Write down the differential equation governing the time evolution of the degree ki of node i fort >>> 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 m = 2 different nodes already present in the network. The probability II, that a new link is connected to node i is: N(t-1) II¿ = ki - 1 Ꮓ with Z=(k-1) j=1 where ki is the degree of node i, and N(t - 1) is the number of nodes in the network at timet - 1.
Write down the differential equation governing the time evolution of the degree ki of node i fort >>> 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 m = 2 different nodes already present in the network. The probability II, that a new link is connected to node i is: N(t-1) II¿ = ki - 1 Ꮓ with Z=(k-1) j=1 where ki is the degree of node i, and N(t - 1) is the number of nodes in the network at timet - 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:Write down the differential equation governing the time evolution of the degree ki
of node i fort >>> 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
m = 2 different nodes already present in the network. The probability II, that a new
link is connected to node i is:
N(t-1)
II¿
=
ki - 1
Ꮓ
with Z=(k-1)
j=1
where ki is the degree of node i, and N(t - 1) is the number of nodes in the network at
timet - 1.
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