MO412 - Christian's Blog

Consider a simple scale-free network with \( k_{min} = 1 \), \( \langle k \rangle = 10 \), \( \gamma = 2.5 \) and \( N \) nodes. For which values of \( N \) is it expected to observe the structural cutoff phenomenon? Tip: establish a relationship between \( k_s(N) \) and \( k_{max} \). \( N > 2500 \) \( N < 2500 \) \( N > 1000 \) \( N < 1000 \) None of the above Original idea by: Christian Konishi

Consider the following directed graph: In order to apply the Kosaraju-Sharir algorithm, a DFS was performed and the following start and end times were computed: What will be the first strongly connected component detected in this graph? {7}. {4, 5, 6}. {2, 3, 4}. {1, 2, 3}. None of the above Original idea by: Christian Konishi

Assume that a certain network can be approximated by a particular continuous model, and its number of nodes increases at a rate given by the function \( f(t) = e^{\frac{-t}{1000}+10} \). Given that at \( t=0 \), the number of nodes is 0, what is the approximate maximum number of nodes that this network can have? Tip: get an expression for the number of nodes \( N(t) \) and its limit when \( t \rightarrow \infty \). \( e^{\frac{-1}{3}+10} \). \( 1000\ e^{\frac{-1}{3}+10} \). \( 1000\ e^{10} \). \( e^{10} \). None of the above Original idea by: Christian Konishi

Suppose that we are trying to project a distributed system with random architecture. Every time a new computer \( i \) tries to ingress, for each existing machine \( j \) in the network, a connection \( \left( i, j \right) \) may be created with a probability of \( p \). For this architecture, it is desired that any computer must be reachable from any node, however a few exceptions are tolerable. Also, \( p \) cannot be too big, because of cost reasons. Which one of these options presents a reasonable choice for the value of \( p \)? Assume that \( N \) is the number of expected computers in the network, and it is large enough to apply the law of large numbers. \( p = \frac{1}{N} \), because it is the critical point. \( p = \frac{ln\ N}{N} \), because it is the boundary between the supercritical and the connected regime. \( p = \frac{ln\ N}{ln\ \langle k \rangle} \), because it is the boundary between the supercritical and the connected regime. \( p

The following network represents the way the computers in a distributed system are connected to each other. The communication between computers flows through the links in any direction. The system started presenting performance issues, after some investigation, the cause was determined to be due to nodes communication bottleneck. By looking at the network, which one of the options presents both a proper hypothesis for this bottleneck cause and a reasonable solution for this? Node 6 has a high clustering coefficient, which indicates a network vulnerability for failures. By removing node 6, the system communication should improve. Node 4 may be overloaded since many nodes depends on it to communicate with each other. By adding a link (3,4), performance should improve. Link (3,5) may be overloaded, since it is a bridge and if removed, two large connected components is created. By adding a link (2,6), performance should improve. The network diameter is 5 c

Consider the bipartite network in which the nodes \( A \) are animals and the nodes \( E \) are environments. If an animal \( a \) lives in an environment \( e \), then there is a link between \( a \) and \( e \). Take the following image as an example of part of this network. If a scientist wants to know possible interactions among animals, they need to know which ones live together. One way to determine this it to: Get all nodes of \( A \), only the ones with the same degree live in the same environment Calculate the probabilistic distribution of degrees, if \( p_k \) is the same for two values \( k_1 \) and \( k_2 \), then the nodes of \( A \) with degree \( k_1 \) live together with the ones of \( k_2 \) Check which nodes of \( A \) are connected, if they are, they share the same environment Calculate the projection \( A \), if two animal nodes are connected in the projection, then they share the same environment None of the above