How to Analyse Social Network? : Part 2

Power Laws and Rich-Get-Richer Phenomena

Thank you for all referred contexts and figures

Introduction

A person's behavior or decisions always depend on the choices made by other people The choices of other people convey information that is

useful in the decision-making process. Conclusion: behavior is correlated across a population

2Source:http://www.khaosod.co.th/view_news.php?newsid=TURObWIzSXdNVEl3TURJMU5RPT0=

The fire caused a panic in the city…!!

Introduction

Popularity is a phenomenon characterized by extreme imbalances:

A few people achieve wider visibility A very, very few attain global name recognition

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…compared to normal people

Introduction

Power laws seem to dominate in cases where the quantity being measured can be viewed as a type of popularity.

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Remark: Scale-Free Networks

S ome nodes are more

highly connecte d than

others are, which are called popular nodes .Popularity: High in-Degree value

Rich-Get-Richer Models

Decision-making: People have a tendency to copy

the decisions of people who act

before them.

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Ice Bucket Challenge

Rich-Get-Richer Models

Example: Creation of links among Web pages Pages are created in order, and named 1, 2, 3, …..,N. When page j is created, it produces a link to an earlier Web page

according to the following probabilistic rule

With probability p, page j chooses a page i uniformly at random from among all earlier pages, and creates a link to this page i.

With probability 1-p, page j instead chooses a page i uniformly at random from among all earlier pages, and creates a link to the page that i points to.

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Rich-Get-Richer Models

Example: Creation of links among Web pages After finding a random earlier page i in the population, the author of

page j does not link to i, but instead copies the decision made by the author of page i -- linking to the same page that i did.

This describes the creation of a single link from page j; one can repeat this process to create multiple, independently generated links from page j.

With probability 1 - p, page j chooses a page i with probability proportional to i's current number of in-links, and creates a link to i.

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Rich-Get-Richer Models

Why do we call this a “rich-get-richer” rule? Because: the probability that page i experiences an increase in

popularity is directly proportional to i's current popularity.

This phenomenon is also known as preferential attachment Links are formed “preferentially” to pages that already have high

popularity.

The copying model provides an operational story for why popularity should exhibit such rich-get-richer dynamics. The more well-known someone is, the more likely you are to hear

their name

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Applying “Preferential Attachment Concept” in Social Network Application”

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Example: What are recommender systems? Recommender systems are systems which provide

recommendations to a user Too much information (information overload) Users have too many choices

Recommend different products for users, suited to their tastes. Assist users in finding information Reduce search and navigation time

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Case Study: Amazon

www.amazon.com

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Personalized Product Recommendation?

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Which Sources of Information? Sources of information for recommendations:

Browsing and searching data Purchase data Feedback provided by the users Textual comments Expert recommendations E-mail Rating

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Type of Recommendations

User-based “Users who bought X like Y.” Each user is represented by a vector indicating

his ratings for each product. Users with a small distance between each other

are similar. Find a similar user and recommend things they

like that you haven’t rated.

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Type of Recommendations

Item-to-item Content-based One item is recommended based on the user’s

indication that they like another item. If you like Lord of the Rings, you’ll like Legend.

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Type of Recommendations

Population-based => TREND!! The most popular news articles, or searches, or

downloads Frequently add content No user tracking needed.

Population-based Recommender System Preferential attachment is like some

trendiness force: a item that is well known in the market would have

a greater probability to be chosen by a user. Movie Song Etc.

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Population-based Recommender System REVIEW: Preferential Attachment

Nodes with higher degrees (i.e., with more links) acquire new links at higher rates than low-degree nodes

The probability that a link will connect a new node j with another existing node i is linearly proportional to the actual degree of i:

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Population-based Recommender System A rating network is a bipartite network

between persons and items they have rated. The nodes are persons and items, and each edge

connects a person with an item, and is annotated with a rating.

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Source: http://en.wikipedia.org/wiki/Bipartite_graph

U: NodesV: Items

Population-based Recommender System A higher probability of selecting a popular

item than an unknown one goods being sold depend on trendiness Items that are well-known will have a higher

probability of being bought

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iPhone 6Nokia 225

Population-based Recommender System REVIEW: Preferential Attachment

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Small number of well-known items compared to regular ones

Rich-Get-Richer Models

How to recommend items? User Similarity Measure

Distance Similarity: Euclidean Distance, City Block Distance..etc Distance between a target user and any other users.

A vector is created for each user, accounting for his/her selected items. the closeness of users related with that item

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Rich-Get-Richer Models

How to recommend items? Items in the top of the list are the best for the

target user, and should be submitted to his/her attention Population-based Recommended Items

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Problem: New Items?

Reference

David Easley and Jon Kleinberg, Networks, Crowds, and Markets: Reasoning About a Highly Connected World, Cambridge University Press, 2010.

Zanin, M., Cano P., Celma Ò., & Buldú J. M.,Preferential attachment, aging and weights in recommendation systems, International Journal of Bifurcation and Chaos, vol. 19, iss.2, pp.755-763,2009.

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