代写MTH6142 Complex Networks Assessed Coursework 5代做Python程序

MTH6142 Complex Networks

Assessed Coursework 5

Consider the following growing network model in which each node i is assigned an attractiveness  drawn from a distribution π(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 Πi given by

where

Provide the mean-field solution of the model by considering the following two points.

(A) Assume that

where a indicates the average of a over the distribution π(a).

Derive the time evolution ki = ki(t) of the expected degree ki of a node i in the mean-field approximation. [2 MARKS]

(B) Assume that

and that 

Derive the degree distribution P(k) of the network for large times, i.e.  in the mean-field approximation. [2 MARKS]