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]