Question 9 - The crazy Network

Pedro was really interested in last algorithms class. So he started thinking and creating crazy networks. He proposed this: 

• First, he restricted the BA model, so when adding a new node with \(m\) links, each link has to go to a different node (probabilities have to be calculated at each new link, excluding the nodes already linked to the new one). 
• Now he established that \(m\) should follow the t-Fibonacci sequence. Where for a given \(t\) we have \(tFib(t) = t * Fib(t)\). (\(Fib(t)\) is the standard Fibonacci sequence)
• Finally, for each node he adds, he deletes \(2^t\) nodes. 

Of course, at this rate, soon he could not add anymore node to the network preserving the constraints he impose. 

Pedro started the network with 9876 nodes. Can you help him find the time \(t\) where he will not be able to follow the crazy network constraints?

1. \(t = 10\)
2. \(t = 11\)
3. \(t = 12\)
4. \(t = 13\)
5. None of the above.

  Original idea by: Pedro Henrique Di Francia Rosso

Question 8 - BA Model

On the Non-linear preferential attachment, our probability becomes \(Π(k) \sim k^\alpha\). 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?

1. \(22026\) nodes; \(10000\) nodes; \(100\) nodes respectively
2. \(1024\) nodes; \(1000\) nodes; \(100\) nodes respectively
3. \(22026\) nodes; \(10000\) nodes; \(200\) nodes respectively
4. \(1024\) nodes; \(1000\) nodes; \(200\) nodes respectively
5. None of the above.

  Original idea by: Pedro Henrique Di Francia Rosso