Consider the following growing network model in which each node i is assigned an attractiveness a; EN+ drawn from a distribution (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 II, given by where ai Π; = Z' Z = Σ aj. j=1,...,N(t−1) Provide the mean-field solution of the model by considering the following two points. (A) Assume that Zāt, where a indicates the average of a over the distribution (a). Derive the time evolution ki = ki(t) of the expected degree k; of a node i in the mean-field approximation. (B) Assume that π(a) = { 1 for a = 1, 0 for a 1, and that Z ~āt. Derive the degree distribution P(k) of the network for large times, i.e. t1, in the mean-field approximation.

Consider the following growing network model in which each node i is assigned an attractiveness a; EN+ drawn from a distribution (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 II, given by where ai Π; = Z' Z = Σ aj. j=1,...,N(t−1) Provide the mean-field solution of the model by considering the following two points. (A) Assume that Zāt, where a indicates the average of a over the distribution (a). Derive the time evolution ki = ki(t) of the expected degree k; of a node i in the mean-field approximation. (B) Assume that π(a) = { 1 for a = 1, 0 for a 1, and that Z ~āt. Derive the degree distribution P(k) of the network for large times, i.e. t1, in the mean-field approximation.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 29EQ

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