Consider the following growing network model in which each node i is assigned an attractiveness a; EN+ drawn from a distribution (a). Let N(t) denote the total number of nodes at time t. At time t = 1 the network is formed by two nodes joined by a link. - At every time step a new node joins the network. Every new node has initially a single link that connects it to the rest of the network. At every time step t the link of the new node is attached to an existing node i of the network chosen with probability II, given by where ai Π; = Z' Z = Σ aj. j=1,...,N(t−1) Provide the mean-field solution of the model by considering the following two points. (A) Assume that Zāt, where a indicates the average of a over the distribution (a). Derive the time evolution ki = ki(t) of the expected degree k; of a node i in the mean-field approximation. (B) Assume that π(a) = { 1 for a = 1, 0 for a 1, and that Z ~āt. Derive the degree distribution P(k) of the network for large times, i.e. t1, in the mean-field approximation.

A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Consider the following growing network model in which each node i is
assigned an attractiveness a; EN+ drawn from a distribution (a).
Let N(t) denote the total number of nodes at time t.
At time t = 1 the network is formed by two nodes joined by a link.
-
At every time step a new node joins the network. Every new node has
initially a single link that connects it to the rest of the network.
At every time step t the link of the new node is attached to an existing
node i of the network chosen with probability II, given by
where
ai
Π;
= Z'
Z =
Σ aj.
j=1,...,N(t−1)
Provide the mean-field solution of the model by considering the
following two points.
(A) Assume that
Zāt,
where a indicates the average of a over the distribution (a).
Derive the time evolution ki = ki(t) of the expected degree k; of a node
i in the mean-field approximation.
(B) Assume that
π(a) = {
1 for a = 1,
0 for a 1,
and that Z ~āt.
Derive the degree distribution P(k) of the network for large times, i.e.
t1, in the mean-field approximation.
Transcribed Image Text:Consider the following growing network model in which each node i is assigned an attractiveness a; EN+ drawn from a distribution (a). Let N(t) denote the total number of nodes at time t. At time t = 1 the network is formed by two nodes joined by a link. - At every time step a new node joins the network. Every new node has initially a single link that connects it to the rest of the network. At every time step t the link of the new node is attached to an existing node i of the network chosen with probability II, given by where ai Π; = Z' Z = Σ aj. j=1,...,N(t−1) Provide the mean-field solution of the model by considering the following two points. (A) Assume that Zāt, where a indicates the average of a over the distribution (a). Derive the time evolution ki = ki(t) of the expected degree k; of a node i in the mean-field approximation. (B) Assume that π(a) = { 1 for a = 1, 0 for a 1, and that Z ~āt. Derive the degree distribution P(k) of the network for large times, i.e. t1, in the mean-field approximation.
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