Consider a network described by the Bianconi–Barabási model, with degree dynamics following a power law. Consider also a node A joining this network at time t=10 and another node B joining the network at time t=100. If node B has twice the fitness of node A, at which time point t will the degree of node B start to be greater than or equal to the degree of node A?
Original idea by: Meer Muhammad Khan
Aging can influence the topology of a network. Denote by \( \Pi(k_i,t-t_i) \) the probability of a new node arriving at time \( t \) connecting to node \( i \) of degree \( k_i \), where \( t_i \) is the time node \( i \) was added to the network. Aging can be modeled by choosing \( \Pi(k_i,t-t_i) \sim k_i(t-t_i)^{-\nu} \), where \( \nu \) is a tunable parameter. In this case, what is not correct to state regarding the tunable parameter \( \nu \):
Original idea by: Marcelo Silva
You are analyzing a network that follows the Bianconi-Barabási Model. You know that a node A has degree 5 and fitness 0.2, and a node B has degree 16 and fitness 0.5. How much more likely is a new node to connect to B than to A?
Original idea by: Luan Silveira
The sequence of images below shows some steps for generating the network based on the Barabási-Albert model. Consider that starting with \( m_0 \) nodes, the empty circles mark the newly added node to the network in each timestep \( t \), and the new node's number of links is \( m = 2 \).
Analyze the following statements about the network:
1. Choosing which node will be connected uses the preferential attachment criteria.
2. After \( t \) timesteps, this model generates a network with \( t + m_0 \) nodes.
3. The degree distribution of this network follows the Poisson distribution.
4. This network has the scale-free property.
5. The diameter of this network with N nodes can be estimated by the
equation \( D \sim \frac{\ln N}{\ln \ln N} \)
Select the correct alternative:
Original idea by: Rubens de Castro Pereira
In the Non-linear preferential attachment, the preferential attachment probability becomes \( \Pi(k) \sim k^{\alpha} \). For a starting network with 1 node, approximately how many nodes we have to add on the network for it to achieve \( k_{max} = 100 \), considering the cases where \( \alpha \) is 0.5, 1, and 1.5, respectively?
Original idea by: Pedro Henrique Di Francia Rosso
Consider a scale-free network that has evolved according to the BA model and choose the best possible option given below. Let P and Q be two nodes such that the rate at which node P acquires new links is twice the rate at which node Q acquires new links. If node P joined the network at time 100 units, find the time at which node Q joined the network.
Original idea by: Muhammad Idrees
By considering the Barabási–Albert model for a network, the degree of a node is k=50, and it joined the network at time t=10, what will be the degree of another node which joined the network at time t=100?
Original idea by: Meer Muhammad Khan
Suppose you are simulating a network that follows the Barabási-Albert model. At each step, you add 5 links to this network. One day, you were feeling sleepy, and you decided to stop for coffee. When you returned to your work, you noted that your simulation was at time step 100 thousand, and there was a node whose degree was 500. Puzzled, you wonder when this node was introduced to the network. After some calculations, you figured it out! Which step did you find?
Consider
an undirected network that follows the Barabási-Albert model with \( m = 4\). What is the
network's expected diameter at time \( t = 1000 \) (round to 1 decimal place)?
A critical point for a function \( f \) is a value \( c \) in the domain of \( f \) where \( f'(c) = 0 \) or \( f'(c) \) does not exist. Choose the statement that presents the critical points of $$ f(x) = x^{3/5}(4 - x) $$
Original idea by: Rubens de Castro Pereira
Choose the correct option regarding the following question related to ordinary differential equations. A racer accelerates from a full stop so that her speed \( t \) seconds after starting is \( 10t \) m/s. How far will her car go in 4 seconds?
Original idea by: Muhammad Idrees
The degree of given differential equation \( y' + y = 5/y' \) will be?
Original idea by: Meer Muhammad Khan
The function \( f(x) = x^3 - 6x^2 + 9x - 2 \) has a local maximum value at the point:
Suppose you are analyzing a scale-free network with currently 10000 nodes, degree exponent \( \gamma = 3.0 \), minimum degree of 10 and maximum degree of 1000. However, you know this network is growing over time at a rate of 1000 nodes/year. Assuming that the minimum degree is growing at 10 nodes/year while the degree exponent remains constant, what is currently the rate of change for the largest hub in this network?
(A) 5 nodes/year
(B) 55 nodes/year
(C) 550 nodes/year
(D) 5500 nodes/year
(E) None of the above.
Original idea by: Luan Silveira
Consider functions \( f \) and \( g \) satisfying:
\( f'' = x^{-1/3} \)
\( g = 4 x^5 + 2 x^3 - 3 x + 5 \)
Which option below is compatible with the preceding equations?
Consider the following networks: Which of the following options correctly ranks these networks from most robust to least robust agai...
