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universal10 - Universal Network Structure and Generative...

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Universal Network Structure and Generative Models Networked Life CIS 112 Spring 2010 Prof. Michael Kearns
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A Little Warm-Up… C o ns ide r yo urs e lf “c o nne c te d” to  a nyo ne  in c la s s  who s e   firs t na m e  yo u kno w (a s s um e  s ym m e tric ) O n the  re s ulting  ne two rk, le t’s  e xa m ine : The  de g re e  dis trib utio n The  num b e r a nd s ize  o f c o nne c te d c o m po ne nts The  dia m e te r The  “c lus te ring  c o e ffic ie nt”
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“Natural” Networks and Universality Consider the many kinds of networks we have examined: social, technological, business, economic, content,… These networks tend to share certain  informal  properties: large scale; continual growth distributed, organic growth: vertices “decide” who to link to interaction (largely) restricted to links mixture of local and long-distance connections abstract notions of distance: geographical, content, social,… Do natural networks share more  quantitative  universals? What would these “universals” be? How can we make them precise and measure them? How can we explain their universality? This is the domain of  network science
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Some Interesting Quantities Connected components: how many, and how large? Network diameter: the small-world phenomenon Clustering: to what extent do links tend to cluster “locally”? what is the balance between local and long-distance connections? what roles do the two types of links play? Degree distribution: what is the typical degree in the network? what is the overall distribution? Etc. etc. etc.
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A “Canonical” Natural Network has… Few  connected components: often only 1 or a small number (compared to network size) Small  diameter: often a constant independent of network size (like 6…) or perhaps growing only very slowly with network size typically look at average; exclude infinite distances high  degree of edge clustering: considerably more so than for a random network in tension with small diameter heavy-tailed  degree distribution: a small but reliable number of high-degree vertices quantifies Gladwell’s connectors often of  power law  form
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Some Models of Network Formation Random graphs (Erdos-Renyi model): gives few components and small diameter does not give high clustering or heavy-tailed degree distributions is the mathematically most well-studied and understood model Watts-Strogatz and related models: give few components, small diameter and high clustering does not give heavy-tailed degree distributions Preferential attachment: gives few components, small diameter and heavy-tailed distribution
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This note was uploaded on 02/19/2010 for the course CIS 112 taught by Professor Kearns during the Spring '06 term at UPenn.

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universal10 - Universal Network Structure and Generative...

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