What power law and rich get richer phenomena means in the world of network and how does it affect in the social networks for web page popularity especially in the facebook platform?

1. Power Laws and Rich-Get-Richer Phenomena
Faculty of Electrical and Computer Engineering
University of Prishtina “Hasan Prishtina”
Prishtina, Kosova
Ajshe Nazmi Klinaku
ajshe1klinaku@gmail.com
Prishtinë

2. Introduct
• Popularity
• Our Model
• Powers Law
• Long Tail
• Richer-get-richer phenomena
• Result
• Conclusion
• References

3. Popularity
•How do we measure it?
•How does it effect the network?
•Basic network models with popularity

4. Popularity Example
• Books
• Movies
• Music
• Websites

5. Our Model
Consider the creation of web pages.
The web as a directed graph
• Nodes are web pages
• Edges are hyperlinks
• #in-links = popularity
• 1 out-link per page
When a new web page is designed, it
includes links to existing web pages.

6. Our Main Question
• As a function of k, what fraction of pages
on the web have k in-links?

7. Expected
Distribution
Normal distribution
 Each link addition is an
experiment

8. Power Laws
• When people measured the distribution of links on the Web,
however, they found something very different
• Fraction of Web pages that have k in-links is approximately
proportional to 1/k2 [1]
• Why is this so different from the normal distribution?
• The crucial point is that 1/k2 decreases much more slowly as k
increases,
• So pages with very large numbers of in-links are much more
common than we’d expect with a normal distribution.
• A function that decreases ask to some fixed power, such as 1/k2 in the
present case, is called a power law.

9. Power Laws vs Normal Distribution
• Normal distribution – many independent
experiments
• Power laws – if the data measured can be
viewed as a type of popularity

10. What causes power laws?
• Correlated decisions across a population
• Human tendency to copy decision

11. Examples
• Telephone numbers that
receive k calls per day
• Books bought by k people
• Scientific papers that receive
k citations
• Web page that recieve k in-
links

12. LONG TAIL
• In search queries it is important to tap the main and most common
search terms
• Tail decreases much slower than the normal distribution

13. Rich – Get - Richer
• A page that gets a small lead over others will
tend to extend this lead
• With probability (1-p), chooses a page k with
probability proportional to k’s #in-links

14. RICH-GET-RICHER PHENOMENA
• We start with m0 nodes, the links between which are chosen arbitrarily as long
as each node has at least one link
• The network develops following two steps :
• Growth : At each time step we add a new node with n(<=m0) links that connect
the new node to m nodes already in the network.
• Preferential attachment: The probability that a link of the new node connects to
node i depends on the degree of node i
Probability[i]=Degree[i]/Sum(degrees of all the nodes).

15. Building a Simple Model
•Webages are created in order 1,2,3,…,N
Dynamic network growth
•When page j is created, with probability:
p: Chooses a page uniformly at random among
all earlier pages and links to it
1-p: Chooses a page uniformly at random among
all earlier pages and link to its link

16. Results

17. Result
Page like network analyze
PageLikeNetwork ICK TECH INSIDER NASA HISTORY
ID 2.48822E+14 3.52751E+14 54971236771 2.01566E+15
Analyses_Period First Second First Second First Second First Second
Nodes(Crawl Depth 2) 1111 1110 61 61 936 935 1 1
Edges(Crawl Depth 2) 6134 6111 1052 1052 12567 12565 0 0
Post_Activity 0.08 0.07 3.83 3.11 0.3 0.35 0.01 0.01
Talking_About_Count 404 947 716843 486.276 100613 226332 246 195
Fan_Count 51.047 51.272 14.073.621 14.109.586 21.168.297 21.285.967 43.411.692 43.441.164
Follow_Page 50.047 50.999 14.291.857 14.333.411 21.244.860 21.373.652 43.411.692 43.441.501
Link 2 3 2 2 2 3 3 3
Video 3 2 1 1 1 2 2 2
Photo 1 1 3 3 3 1 1 1

18. Result
Post page Analyze
Post_Page_Analyses Analyse Posts Like Reactions Comments Share
ICK First 999 58981 63317 1420 2303
Second 50 2723 2820(56.4avg) 46(0.92avg) 154
Tech Insider First 999 3066780 683919 456536 819186
Second 50 5918 6756(135.12) 714(14.28) 1834
NASA First 999 3329112 3971234 474995 859250
Second 50 156534 184099(3681.98 avg) 168700(337.4avg) 20488
History First 108 4186 5076 391 273
Second 50 2755 4117(82.34avg) 373(7.46avg) 219

19. Conclusions
• We see how big the Share number is, the greater the
number of links Likes is, that reflect the popularity of
the site.
• Share has an impact and a leading role in the
popularity of the site.
• Also such a conclusion is derived from Facebook
analytics firm PageLever

20. References
• David Easley, Jon Kleinberg."Networks, Crowds, and Markets: Reasoning about a Highly Connected World". Cambridge
University Press, 2010.
• Michalis Faloutsos, Petros Faloutsos and ChristosFaloutsos."On Power-law Relationships of the internet topology".
• Steffen Staab, Christoph Ringelstein." Network Theory and Dynamic Systems Power Laws and Rich-get-
Richer."University of Koblenz ▪ Landau, Germany,2012.
• Hongyu Zhang. "Discovering power laws in computer programs " Copyright © 2017 Elsevier B.V. or its licensors or
contributors. ScienceDirect ® is a registered trademark of Elsevier B.V.
• Koonin Eugene, Wolf Yuri, Karev Georgy "Power Laws, Scale-Free Networks and Genome Biology ". Springer 2006.
• Gal Oestreicher-Singer,Arun Sundararajan "Recommendation networks and the long tail of electronic
commerce".Research article in New York,USA, 2007.
• Oliver Hinz, Jochen Eckert“The Impact of Search and Recommendation Systems on Sales in Electronic
Commerce”.Bussiness and Information Systems Engineering,Springer,2010.
• James P Bagrow,Jie Sunand Daniel ben-Avraham: " Phase transition in the rich-get-richer mechanism due to finite-size
effects". Journal of physics, mathematical and theoretical,2008
• Goldman Sachs http://www.barrons.com/articles/rich-get-richer-as-google-and-facebook-dominate-web-ads-
1443851396
• https://en.wikipedia.org/wiki/Barab%C3%A1si%E2%80%93Albert_model
• Studying Facebook via Data Extraction: The Netvizz Application
http://thepoliticsofsystems.net/permafiles/rieder_websci.pdf