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 Π II₁ = z = Σ αγ j=1,...,N(t-1) Provide the mean-field solution of the model by considering the following two points. (A) Assume that Zat, where a indicates the average of a over the distribution (a). Derive the time evolution k = k(t) of the expected degree k; of a node i in the mean-field approximation.
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 Π II₁ = z = Σ αγ j=1,...,N(t-1) Provide the mean-field solution of the model by considering the following two points. (A) Assume that Zat, where a indicates the average of a over the distribution (a). Derive the time evolution k = k(t) of the expected degree k; of a node i in the mean-field approximation.
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: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
Π II₁ =
z =
Σ αγ
j=1,...,N(t-1)
Provide the mean-field solution of the model by considering the
following two points.
(A) Assume that
Zat,
where a indicates the average of a over the distribution (a).
Derive the time evolution k = k(t) of the expected degree k; of a node
i in the mean-field approximation.
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