[math-fun] 'degrees of separation' hopscotch
According to the NYTimes, http://well.blogs.nytimes.com/2011/11/22/four-degrees-of-separation/ "Using data on the links among 721 million Facebook users, a team of scientists discovered that the average number of acquaintances separating any two people in the _United States_ was *** 4.37, *** and that the number separating any two people *** in the _world_ was 4.74. *** As John Markoff and Somini Sengupta report in todayÂs New York Times, the findings highlight the growing power of the emerging science of social networks: "The original Âsix degrees finding, published in 1967 by the psychologist Stanley Milgram, was drawn from 296 volunteers who were asked to send a message by postcard, through friends and then friends of friends, to a specific person in a Boston suburb. "The new research used a slightly bigger cohort: 721 million Facebook users, more than *** one-tenth *** of the worldÂs population. The findings were posted on FacebookÂs site Monday night. " ... -------- Here's the new twist from yesterday: http://www.theatlanticwire.com/politics/2013/07/nsa-admits-it-analyzes-more-... "Chris Inglis, the [NSA] agency's deputy director, was one of several government representativesÂincluding from the FBI and the office of the Director of National IntelligenceÂtestifying before the House Judiciary Committee this morning.... But Inglis' statement was new. Analysts look *** "two or three hops" *** from terror suspects when evaluating terror activity, Inglis revealed. Previously, the limit of how surveillance was extended had been described as *** two *** hops. This meant that if the NSA were following a phone metadata or web trail from a terror suspect, it could also look at the calls from the people that suspect has spoken withÂone hop. And then, the calls that second person had also spoken withÂtwo hops. Terror suspect to person two to person three. Two hops. And now: *** A third hop. *** " ... "For a sense of scale, researchers at the University of Milan found in 2011 that everyone on the Internet was, on average, 4.74 steps away from anyone else. The NSA explores relationships up to *** three *** of those steps." "Inglis' admission didn't register among the members of Congress present, ..." --- This reminds me of a possibly apocryphal story about a debate in Congress over whether Reagan's 'Star Wars' program would work. When one of the scientists testified that some number would have to be something like 10^20 for the system to work, and the best they'd been able to achieve by that date was 10^10, one of the Congressmen remarked "then we're halfway there!"
The conclusion is seriously questionable since Facebook "friends" are often barely acquainted if at all. Many people evidently have "friends" lists in the high multi-hundreds. To compare 1967 with today, it would be best to use the same methodology that Milgram used, or at least, or a very similar one. See < https://en.wikipedia.org/wiki/Small-world_experiment#The_experiment >. --Dan On 2013-07-18, at 9:49 PM, Henry Baker wrote:
According to the NYTimes,
http://well.blogs.nytimes.com/2011/11/22/four-degrees-of-separation/
"Using data on the links among 721 million Facebook users, a team of scientists discovered that the average number of acquaintances separating any two people in the _United States_ was *** 4.37, *** and that the number separating any two people *** in the _world_ was 4.74. *** As John Markoff and Somini Sengupta report in today’s New York Times, the findings highlight the growing power of the emerging science of social networks:
"The original “six degrees” finding, published in 1967 by the psychologist Stanley Milgram, was drawn from 296 volunteers who were asked to send a message by postcard, through friends and then friends of friends, to a specific person in a Boston suburb.
"The new research used a slightly bigger cohort: 721 million Facebook users, more than *** one-tenth *** of the world’s population. The findings were posted on Facebook’s site Monday night.…"
Wikipedia mentions this intriguing fact: PSL(2, p) acts non-trivially on p points if and only if p = 2, 3, 5, 7, or 11. It says this observation is due to Galois, 1832. More at < https://en.wikipedia.org/wiki/Projective_linear_group#Action_on_p_points>. Terms are defined at bottom. QUESTION: Does anyone in math-fun understand this? A heuristic explanation of why non-trivial actions of PSL(2,p) exist for primes 2 <= p <= 11 -- and only for these primes -- would be especially welcome. That Wikipedia article explicitly describes actions of PSL(2,p) on a set of p points for all primes p <= 11, but does not seem to address the reasons such don't exist for p >= 13. Perhaps the actions that do exist are mathematical coincidences, made more likely by the lowness of the relevant primes? --Dan _________________________________________________________________________________________ * Just to be clear, for each prime p, PSL(2,p) is the group of formal linear fractional transformations x -> (ax + b)/(cx + d) where a,b,c,d are in the field F_p (=the ring Z/pZ), and ad-bc = 1. This is the quotient of SL(2,p) obtained by identifying each matrix with its negative (irrelevant if p = 2, of course). It's a pleasant calculation to see that #(PSL(2,2)) = 6, and PSL(2,p) = (p^3-p)/2 for p > 2. It's known that PSL(2,p) is simple if and only if p >= 5. PSL(2,2) == S_3, and PSL(2,3) == A_4. PSL(2,4) == PSL(2,5) == A_5, and PSL(2,7), are the two smallest nonabelian simple groups. To say that a group G acts on a set X means there is a map f: G x X -> X , with f(g,x) denoted by gx, such that for all g and x we have 1x = x and g(hx) = (gh)x. This action is called faithful if gx = x for all x implies that g = 1. The action is called trivial if gx = x for all g and all x.
