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Then in two steps i can reach 2 other nodes repeating

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Unformatted text preview: in common (i.e., the λ websites that are linked to my website are not linked among themselves). Then in two steps, I can reach λ2 other nodes. Repeating the same reasoning (and maintaining the same unrealistic assumption), in d steps I can reach λd other nodes. Now imagine that this network has n = λd nodes. This implies that the “degrees of separation” (average distance) is d= ln n . ln λ 13 Networks: Lecture 1 Introduction Interpreting Small Worlds (continued) But our unrealistic assumption rules out the reasonable triadic relations and clustering phenomena, which are common both in social networks, web links, and other networks. 4 4 2 2 5a 1 5 1 5b 3 3 6 6 Interestingly, however, in Poisson (Erdos-Renyi) random graphs, we will see that average distance can be approximated for large n by d = ln n / ln λ (where λ is the expected degree of a node). This is because triadic relations shown in the figure are relatively rare in such graphs. 14 Networks: Lecture 1 Introduction Interpreting Small Worlds (continued) This last result in fact can be interpreted as stating that Poisson (Erdos-Renyi) random graphs, though mathematically convenient, will not be good approximations to social networks. This can be seen from the above numbers as well. The Karinthy conjecture, under the Poisson assumption, would require that each person should have had approximately 68 “independent” friends. (exp[ln(1, 500, 000, 000)/5] � 68.5). The Milgram conjecture, of six degrees of separation in the 1960s, would require that each person should have had approximately 41 friends. Instead, most people would be connected to others in remote parts through “special links” (or “connectors”), such as their political representatives, village head, or cousin in a different city etc. Models of small world networks try to capture this pattern (albeit not always perfectly). 15 Networks: Lecture 1 Introduction Social and Economic Networks Most “networked” interactions involve a human element, hence much of network analysis must have some focus on social and economic networks (even when the main interest may be on understanding communication networks). E.g., social network structures, such as Facebook, superimposed over the Internet. In this course, social and economic networks will be our main focus. An important feature of social and economic networks is that they are not only characterized by a pattern of linkages, but also by the interactions that take place over the network structure. Will you lend money to your friend? Will you follow their advice? Will you imitate their behavior? Will you trade with other firms that you are potentially “connected to”? Most of these decisions are strategic, hence the use of game theory. 16 Networks: Lecture 1 Introduction A Central Question What are the commonalities in different (social, economic and other) networks? Diffusion of new technologies and spread of epidemics have certain common features when one looks at their dynamics. Does this mean th...
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