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 x, 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 x; 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 1,1 Vi = 1,2..., N, and use the following recursive rule x(n) = Ax (n-1), where n = N. Finally calculate the eigenvector centrality x; of each node i of the network from the limit x(n) xi = lim Στ (n) Can you obtain the same result by directly calculating eigenvalues and eigenvectors of the adjacency matrix?
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 x, 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 x; 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 1,1 Vi = 1,2..., N, and use the following recursive rule x(n) = Ax (n-1), where n = N. Finally calculate the eigenvector centrality x; of each node i of the network from the limit x(n) xi = lim Στ (n) Can you obtain the same result by directly calculating eigenvalues and eigenvectors of the adjacency matrix?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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 x, 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 x; 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 1,1 Vi = 1,2..., N, and use
the following recursive rule
x(n) = Ax (n-1),
where n = N. Finally calculate the eigenvector centrality x; of each node i of the
network from the limit
x(n)
xi
= lim
Στ
(n)
Can you obtain the same result by directly calculating eigenvalues and
eigenvectors of the adjacency matrix?
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