Consider a network that follows the Barabási-Albert model with \(m_0 = m = 2\). A node \(i\) joined the network at time \(t_i = 375\) and has degree \(k_i(t) = 40\) at the current time step. What is the expected diameter of this network now? Round to two decimal places.
Original idea by: Carlos Trindade
A secret organization suspects that some gang members form closed communication groups, that is, groups in which any member can send or receive messages from other members of the group, passing only through members of that group. To understand the communication traffic between these members, the organization created the following graph.
What are the closed communication groups of the gang members?
A) {Michael, Sophie, Daniel, Olivie, Lucas, Emma, Ryan}, {Chloe, Ethan}
B) {Michael, Sophie, Daniel}, {Olivie, Lucas}, {Emma, Ryan, Chloe}, {Ethan}
C) {Chloe}, {Ethan}, {Michael, Sophie, Daniel, Olivie, Lucas}, {Emma, Ryan}
D) {Michael, Sophie, Daniel, Olivie}, {Lucas, Emma, Ryan}, {Chloe}, {Ethan}
E) None of the above
Original idea by: Angelica Santos
In the study of Network Flow, the Ford-Fulkerson algorithm relies on constructing a Residual Network
Which of the following pairs of Original Network
Pair I:
The alternative that contains a correct statement is:
Original idea by: Yuri S. Costa
During a peak usage period following a power outage, the network of a community center began operating using packet switching, where data is divided into small packets and transmitted across the network, and bandwidth is provided on demand.
The infrastructure is composed of devices interconnected by links with limited capacities. Each link has a maximum transmission capacity in (MB/s), representing how much data can be transmitted. The main objective of the system is to maximize the total flow of data (packets) sent from S to T.
Since packets can follow different paths to reach the destination, the network can be modeled as a directed graph. To ensure the best possible performance, we aim to determine the maximum flow of packets that can be transmitted from the source to the destination. A schematic drawing of the community center's network with capacities is given below.
Original idea by: Tássia Martins
Cynthya is a big fan of the band Suplaz, which rarely performs in Brazil. She just found out they’ll be at Rock in Rio festival and is thrilled. She is the manager of the Capanema Suplaz fan group, so she decides to organize transportation to help as many fans as possible from her city attend the concert.
Since the trip is long, she sets up a system of shared rides and transfers: fans can travel between cities using different cars, vans, or buses, meeting at stops along the way. At each segment of the trip, there is a limit to how many people can be transported, depending on vehicle availability. Fans may switch vehicles at intermediate stops.
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The figure above represents this transportation system as a directed network. Capanema is the source (S) and Rio is the sink (T). The intermediate nodes (A, B, C, and D) represent transfer points, like hotels or meeting locations. Each edge has a capacity, indicating the maximum number of people that can travel along that segment.
What is the maximum flow from Capanema to Rio, which corresponds to the maximum number of fans that can be transported?
A) 8
B) 10
C) 20
D) 12
E) None of the above
Original idea by: Melissa Araújo
As a resistance leader, you have obtained a schematic map of the dictatorship communication cables. Because of the technology used, the signal travels one-way only, as depicted in the map. To launch your attack, you must completely cut all paths from the Capital (s) to the Military Base (t).
Each one-way cable is guarded. The cost to destroy a cable (in C4 charges and personnel) is shown in the map. You have a total budget of B = 10.
Which of the following statements is true regarding your sabotage plan?
a) The cheapest way to disconnect the base costs 11, so your budget of B = 10 is not enough to go through with the plan.
b) You can successfully implement the plan for a cost of 9 by destroying the cables (a, t) and (b, t).
c) The most efficient plan is to hit cables (s, b) and (b, a), which only costs 10 and leaves the base isolated.
d) You would need to destroy every cable connected to the Capital, costing you exactly 18.
e) None of the above
Original idea by: Luis Alberto Vásquez Vargas
On Easter Sunday, due to high demand, airlines had to limit the number of planes that could fly over certain regions. Help Zula Airlines maximize the number of planes that could travel on April 4, 2026, at 2 PM, through the areas shown in the figure below, where SP is the source node and AM is the sink node. In this network flow problem, what would be the residual network formed from the flows and capacities shown?
a.)
b.)
c.)
d.)
e) None of the above
Original idea by: Julia de Pietro Bigi
