The influence of search engines on preferential attachment

1
The influence of search engines on preferential
attachment

• Dan Li
• CS3150
• Spring 2006

2
The paper

• The influence of search engines on preferential
  attachment
• Soumen Chakrabarti, Alan Frieze and Juan Vera

3
Background

• The evolution of social networks through time
• Web graph
• Models
• Preferential Attachment
• Copying Model

4
Background

• Evolution of the Web
• Power-law
• Preferential attachment( Barabasi and Albert)

• Copying Model
• The author of a newborn page u picks a random
  reference page v from the web, and with some
  probability, copies out-links from v to u.
• Power-law power 2
• Organic Evolution
• NO POWERFUL CENTRAL ENTIRY!

5
The New Problem

• How the page authors find existing pages and
  create links to them?
• Highly popular search engines limit the attention
  of the page authors to a small set of celebrity
  pages.
• Page authors frequently use search engines to
  locate pages, and include the HOT pages they
  visit (with probability p)

6
The New Problem

• The evolution of the Web graph has been
  influenced permanently and pervasively by the
  existence of search engines.
• A search engine ranks a page highly,
• Authors find the page more often, some of them
  link to it, raising its in-degree and Pagerank,
  which leads to a further improvement or
  entrenchment of its rank.

7
The Results in This Paper

• The celebrity nodes eventually accumulate a
  constant fraction of all links created with high
  probability
• The degree of the other nodes still follow a
  power-law distribution with a steeper power

8
The New Model

• Modeling how the web graphs evolves if the author
  use search engine to decide on links that they
  insert into new pages.
• How the degree distribution deviates from the
  traditional model

9
The New Model

• Undirected Web Graph
• Query to the Search Engine is fixed
• The search Engine returns a fix number of URLs
  ordered by their degree at the previous time-step
• Limit the analysis to one topic at a time with
  out loss of generality??
• Comments A new page may involve multiple topics
  at the same time and include different number of
  links for each topic.

10
The New Model

• Growth process
• Generates a sequence of graphs Gt, t 1,2,3,
• At time t, the Graph Gt (Vt, Et) has t vertices
  and mt edges.
• Parameters
• p a probability
• N maximum number of celebrity nodes listed by
  the search engine

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The New Model Comments

• Comments
• The number of links each new page creates is
  fixed? Is this real? How does this affect the
  results?
• Intuitively, the page author may not have a
  number in mind of how many links he wants to
  include, he will only determine whether a link
  will be included based on the content of that
  link

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Some Notations in the new model
13
Formal Definition of Process P
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The New Model

• In both cases yi is selected by preferential
  attachment within the target subset of old nodes,
  i.e. for x in U

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The New Model - Comments

• The m random edges may have duplicate vertices.
  For different i, the same vertex may be selected!
  When t is smaller than m, we have a lot of loops.
• Should we not start from one vertex? Instead, we
  can start from m vertices or N vertices and the
  initial web graph is created at random.
• With high probability, the oldest links become
  celebrity page.
• What happens in the real world?
• A page becomes hot not only by random, but also
  due to its contents, can we model this??

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The simulation results

• Very different from the standard preferential
  attachment!
• The celebrities is far from the Power-Law
  straight line in log-log plot.
• As p increases, the power increases as well!
• P Simulated power Computed power
• P 0 2.8 3
• P 0.3 3.96 3.857
• P 0.6 5.9 6
• The celebrities command a constant fraction of
  the total degree over all nodes, this fraction
  grows with p.

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The simulation results
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Results
19
Theorem 1
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Interpretations

• Celebrities capture a large? (depends on the
  constant) fraction of links.
• Non-celebrities follow a power-law degree
  distribution with a power steeper than in
  preferential attachment.

21
The Proof

• The celebrity list becomes fixed whp after some
  time tf
• Once the celebrity list is fixed, process P looks
  very similar to an analogous process P
• In each step, P takes the N oldest vertices as
  St, instead of the N largest-degree vertices.
• This is quite reasonable, basically, the oldest
  vertices have higher degree, since they have
  longer time to be included

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Coupling Gt and Gt
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Analysis of the degree distribution of Gt
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Basic Proof to Lemma 2

• Finding recurrence of

• Finding a similar recurrence

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Lemma 3
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Basic Proof to Lemma 3
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The celebrity list get fixed

• WHP, adding m edges to a single non-celebrity
  will not make it a celebrity.
• The total degree of celebrities is concentrated
  to a constant fraction of all edges ever added to
  the graph

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List-fixing Lemma
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Proof to Lemma 4
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Lemma 5
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Lemma 6

• With low degree, the celebrity has low degree

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Lemma 7

• With low probability, the non-celebrity has high
  degree

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Lemma 8

• With low probability, the gap will keep small

34
Proof of Theorem 1

• Lst tf to be the last time that St changes in
  the process P

35
Proof of Theorem 1 cont.
36
Proof of Theorem 1 cont.
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Proof of Theorem 1 cont.
38
Proof of Theorem 1 cont.
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Proof of Theorem 1 cont.
40
Conclusions

• Modeling the influence of a search engine within
  the preferential attachment framework leads to a
  qualitative change in the familiar power-law
  degree distribution.
• Each of a clot of celebrities captures a constant
  fraction of the total degree of the graph, and
  the degree of the remaining nodes follow a
  steeper power law.

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Is this Model real?

• The model differs from the reality.
• Edges are undirected?
• Outlinks are not modified after creation
• Pages do not die
• No topic-based clustering

42
Comments

• This model is used on to one topic
• There may be interactions between topics
• The author may include links for different topics
  into the same page
• The number of links on a page is fixed, which is
  not the real case

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Thank you!
Have a nice summer!