Question 2. Random Networks


Imagine a Cayley Tree, constructed starting from a central node of degree k. Each node at distance d from the central node has degree k, until the nodes reachable at distance P that have degree one, as shown in figure above. Considering that the number of nodes reachable in t steps from the central node is computed as k(k-1)t-1, and its average number of links <L> = N-1, being N the number of nodes, what is the probability of conexion between nodes if k=4 and t=4?
 

A. 1/161

B. 2/161

C. 1/41

D. 2/41

E. None of above

Original idea by: Germán Darío Buitrago Salazar

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Question 6. Modularity

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  Imagine a Cayley Tree, constructed starting from a central node of degree k. Each node at distance d from the central node has degree k, until the nodes reachable at distance P that have degree one, as shown in figure above. Consider that there are k+1 communities, where each community corresponds to a leaf of the network and the central node is other community. Considering that the number of nodes reachable in t steps from the central node is computed as k(k-1) t-1 , and  number of links L = N-1, being N the number of nodes, what is the modularity if k=3 and t =4? Round it up to three decimals places. Tip: Try to model the total number of links within the community , L c, and the total degree of the nodes in the community k c as a function of k and t. A. 0.621 B. 0.523 C. 0.785 D. 0.591 E. None of the above. Original idea by: Germán Darío Buitrago Salazar
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Question 3. Calculus

Consider a random network with constant probability of conexion p = 0.1 at all times and a number of nodes that increases over time according to N(t)=10t. What would it be the function that estimates the variation of average links over time (d<L> /dt) a . 10t - 1 b. 0.1 * [10t - 1] c. 10t - 1/2 d. 0.1 * [10t - 1/2] e. None of above Original idea by: Germán Darío Buitrago Salazar
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