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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) 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. (f) Suppose now that you iterate the growing process for a finite number of time steps, until you produce a final network with N = 106 nodes (and a minimum degree m = 2). Denote as K the natural cutoff in the network. Treating the degree k as a continuous variable, evaluate the natural cutoff K, the normalisation constant in the degree distribution, the average degree (k), and (k²). (g) Calculate the probability of finding a node with 1000 links in the network obtained in point (f). Calculate the probability of finding a node with 1000 links in a Erdös-Rènyi random graphs with the same number of nodes and links as in the network obtained in point (f).

= 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. (f) Suppose now that you iterate the growing process for a finite number of time steps, until you produce a final network with N = 106 nodes (and a minimum degree m = 2). Denote as K the natural cutoff in the network. Treating the degree k as a continuous variable, evaluate the natural cutoff K, the normalisation constant in the degree distribution, the average degree (k), and (k²). (g) Calculate the probability of finding a node with 1000 links in the network obtained in point (f). Calculate the probability of finding a node with 1000 links in a Erdös-Rènyi random graphs with the same number of nodes and links as in the network obtained in point (f).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 2EQ: 2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed...
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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) 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.
Transcribed Image Text:= 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.
(f) Suppose now that you iterate the growing process for a finite number of time steps, until you produce a final network with N = 106 nodes (and a minimum degree m = 2). Denote as K the natural cutoff in the network. Treating the degree k as a continuous variable, evaluate the natural cutoff K, the normalisation constant in the degree distribution, the average degree (k), and (k²). (g) Calculate the probability of finding a node with 1000 links in the network obtained in point (f). Calculate the probability of finding a node with 1000 links in a Erdös-Rènyi random graphs with the same number of nodes and links as in the network obtained in point (f).
Transcribed Image Text:(f) Suppose now that you iterate the growing process for a finite number of time steps, until you produce a final network with N = 106 nodes (and a minimum degree m = 2). Denote as K the natural cutoff in the network. Treating the degree k as a continuous variable, evaluate the natural cutoff K, the normalisation constant in the degree distribution, the average degree (k), and (k²). (g) Calculate the probability of finding a node with 1000 links in the network obtained in point (f). Calculate the probability of finding a node with 1000 links in a Erdös-Rènyi random graphs with the same number of nodes and links as in the network obtained in point (f).
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