= 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 m = 2 different nodes already present in the network. The probability II; that a new link is connected to node i is: N(t-1) ki 1 Π = Z - with Z = Σ (ky - 1) j=1 where ki is the degree of node i, and N(t − 1) is the number of nodes in the network at time t — 1. (c) Write down the differential equation governing the time evolution of the degree ki of node i fort 1 in the mean-field approximation. Solve this equation with the initial condition ki (ti) m, where t; is the time of arrival of node i. = (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.