(d) Derive the degree distribution P(k) of the network for t» 1 in the mean-field approximation. Does the model produce scale-free networks? If so, what is the value of the degree exponent y? (e) Write down the master equation of the model, i.e. the equation that describes the evolution of the average number Nk (t) of nodes that at time t have degree k. Consider the following model to grow simple networks. At time t = 1 we start with a complete network with no = 6 nodes. At each time step t> 1 a new node is added to the network. The node arrives together with m = = 2 new links, which are connected to 2 different nodes already present in the network. The probability II; that a new link is connected to node i is: m = N(t-1) Π ki - 1 Z with Z = Σ (ky - 1) j=1 I where ki is the degree of node i, and N(t - 1) is the number of nodes in the network at time t-1.
(d) Derive the degree distribution P(k) of the network for t» 1 in the mean-field approximation. Does the model produce scale-free networks? If so, what is the value of the degree exponent y? (e) Write down the master equation of the model, i.e. the equation that describes the evolution of the average number Nk (t) of nodes that at time t have degree k. Consider the following model to grow simple networks. At time t = 1 we start with a complete network with no = 6 nodes. At each time step t> 1 a new node is added to the network. The node arrives together with m = = 2 new links, which are connected to 2 different nodes already present in the network. The probability II; that a new link is connected to node i is: m = N(t-1) Π ki - 1 Z with Z = Σ (ky - 1) j=1 I where ki is the degree of node i, and N(t - 1) is the number of nodes in the network at time t-1.
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:(d) Derive the degree distribution P(k) of the network for t» 1 in the mean-field
approximation. Does the model produce scale-free networks? If so, what is the
value of the degree exponent y?
(e) Write down the master equation of the model, i.e. the equation that describes the
evolution of the average number Nk (t) of nodes that at time t have degree k.
Transcribed Image Text:Consider the following model to grow simple networks. At time t = 1 we start with a
complete network with no = 6 nodes. At each time step t> 1 a new node is added to
the network. The node arrives together with m = = 2 new links, which are connected to
2 different nodes already present in the network. The probability II; that a new
link is connected to node i is:
m =
N(t-1)
Π
ki - 1
Z
with Z =
Σ (ky - 1)
j=1
I
where ki is the degree of node i, and N(t - 1) is the number of nodes in the network at
time t-1.
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