Here's a wonderful sequence of articles by Tim Silverman depicting modular curves. In particular, he considers actions of the modular group on Z/nZ. https://www.google.com/search?q=site%3Agolem.ph.utexas.edu%2Fcategory%2F+"pictures+of+modular+curves" On Fri, Jul 19, 2013 at 2:40 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Wikipedia mentions this intriguing fact:
PSL(2, p) acts non-trivially on p points if and only if p = 2, 3, 5, 7, or 11.
It says this observation is due to Galois, 1832. More at < https://en.wikipedia.org/wiki/Projective_linear_group#Action_on_p_points>.
Terms are defined at bottom.
QUESTION: Does anyone in math-fun understand this? A heuristic explanation of why non-trivial actions of PSL(2,p) exist for primes 2 <= p <= 11 -- and only for these primes -- would be especially welcome.
That Wikipedia article explicitly describes actions of PSL(2,p) on a set of p points for all primes p <= 11, but does not seem to address the reasons such don't exist for p >= 13. Perhaps the actions that do exist are mathematical coincidences, made more likely by the lowness of the relevant primes?
--Dan _________________________________________________________________________________________ * Just to be clear, for each prime p, PSL(2,p) is the group of formal linear fractional transformations
x -> (ax + b)/(cx + d)
where a,b,c,d are in the field F_p (=the ring Z/pZ), and ad-bc = 1. This is the quotient of SL(2,p) obtained by identifying each matrix with its negative (irrelevant if p = 2, of course).
It's a pleasant calculation to see that #(PSL(2,2)) = 6, and PSL(2,p) = (p^3-p)/2 for p > 2. It's known that PSL(2,p) is simple if and only if p >= 5.
PSL(2,2) == S_3, and PSL(2,3) == A_4. PSL(2,4) == PSL(2,5) == A_5, and PSL(2,7), are the two smallest nonabelian simple groups.
To say that a group G acts on a set X means there is a map
f: G x X -> X , with f(g,x) denoted by gx,
such that for all g and x we have 1x = x and g(hx) = (gh)x. This action is called faithful if gx = x for all x implies that g = 1. The action is called trivial if gx = x for all g and all x.
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Thanks, Mike!!! The articles, posted in late 2010 and early 2011, look terrific and will keep me occupied for a long time. On the other hand, I was able to find only articles I through XI, and XI promised there would be two more. Do you know how many are available? --Dan On 2013-07-19, at 7:47 AM, Mike Stay wrote: Here's a wonderful sequence of articles by Tim Silverman depicting modular curves. In particular, he considers actions of the modular group on Z/nZ. < https://www.google.com/search?q=site%3Agolem.ph.utexas.edu%2Fcategory%2F+"pictures+of+modular+cu rves"
On Jul 19, 2013 12:09 PM, "Dan Asimov" <dasimov@earthlink.net> wrote:
Thanks, Mike!!! The articles, posted in late 2010 and early 2011, look
terrific and will keep me occupied for a long time.
On the other hand, I was able to find only articles I through XI, and XI
promised there would be two more. Do you know how many are available? They were promised, and I think he has notes for what he wants to talk about, but they haven't been posted yet.
--Dan
On 2013-07-19, at 7:47 AM, Mike Stay wrote:
Here's a wonderful sequence of articles by Tim Silverman depicting modular curves. In particular, he considers actions of the modular group on Z/nZ.
< https://www.google.com/search?q=site%3Agolem.ph.utexas.edu%2Fcategory%2F+"pictures+of+modular+cu rves"
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