en>Red Act: /* Algebraic and geometric properties of cross and wedge products */ Minor grammar tweak.

2014-08-14T14:16:06Z

Algebraic and geometric properties of cross and wedge products: Minor grammar tweak.

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	The '''friendship paradox''' is the phenomenon first observed by the sociologist Scott L. Feld in 1991 that most people have fewer friends than their friends have, on average.<ref>{{citation\|first=Scott L.\|last=Feld\|year=1991\|title=Why your friends have more friends than you do\|journal=[[American Journal of Sociology]]\|volume=96\|issue=6\|pages=1464–1477\|jstor=2781907}}.</ref> It can be explained as a form of [[sampling bias]] in which people with greater numbers of friends have an increased likelihood of being observed among one's own friends. In contradiction to this, most people believe that they have more friends than their friends have.<ref>{{citation\|last1=Zuckerman\|first1=Ezra W.\|first2=John T.\|last2=Jost\|year=2001\|url=http://www.psych.nyu.edu/jost/Zuckerman%20&%20Jost%20(2001)%20What%20Makes%20You%20Think%20You%27re%20So%20Popular1.pdf\|title=What makes you think you’re so popular? Self evaluation maintenance and the subjective side of the “friendship paradox"\|journal=Social Psychology Quarterly\|volume=64\|issue=3\|pages=207–223\|doi=10.2307/3090112}}.</ref>

	The same observation can be applied more generally to social networks defined by other relations than friendship: for instance, most people's sexual partners have (on the average) a greater number of sexual partners than they have.<ref>{{citation\|first=Satoshi\|last=Kanazawa\|authorlink=Satoshi Kanazawa\|journal=[[Psychology Today]]\|url=http://www.psychologytoday.com/blog/the-scientific-fundamentalist/200911/why-your-friends-have-more-friends-you-do\|year=2009\|title=The Scientific Fundamentalist: A Look at the Hard Truths About Human Nature\|contribution=Why your friends have more friends than you do}}.</ref><ref>{{citation\|url=http://www.guardian.co.uk/lifeandstyle/2010/jan/30/change-your-life-friends-popular\|title=This column will change your life: Ever wondered why your friends seem so much more popular than you are? There's a reason for that\|first=Oliver\|last=Burkeman\|authorlink=Oliver Burkeman\|date=30 January 2010\|journal=[[The Guardian]]}}.</ref>

	~~==Mathematical explanation==~~		There are many types of bike brakes for different applications. If you have toe clips on your pedals, you'll need to spend a little extra time with this exercise. The trail then follows the side of the glacial Lake Pukaki through to Twizel, the town build in the 60s only to house workers from the Waitaki Hydroelectricity Scheme. Mountain bike frames spend a lot of time on roads these days, too. If you've ridden for years and years and know what you need, buying online does make sense. <br><br>On the road to the airport find your way to Lake Caburgua and then Trancura River. If you have any questions pertaining to in which and how to use [http://snakzstock.com/dressup4ever/profile/954401/MoJVL.html Biking for modern life mountain bike sizing.], you can contact us at the web site. You can install an electric motor on your regular bike in about an hour or hire a bike mechanic to do it. Here are the answers and meanings to each of the things you encountered in this dream walk. Then, the front Shock absorbers are used as a headset mechanism that connects the front fork to the stem or the handlebars and spokes. He is known for his expertise on the subject and on other Business and Finance related articles. <br><br>It seems there is still ongoing trail expansion, so I predict a great future for the Gold Canyon trails. That's a level of quality you just don't get from many brands these days. Buy a bike that has a top quality body made outside of steel, aluminum, carbon fiber or titanium. Hence, having the bicycle that suits you perfectly is much easier now. If you really want a quality bike which will hold up to the rough terrain with out costing a fortune, this is a great choice. <br><br>There are still other types of bikes that require different types of tires. Then, the shorter and wider teeth as well as the ramps and pegs in a bike are used in sprocket technology to ease the transition of the gear. Just remember your heart is a muscle and can be developed the more you practice. With a Shimano 105 groupset and FSA Gossamer chain set, it's a great choice for under. Next to a good mountain bike helmet, mountain bike gloves to protect your hands are the best accessory you can invest in. <br><br>(many commute bikes come with front suspension and disc brakes these days). The bike should be examined to check it is working correctly. Trying out various sizes is the first step in choosing the correct folding bike. Many of these trails and stunts are so dangerous that signs are up because of the potential of a serious injury or even death. The popular Whistler mountain bike website now has an extension into iphone.
	~~In spite~~ of ~~its apparently [[paradox]]ical nature~~, ~~the phenomenon is real, and can be explained as~~ a ~~consequence of the general mathematical properties of [[social network]]s~~. The ~~mathematics behind this are directly related to~~ the ~~[[arithmetic-geometric mean inequality]] and~~ the ~~[[Cauchy–Schwarz inequality]].~~

