please solve it on paper 2.Let P be a k k stochastic matrix. Show that ₁ P = 1 for all 1 ≤ i ≤ k. Herej=1Pr denotes the entry in the i-th row and j-th column of the n-th power of matrix P.

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Answered: 2. Let P be a k k stochastic matrix.…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 70EQ

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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) II₁ = ki - 1 Z == with Z = (k; -1) j=1 where k; is the degree of node i, and N(t - 1) is the number of nodes in the network at timet - 1. (e) Write down the master equation of the model, i.e. the equation that describes the evolution of the average number N(t) of nodes that at time t have degree k.
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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) II¿ = ki - 1 Ꮓ with Z=(k-1) j=1 where ki is the degree of node i, and N(t - 1) is the number of nodes in the network at timet - 1.
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2. Let P be a k k stochastic matrix. Show that ₁ P = 1 for all 1 ≤ i ≤ k. Here j=1 Pr denotes the entry in the i-th row and j-th column of the n-th power of matrix P.