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(d) Derive the degree distribution P(k) of the network for t» 1 in the mean-field approximation. Does the model produce scale-free networks? If so, what is the value of the degree exponent y? (e) Write down the master equation of the model, i.e. the equation that describes the evolution of the average number Nk (t) of nodes that at time t have degree k. 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.

(d) Derive the degree distribution P(k) of the network for t» 1 in the mean-field approximation. Does the model produce scale-free networks? If so, what is the value of the degree exponent y? (e) Write down the master equation of the model, i.e. the equation that describes the evolution of the average number Nk (t) of nodes that at time t have degree k. 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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 28EQ
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(d) Derive the degree distribution P(k) of the network for t» 1 in the mean-field approximation. Does the model produce scale-free networks? If so, what is the value of the degree exponent y? (e) Write down the master equation of the model, i.e. the equation that describes the evolution of the average number Nk (t) of nodes that at time t have degree k.
Transcribed Image Text:(d) Derive the degree distribution P(k) of the network for t» 1 in the mean-field approximation. Does the model produce scale-free networks? If so, what is the value of the degree exponent y? (e) Write down the master equation of the model, i.e. the equation that describes the evolution of the average number Nk (t) of nodes that at time t have degree k.
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.
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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