	~~Formally~~, ~~Feld assumes that a social network is represented by an [[undirected graph]] {{math\|1=''G'' = (''V'',''E'')}}, where~~ the ~~set {{mvar\|V}} of [[vertex (graph theory)\|vertices]] corresponds to the people~~ in the ~~social network, and the set {{mvar\|E}} of edges corresponds~~ to the ~~friendship relation between pairs of people~~. ~~That is, he assumes that friendship is~~ a ~~[[symmetric relation]]: if {{mvar\|X}} is a friend~~ of ~~{{mvar\|Y}}~~, ~~then {{mvar\|Y}} is a friend of {{mvar\|X}}~~. He models the average number of friends of a person in the social network as the average of the [[degree (graph theory)\|degrees]] of the [[vertex (graph theory)\|vertices]] in the graph. That is, if vertex {{mvar\|v}} has {{math\|'~~'d''(''v'')}} edges touching it (representing a person who has {{math\|''d''(''v'')}} friends)~~, ~~then the average number {{math\|''μ''}} of friends of a random person in the graph is~~
	:<~~math~~>~~\mu=\frac{\sum_{v\in V} d(v)}{\|V\|}=\frac{2\|E\|}{\|V\|}.~~<~~/math~~>
	~~The average number of friends that a typical friend has can be modeled by choosing, uniformly at random, an edge of~~ the ~~graph (representing a pair of friends) and an endpoint of that edge (one of~~ the ~~friends),~~ and ~~again calculating the degree of the selected endpoint~~. ~~That is, mathematically, it is~~
	:~~<math>\frac{\sum_{uv\in E} d(v)^2}{2\|E\|}=\mu + \frac{\sigma^2}{\mu},<~~/~~math>~~
	~~where <math> {\sigma}^{2} <~~/~~math> is the [[variance~~]~~] of~~ the ~~degrees~~ in ~~the graph~~. ~~For a graph that has vertices of varying degrees (as is typical for social networks), both {{math\|''μ''}} and <math> {\sigma}^{2} </math>~~ are ~~positive, which implies that~~ the ~~average degree~~ of ~~a friend is strictly greater than~~ the ~~average degree of a random node~~.

	~~Another way of understanding how~~ the ~~first term came is~~ as ~~follows. For each friendship uv,~~ a ~~node u mentions~~ that ~~v is a friend and v has d(v) friends. There are d(v) such friends who mention this. Hence~~ the ~~square of d(v) term. We add this for all such friendships in~~ the ~~network from both~~ the ~~u's~~ and ~~v's perspective, which gives the numerator~~. ~~The denominator~~ is the ~~number of total such friendships, which counts to total edges in the network twice (one from the u's perspective~~ and ~~the~~ other ~~from the v's)~~.

	~~After this analysis~~, ~~Feld goes on to make some more qualitative assumptions about the statistical correlation between~~ the number of friends that two friends have, based on theories of social networks such as [[assortative mixing]], and he analyzes what these assumptions imply about the number of people whose friends have more friends than they do. ~~Based on this analysis, he concludes that in real social networks, most people are likely to have fewer friends than the average~~ of ~~their friends~~' ~~numbers of friends~~. ~~However, this conclusion is not~~ a ~~mathematical certainty; there exist undirected graphs (such as the graph formed by removing a single edge from~~ a ~~large [[complete graph]]) that are unlikely to arise as social networks but in which most vertices have higher degree than the average~~ of ~~their neighbors' degrees~~.

	~~==Explanation==~~

	~~People with more friends are more likely to be your friend in~~ the ~~first place;~~ that is~~, they have~~ a ~~higher propensity~~ to ~~make friends in~~ the ~~first place. Another example deals~~ with ~~[[Twitter]]: The people~~ a ~~person follows almost certainly have more followers than they. This~~ is ~~because people are more likely to follow those who are popular than those who are not. Thus, over 98% of users are subject to the friendship paradox~~.<~~ref~~>{{cite paper\|last=Hodas\|first=N.\|coauthors=Kooti, F. and Lerman. L.\|title=Friendship Paradox Reduce: Your Friends Are More Interesting Than You\|work=The 7th International AAAI Conference on Weblogs and Social Media\|year=2013\|arxiv=1304.3480}}<~~/ref~~>

