I (a) Find an expression for the number of nodes, N(t), and the number of links, L(t), in the network as a function of time t. Find an expression for the value of Z as a function of time t. (b) What is the average node degree (k) at time t? What is the average node degree in the limit t→ ∞? (c) Write down the differential equation governing the time evolution of the degree ki of node i for t≫ 1 in the mean-field approximation. Solve this equation with the initial condition ki(t) = m, where t; is the time of arrival of node i. 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.
I (a) Find an expression for the number of nodes, N(t), and the number of links, L(t), in the network as a function of time t. Find an expression for the value of Z as a function of time t. (b) What is the average node degree (k) at time t? What is the average node degree in the limit t→ ∞? (c) Write down the differential equation governing the time evolution of the degree ki of node i for t≫ 1 in the mean-field approximation. Solve this equation with the initial condition ki(t) = m, where t; is the time of arrival of node i. 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
Related questions
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Transcribed Image Text:I
(a) Find an expression for the number of nodes, N(t), and the number of links, L(t),
in the network as a function of time t. Find an expression for the value of Z as a
function of time t.
(b) What is the average node degree (k) at time t? What is the average node degree
in the limit t→ ∞?
(c) Write down the differential equation governing the time evolution of the degree ki
of node i for t≫ 1 in the mean-field approximation. Solve this equation with the
initial condition ki(t) = m, where t; is the time of arrival of node i.
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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