= 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: ki – 1 II¿ Z N(t-1) with Z = (k; - 1) j=1 where k¿ is the degree of node i, and N(t − 1) is the number of nodes in the network at time t - 1. (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 k¿(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: ki – 1 II¿ Z N(t-1) with Z = (k; - 1) j=1 where k¿ is the degree of node i, and N(t − 1) is the number of nodes in the network at time t - 1. (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 k¿(t;) = m, where t¿ is the time of arrival of node i.
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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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:
ki – 1
II¿
Z
N(t-1)
with Z = (k; - 1)
j=1
where k¿ is the degree of node i, and N(t − 1) is the number of nodes in the network at
time t - 1.
(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 k¿(t;) = m, where t¿ is the time of arrival of node i.
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