network. Consider the following model for a growing simple We adopt the following notation: N and...

  

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network. Consider the following model for a growing simple We adopt the following notation: N and L indicate respectively the total number of nodes and links of the network, A,, indicates the generic element of the adjacency matrix A of the network, k, indicates the degree of node i and (k) indicates the average degree of the network. At time = 1 the network is formed by a no = 6 nodes mo= 6 links. At every time step!> 1 the network evolves according to the following rules: - A link (r,s) between a node r and a nodes is chosen randomly with uniform probability Ars (rs) L and is removed from the network. - A single new node joins the network and is connected to the rest of the network by m links with m fixed to a time-independent integer constant satisfying 2 1 in the mean-field, continuous approximation. [6] e) Show that the degree distribution derived in the mean-field approximation is power-law in the limit of large network sizes. 1151 [6] f) Indicate with the power-law exponent of the degree distribution found in the mean-field approximation. Derive the dependence of y on m. g) For which values of m is the network scale-free? [3] [4] network. Consider the following model for a growing simple We adopt the following notation: N and L indicate respectively the total number of nodes and links of the network, A,, indicates the generic element of the adjacency matrix A of the network, k, indicates the degree of node i and (k) indicates the average degree of the network. At time = 1 the network is formed by a no = 6 nodes mo= 6 links. At every time step!> 1 the network evolves according to the following rules: - A link (r,s) between a node r and a nodes is chosen randomly with uniform probability Ars (rs) L and is removed from the network. - A single new node joins the network and is connected to the rest of the network by m links with m fixed to a time-independent integer constant satisfying 2 1 in the mean-field, continuous approximation. [6] e) Show that the degree distribution derived in the mean-field approximation is power-law in the limit of large network sizes. 1151 [6] f) Indicate with the power-law exponent of the degree distribution found in the mean-field approximation. Derive the dependence of y on m. g) For which values of m is the network scale-free? [3] [4]