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2. Watts-Strogatz (WS) Model a) Explain why the WS model can generate networks closer to real ones than the G(N,p) model.

2. Watts-Strogatz (WS) Model a) Explain why the WS model can generate networks closer to real ones than the G(N,p) model.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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2. Watts-Strogatz (WS) Model a) Explain why the WS model can generate networks closer to real ones than the G(N,p) model. The WS initial configuration is a k-regular circular graph, meaning that all nodes are arranged b) on a circle and each node has exactly k neighbours, with k being an even number. Show that if k « N, the average clustering coefficient for this initial configuration for a general k and a graph with N nodes is given by 3(k – 2) (C) = 4(k – 1)' Suggestions: Fix a node and count the number of links for each of its neighbours. • Decide how many of them should be used to calculate the clustering coefficient of the node. Remember this is a regular graph. If k « N, one can assume that the most distant neighbours at each side of a fixed node do not touch themselves. Plot the quantity (C)/(k) against the number of nodes in the network (varying from 1 to 100) c) for the fixed value of (k) = 6 for both the regular network of item (b) and a G(N,p) model with p = 1/2. If we plot in the same figure the result of a WS model with initial configuration the regular graph with k = 6 and a rewiring probability less than 1, where do you expect that line to be? Explain your answer based on how the algorithm works. %3D
Transcribed Image Text:2. Watts-Strogatz (WS) Model a) Explain why the WS model can generate networks closer to real ones than the G(N,p) model. The WS initial configuration is a k-regular circular graph, meaning that all nodes are arranged b) on a circle and each node has exactly k neighbours, with k being an even number. Show that if k « N, the average clustering coefficient for this initial configuration for a general k and a graph with N nodes is given by 3(k – 2) (C) = 4(k – 1)' Suggestions: Fix a node and count the number of links for each of its neighbours. • Decide how many of them should be used to calculate the clustering coefficient of the node. Remember this is a regular graph. If k « N, one can assume that the most distant neighbours at each side of a fixed node do not touch themselves. Plot the quantity (C)/(k) against the number of nodes in the network (varying from 1 to 100) c) for the fixed value of (k) = 6 for both the regular network of item (b) and a G(N,p) model with p = 1/2. If we plot in the same figure the result of a WS model with initial configuration the regular graph with k = 6 and a rewiring probability less than 1, where do you expect that line to be? Explain your answer based on how the algorithm works. %3D
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