	~~==Applications==~~
	~~The analysis~~ of ~~the friendship paradox implies~~ that ~~the friends~~ of ~~randomly selected individuals are likely to have higher than average [[centrality]]~~. ~~This observation has been used as a way to forecast and slow the course of [[epidemic]]s~~, ~~by using this random selection process to choose individuals to immunize or monitor for infection while avoiding~~ the ~~need for a complex computation of~~ the ~~centrality of all nodes~~ in the network.<ref>{{citation\|title=Efficient immunization strategies for computer networks and populations\|first1=Reuven\|last1=Cohen\|first2=Shlomo\|last2=Havlin\|first3=Daniel\|last3=ben-Avraham\|arxiv=cond-mat/0207387\|journal=Phys. Rev. Lett.\|volume=91\|at=247901\|year=2003\|doi=10.1103/PhysRevLett.91.247901\|pmid=14683159}}.</ref><ref>{{citation\|last1=Christakis\|first1=N. A.\|last2=Fowler\|first2=J. H.\|year=2010\|title=Social network sensors for early detection of contagious outbreaks\|journal=PLoS ONE\|volume=5\|issue=9\|at=e12948\|doi=10.1371/journal.pone.0012948\|pmid=20856792\|pmc=2939797}}.</ref><ref>{{citation\|title=Using the friendship paradox to sample a ~~social network\|first=Mark\|last=Wilson\|date=November 2010\|journal=Physics Today\|doi=10.1063/1.3518199}}.</ref> <br />~~
	~~A PLoS One study found that those~~ in the ~~center~~ of ~~their social networks can detect flu outbreaks almost 2 weeks before traditional surveillance measures can~~. ~~They found that using the Friendship paradox to analyze the health of central friends~~ is ~~"an ideal way to predict outbreaks, but detailed information doesn't exist for most groups,~~ and ~~to produce it would~~ be ~~time-consuming~~ and ~~costly." <ref>{{cite news\|last=Schnirring\|first=Lisa\|title=Study: Friend~~ '~~sentinels' provide early flu warning\|url=http://www~~.~~cidrap.umn.edu/cidrap/content/influenza/general/news/sep1610friends.html\|newspaper=CIDRAP News\|date=Sep 16~~, ~~2010}}</ref>~~

	~~The "generalized friendship paradox" states that the friendship paradox applies~~ to ~~other characteristics as well. For example,~~ your ~~friends~~ are ~~on average likely to be richer than~~ you~~<ref>{{cite journal\|last=Eom\|first=Young-Ho\|coauthors=Jo, Hang-Hyun\|title=Generalized friendship paradox~~ in ~~complex networks\|journal=eprint arXiv\|date=01/2014\|url=http://arxiv~~.~~org/abs/1401.1458\|accessdate=17 January 2014}}~~<~~/ref~~><~~ref~~>~~{{cite web\|last=Dickerson\|first=Kelly\|title=Why Your Friends Are Probably More Popular, Richer,~~ and ~~Happier Than You\|url=http://www~~.~~slate.com/blogs/business_insider/2014/01/16/friendship_paradox_why_are_my_friends_better_off_than_me.html\|work=[[Slate Magazine]]\|publisher=[[~~The ~~Slate Group]]\|accessdate=17 January 2014}}</ref>, or your followers on Twitter have more followers than you~~.~~<ref>{{cite journal\|last=Hodas\|~~first~~=Nathan\|coauthors=Kooti, Farshad\|title=Friendship Paradox Redux: Your Friends~~ are ~~More Interesting than You\|journal=eprint arXiv\|date=05/2013\|url=http://arxiv~~.~~org/abs/1304~~.~~3480\|accessdate=21 January 2014}}</ref>~~

	~~==See also==~~
	* [[Self-evaluation maintenance theory]]

	~~==References==~~
	~~{{reflist}}~~

	~~==External links==~~

	* {{cite news\|last=Strogatz\|first=Steven\|title=Friends You Can Count On\|url=http://opinionator.blogs.nytimes.com/2012/09/17/friends-you-can-count-on/\|accessdate=17 January 2013\|newspaper=New York Times\|date=September 17, 2012}}

	~~{{Social networking}}~~

	~~[[Category:Statistical paradoxes]]~~
	~~[[Category:Social networks]]~~
	~~[[Category:Graph theory]]~~

88.116.101.242: /* List of analogous formulas */

2013-05-14T17:25:00Z

List of analogous formulas

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en>Scottywong: undo bad correction of math equation by bot

2012-02-15T17:22:46Z

undo bad correction of math equation by bot

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Comparison of vector algebra and geometric algebra - Revision history

en>Red Act: /* Algebraic and geometric properties of cross and wedge products */ Minor grammar tweak.

88.116.101.242: /* List of analogous formulas */

en>Scottywong: undo bad correction of math equation by bot