of nodes depicted in the picture. Choose the option that correctly answers the following two questions: \(1) In which row are the network(s) **Efficient**? \(2) In which row are the network(s) **Pairwise stable**? Immagine che contiene schermata, oscurità, cerchio Descrizione generata automaticamente \(1) c, (2) a \(1) b, (2) a \(1) b, (2) c \(1) c, (2) b **11.** Domanda 11 In the pairwise stable structure of network(s) described in this video for Islands connections model (Jackson&Rogers 2004), which of the following properties are correct? Diameter increases proportionally to the total number of nodes, as the number of islands goes up (fixing the number of nodes in each island) (Relatively) high clustering (Relatively) low diameter Average degree increases proportionally to the total number of nodes, as the number of islands goes up (fixing the number of nodes in each island) **12.** Domanda 12 (Lecture 4.10) Consider the networks with 3 nodes and the utilities as shown in the picture. Which network(s) are Nash stable?  b d a c **13.** Domanda 13 (Lecture 4.10) Consider the networks with 3 nodes as shown in the picture. Which network(s) are Pairwise Nash stable? Immagine che contiene schermata, cerchio, oscurità Descrizione generata automaticamente b d a c **14.** Domanda 14 Consider connections model with δ−δ2*δ*−*δ*2 \< c*c* \< δ*δ*, and the following process described by A. Watts (2001): - a link is picked uniformly at random; - the link is added if that weakly benefits both players (at least one strictly); - the link is deleted if that strictly benefits either player; - go back to the first bullet point and repeat. Which statement(s) are true? \[This is to guide you through the Proposition (A. Watts (2001)), as covered in this video.\] Star networks involving all nodes are efficient. When the above process reaches a star network with all players, it may still move forward to other networks. Star networks involving all nodes are pairwise stable. As the number of nodes n grows, the probability that the above process stops at a star network goes to 0. **15.** Domanda 15 Consider all undirected networks with 3 nodes as depicted in the picture. In which row(s) are the network(s) **Pairwise Nash stable**?  b d a c **16.** Domanda 16 Consider the discussion of improving paths with error ϵ=0.05*ϵ*=0.05 on networks with 3 nodes (as described in this video, with utilities as a function of the network depicted in the picture). Starting from the green network in the middle of third row, what is the **probability of staying at the same network** after one step? Immagine che contiene schermata, cerchio, Policromia Descrizione generata automaticamente 0 1 0.15 0.65 **17.** Domanda 17 Consider a network formation game with directed networks on 2 nodes and with resulting utilities depicted in the picture as a function of the directed links formed (with arrows pointing from the node that formed the link to the node at which it directed the link). Which network(s) are the outcomes(s) of (pure) Nash equilibrium?  b d a c **18.** Domanda 18 Consider the directed connections model with no decay (as described in this video). What is the **payoff to each nodes** in the wheel network in the picture, with n=5 and c=3? Immagine che contiene luna, cerchio, sfera, Oggetto astronomico Descrizione generata automaticamente 3 1 2 5