Consider the directed network G = (V, E) with N = 5 nodes and L = 8 links, in which
node 1 points to nodes 2 and 3, node 2 points to node 4, node 3 points to nodes 2 and
4, node 4 points to node 2, and node 5 points to nodes 3 and 4.
(a) Draw the network and write down its adjacency matrix A.
(b) How many weakly-connected components and how many non-trivial (i.e. with
more than one node) strongly-connected components are there in the network?
List all the nodes belonging to each one of these components. List all the nodes
belonging, respectively, to the in-component and the out-component of each of the
non-trivial strongly-connected components.
(c) Determine the in-degree sequence {kin, kin, kin, kin, kin} and the out-degree
sequence {kout, kout, kout, kout, kout} of the network. Write down the average node
in-degree, the average node out-degree, the node in-degree distribution Pin(k) and
the node out-degree distribution Pout (k).
(d) Calculate the normalised in-degree centrality x₁ of each node of the network and
rank the nodes, from the most to the least central, according to their in-degree
centrality.
(e) Calculate the eigenvector centrality x₁ of each node of the network and rank the
nodes, from the most to the least central, according to their eigenvector centrality.
To obtain the eigenvector centrality, start from the initial guess x(0) = 1 where
1 is the N-dimensional column vector of elements 1₁ = 1 Vi = 1,2..., N, and use
the following recursive rule
x(n) = Ax(n-1),
where n ∈ N. Finally calculate the eigenvector centrality x₁ of each node i of the
network from the limit
x(n)
x₁ = lim
n→∞
Σ
(n)
Can you obtain the same result by directly calculating eigenvalues and
eigenvectors of the adjacency matrix?
(f) State the definition of the Katz centrality. Calculate the Katz centrality x₁ of each
node of the network and rank the nodes, from the most to the least central,
according to their Katz centrality. Do you get a different ranking from that
obtained based on the eigenvector centrality? Explain why.
(g) Calculate the NX N matrix d whose element dij is the length of the shortest
paths from node i to node j. Calculate the efficiency centrality 2 of each node of
the network and rank the nodes, from the most to the least central, according to
their efficiency centrality.