Quiz Question #2

As the average degree \(\langle k \rangle\) of an Erdős-Rényi random network increases, the network's topology undergoes distinct phases of evolution. Which of the following statements accurately characterizes the network exactly at its critical point (\(\boldsymbol \langle k \rangle = 1\))?

A) The network lacks a giant component, and the size of the largest cluster scales logarithmically with the total number of nodes (\(\mathrm{ln} \ N\)). 

B) A giant component emerges that contains a finite fraction of the network's nodes, and the distribution of cluster sizes is exponential. 

C) The size of the largest component scales as \(N^{\frac{2}{3}}\), containing a vanishing fraction of all nodes, and the cluster size distribution follows a power law. 

D) The giant component absorbs all isolated nodes and clusters, rendering the network fully connected. 

E) None of the above.


Original idea by: João Vianini

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Quiz Question #1

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Figure 1(a) below shows a drawing of a network \(N\). Figures 1(b), 1(c), and 1(d) shows the matrices \(\boldsymbol A\), \(\boldsymbol D\), \(\boldsymbol S\), respectively, and Figure 1(e) shows the vector \(\boldsymbol{k}\).         Regarding the figures above, consider the following six propositions: For matrix \(\boldsymbol A\) (shown in Figure 1(b)) entry \(a_{ij} = 1\) if and only if there is a link in network \(N\) connecting node \(i\) to node \(j\). Therefore, \(\boldsymbol A\) is an adjacency matrix  of network \(N\). Matrix \(\boldsymbol D\) (shown in Figure 1(c)) can be regarded as a matrix indicating second neighbors (nodes at distance two) of network \(N\), since entry \(d_{ij} \neq \infty\) if and only if nodes \(i\) and \(j\) are second neighbors.     Matrix \(\boldsymbol D\) (shown in Figure 1(c)) can be regarded as distance matrix of network \(N\)   , since if  \(d_{ij} = \infty\) there is no path bet...
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